id sid tid token lemma pos iajs-3553 1 1 316 316 NUM iajs-3553 1 2 Β© Β© PROPN iajs-3553 1 3 2025 2025 NUM iajs-3553 1 4 the the DET iajs-3553 1 5 author(s author(s NOUN iajs-3553 1 6 ) ) PUNCT iajs-3553 1 7 . . PUNCT iajs-3553 2 1 published publish VERB iajs-3553 2 2 by by ADP iajs-3553 2 3 college college NOUN iajs-3553 2 4 of of ADP iajs-3553 2 5 education education NOUN iajs-3553 2 6 for for ADP iajs-3553 2 7 pure pure ADJ iajs-3553 2 8 science science NOUN iajs-3553 2 9 ( ( PUNCT iajs-3553 2 10 ibn ibn PROPN iajs-3553 2 11 al al PROPN iajs-3553 2 12 - - PUNCT iajs-3553 2 13 haitham haitham PROPN iajs-3553 2 14 ) ) PUNCT iajs-3553 2 15 , , PUNCT iajs-3553 2 16 university university NOUN iajs-3553 2 17 of of ADP iajs-3553 2 18 baghdad baghdad PROPN iajs-3553 2 19 . . PUNCT iajs-3553 3 1 this this PRON iajs-3553 3 2 is be AUX iajs-3553 3 3 an an DET iajs-3553 3 4 open open ADJ iajs-3553 3 5 - - PUNCT iajs-3553 3 6 access access NOUN iajs-3553 3 7 article article NOUN iajs-3553 3 8 distributed distribute VERB iajs-3553 3 9 under under ADP iajs-3553 3 10 the the DET iajs-3553 3 11 terms term NOUN iajs-3553 3 12 of of ADP iajs-3553 3 13 the the DET iajs-3553 3 14 creative creative ADJ iajs-3553 3 15 commons common NOUN iajs-3553 3 16 attribution attribution NOUN iajs-3553 3 17 4.0 4.0 NUM iajs-3553 3 18 international international ADJ iajs-3553 3 19 license license NOUN iajs-3553 3 20 the the DET iajs-3553 3 21 local local ADJ iajs-3553 3 22 bifurcation bifurcation NOUN iajs-3553 3 23 of of ADP iajs-3553 3 24 food food NOUN iajs-3553 3 25 chain chain NOUN iajs-3553 3 26 prey prey NOUN iajs-3553 3 27 - - PUNCT iajs-3553 3 28 predator predator NOUN iajs-3553 3 29 model model NOUN iajs-3553 3 30 with with ADP iajs-3553 3 31 crowleymartin crowleymartin NOUN iajs-3553 3 32 - - PUNCT iajs-3553 3 33 type type NOUN iajs-3553 3 34 of of ADP iajs-3553 3 35 functional functional ADJ iajs-3553 3 36 response response NOUN iajs-3553 3 37 asmaa asmaa PROPN iajs-3553 3 38 abdulhussein abdulhussein PROPN iajs-3553 3 39 aziz1 aziz1 PROPN iajs-3553 3 40 * * PROPN iajs-3553 3 41 and and CCONJ iajs-3553 3 42 azhar azhar PROPN iajs-3553 3 43 abbas abbas PROPN iajs-3553 3 44 majeed majeed PROPN iajs-3553 3 45 2 2 NUM iajs-3553 3 46 1,2department 1,2department NUM iajs-3553 3 47 of of ADP iajs-3553 3 48 mathematics mathematic NOUN iajs-3553 3 49 , , PUNCT iajs-3553 3 50 college college NOUN iajs-3553 3 51 of of ADP iajs-3553 3 52 science science NOUN iajs-3553 3 53 , , PUNCT iajs-3553 3 54 university university NOUN iajs-3553 3 55 of of ADP iajs-3553 3 56 baghdad baghdad PROPN iajs-3553 3 57 , , PUNCT iajs-3553 3 58 baghdad baghdad PROPN iajs-3553 3 59 , , PUNCT iajs-3553 3 60 iraq iraq PROPN iajs-3553 3 61 . . PUNCT iajs-3553 4 1 * * PUNCT iajs-3553 4 2 corresponding correspond VERB iajs-3553 4 3 author author NOUN iajs-3553 4 4 . . PUNCT iajs-3553 5 1 received receive VERB iajs-3553 5 2 : : PUNCT iajs-3553 5 3 4 4 NUM iajs-3553 5 4 june june PROPN iajs-3553 5 5 2023 2023 NUM iajs-3553 5 6 accepted accept VERB iajs-3553 5 7 : : PUNCT iajs-3553 5 8 26 26 NUM iajs-3553 5 9 february february NOUN iajs-3553 5 10 2024 2024 NUM iajs-3553 5 11 published publish VERB iajs-3553 5 12 : : PUNCT iajs-3553 5 13 20 20 NUM iajs-3553 5 14 april april PROPN iajs-3553 5 15 2025 2025 NUM iajs-3553 5 16 doi.org/10.30526/38.2.3553 doi.org/10.30526/38.2.3553 NOUN iajs-3553 5 17 abstract abstract ADJ iajs-3553 5 18 the the DET iajs-3553 5 19 conditions condition NOUN iajs-3553 5 20 under under ADP iajs-3553 5 21 which which PRON iajs-3553 5 22 the the DET iajs-3553 5 23 local local ADJ iajs-3553 5 24 bifurcation bifurcation NOUN iajs-3553 5 25 including include VERB iajs-3553 5 26 ( ( PUNCT iajs-3553 5 27 saddle saddle NOUN iajs-3553 5 28 - - PUNCT iajs-3553 5 29 node node NOUN iajs-3553 5 30 , , PUNCT iajs-3553 5 31 transcritical transcritical ADJ iajs-3553 5 32 , , PUNCT iajs-3553 5 33 and and CCONJ iajs-3553 5 34 pitchfork pitchfork NOUN iajs-3553 5 35 ) ) PUNCT iajs-3553 5 36 of of ADP iajs-3553 5 37 all all DET iajs-3553 5 38 stable stable ADJ iajs-3553 5 39 points point NOUN iajs-3553 5 40 in in ADP iajs-3553 5 41 a a DET iajs-3553 5 42 model model NOUN iajs-3553 5 43 of of ADP iajs-3553 5 44 a a DET iajs-3553 5 45 food food NOUN iajs-3553 5 46 chain chain NOUN iajs-3553 5 47 occur occur NOUN iajs-3553 5 48 are be AUX iajs-3553 5 49 examined examine VERB iajs-3553 5 50 in in ADP iajs-3553 5 51 this this DET iajs-3553 5 52 paper paper NOUN iajs-3553 5 53 . . PUNCT iajs-3553 6 1 the the DET iajs-3553 6 2 growly growly VERB iajs-3553 6 3 - - PUNCT iajs-3553 6 4 martin martin NOUN iajs-3553 6 5 model model NOUN iajs-3553 6 6 and and CCONJ iajs-3553 6 7 the the DET iajs-3553 6 8 emotion emotion NOUN iajs-3553 6 9 of of ADP iajs-3553 6 10 fear fear NOUN iajs-3553 6 11 have have AUX iajs-3553 6 12 both both PRON iajs-3553 6 13 been be AUX iajs-3553 6 14 important important ADJ iajs-3553 6 15 factors factor NOUN iajs-3553 6 16 in in ADP iajs-3553 6 17 the the DET iajs-3553 6 18 development development NOUN iajs-3553 6 19 of of ADP iajs-3553 6 20 a a DET iajs-3553 6 21 functional functional ADJ iajs-3553 6 22 reaction reaction NOUN iajs-3553 6 23 in in ADP iajs-3553 6 24 a a DET iajs-3553 6 25 food food NOUN iajs-3553 6 26 chain chain NOUN iajs-3553 6 27 model model NOUN iajs-3553 6 28 , , PUNCT iajs-3553 6 29 it it PRON iajs-3553 6 30 has have AUX iajs-3553 6 31 been be AUX iajs-3553 6 32 shown show VERB iajs-3553 6 33 that that SCONJ iajs-3553 6 34 a a DET iajs-3553 6 35 transcritical transcritical ADJ iajs-3553 6 36 bifurcation bifurcation NOUN iajs-3553 6 37 and and CCONJ iajs-3553 6 38 a a DET iajs-3553 6 39 pitchfork pitchfork NOUN iajs-3553 6 40 bifurcation bifurcation NOUN iajs-3553 6 41 can can AUX iajs-3553 6 42 be be AUX iajs-3553 6 43 discovered discover VERB iajs-3553 6 44 near near ADP iajs-3553 6 45 to to ADP iajs-3553 6 46 each each PRON iajs-3553 6 47 of of ADP iajs-3553 6 48 the the DET iajs-3553 6 49 sites site NOUN iajs-3553 6 50 𝐴1 𝐴1 PROPN iajs-3553 6 51 , , PUNCT iajs-3553 6 52 a2 a2 PROPN iajs-3553 6 53 , , PUNCT iajs-3553 6 54 𝐴3π‘Žπ‘›π‘‘ 𝐴3π‘Žπ‘›π‘‘ PROPN iajs-3553 6 55 𝐴4 𝐴4 PROPN iajs-3553 6 56 , , PUNCT iajs-3553 6 57 and and CCONJ iajs-3553 6 58 that that SCONJ iajs-3553 6 59 a a DET iajs-3553 6 60 saddle saddle NOUN iajs-3553 6 61 - - PUNCT iajs-3553 6 62 node node NOUN iajs-3553 6 63 bifurcation bifurcation NOUN iajs-3553 6 64 can can AUX iajs-3553 6 65 be be AUX iajs-3553 6 66 found find VERB iajs-3553 6 67 close close ADV iajs-3553 6 68 to to ADP iajs-3553 6 69 the the DET iajs-3553 6 70 point point NOUN iajs-3553 6 71 where where SCONJ iajs-3553 6 72 positive positive ADJ iajs-3553 6 73 equilibrium equilibrium NOUN iajs-3553 6 74 is be AUX iajs-3553 6 75 located locate VERB iajs-3553 6 76 . . PUNCT iajs-3553 7 1 in in ADP iajs-3553 7 2 conclusion conclusion NOUN iajs-3553 7 3 , , PUNCT iajs-3553 7 4 a a DET iajs-3553 7 5 numerical numerical ADJ iajs-3553 7 6 simulation simulation NOUN iajs-3553 7 7 was be AUX iajs-3553 7 8 run run VERB iajs-3553 7 9 in in ADP iajs-3553 7 10 order order NOUN iajs-3553 7 11 to to PART iajs-3553 7 12 illustrate illustrate VERB iajs-3553 7 13 how how SCONJ iajs-3553 7 14 the the DET iajs-3553 7 15 proposed propose VERB iajs-3553 7 16 model model NOUN iajs-3553 7 17 may may AUX iajs-3553 7 18 display display VERB iajs-3553 7 19 bifurcation bifurcation NOUN iajs-3553 7 20 in in ADP iajs-3553 7 21 its its PRON iajs-3553 7 22 behavior behavior NOUN iajs-3553 7 23 . . PUNCT iajs-3553 8 1 this this PRON iajs-3553 8 2 was be AUX iajs-3553 8 3 done do VERB iajs-3553 8 4 in in ADP iajs-3553 8 5 order order NOUN iajs-3553 8 6 to to PART iajs-3553 8 7 show show VERB iajs-3553 8 8 how how SCONJ iajs-3553 8 9 the the DET iajs-3553 8 10 impact impact NOUN iajs-3553 8 11 of of ADP iajs-3553 8 12 parameters parameter NOUN iajs-3553 8 13 on on ADP iajs-3553 8 14 the the DET iajs-3553 8 15 dynamics dynamic NOUN iajs-3553 8 16 of of ADP iajs-3553 8 17 the the DET iajs-3553 8 18 proposed propose VERB iajs-3553 8 19 model model NOUN iajs-3553 8 20 . . PUNCT iajs-3553 9 1 keywords keyword NOUN iajs-3553 9 2 : : PUNCT iajs-3553 9 3 transcritical transcritical ADJ iajs-3553 9 4 pitchfork pitchfork NOUN iajs-3553 9 5 , , PUNCT iajs-3553 9 6 sotomayor sotomayor PROPN iajs-3553 9 7 's 's PART iajs-3553 9 8 theorem theorem ADJ iajs-3553 9 9 , , PUNCT iajs-3553 9 10 local local ADJ iajs-3553 9 11 bifurcation bifurcation NOUN iajs-3553 9 12 , , PUNCT iajs-3553 9 13 global global ADJ iajs-3553 9 14 bifurcation bifurcation NOUN iajs-3553 9 15 . . PUNCT iajs-3553 10 1 1 1 X iajs-3553 10 2 . . X iajs-3553 10 3 introduction introduction NOUN iajs-3553 10 4 bifurcation bifurcation NOUN iajs-3553 10 5 theory theory NOUN iajs-3553 10 6 is be AUX iajs-3553 10 7 the the DET iajs-3553 10 8 scientific scientific ADJ iajs-3553 10 9 study study NOUN iajs-3553 10 10 of of ADP iajs-3553 10 11 how how SCONJ iajs-3553 10 12 a a DET iajs-3553 10 13 moving move VERB iajs-3553 10 14 system system NOUN iajs-3553 10 15 's 's PART iajs-3553 10 16 structure structure NOUN iajs-3553 10 17 changes change NOUN iajs-3553 10 18 over over ADP iajs-3553 10 19 time time NOUN iajs-3553 10 20 . . PUNCT iajs-3553 11 1 changes change NOUN iajs-3553 11 2 that that PRON iajs-3553 11 3 can can AUX iajs-3553 11 4 be be AUX iajs-3553 11 5 qualitative qualitative ADJ iajs-3553 11 6 or or CCONJ iajs-3553 11 7 visual visual ADJ iajs-3553 11 8 in in ADP iajs-3553 11 9 integral integral ADJ iajs-3553 11 10 family family NOUN iajs-3553 11 11 curves curve NOUN iajs-3553 11 12 , , PUNCT iajs-3553 11 13 vector vector NOUN iajs-3553 11 14 fields field NOUN iajs-3553 11 15 , , PUNCT iajs-3553 11 16 differential differential ADJ iajs-3553 11 17 equation equation NOUN iajs-3553 11 18 solutions solution NOUN iajs-3553 11 19 . . PUNCT iajs-3553 12 1 bifurcation bifurcation NOUN iajs-3553 12 2 happens happen VERB iajs-3553 12 3 when when SCONJ iajs-3553 12 4 a a DET iajs-3553 12 5 small small ADJ iajs-3553 12 6 , , PUNCT iajs-3553 12 7 smooth smooth ADJ iajs-3553 12 8 change change NOUN iajs-3553 12 9 in in ADP iajs-3553 12 10 the the DET iajs-3553 12 11 values value NOUN iajs-3553 12 12 of of ADP iajs-3553 12 13 a a DET iajs-3553 12 14 system system NOUN iajs-3553 12 15 's 's PART iajs-3553 12 16 parameters parameter NOUN iajs-3553 12 17 ( ( PUNCT iajs-3553 12 18 bifurcation bifurcation NOUN iajs-3553 12 19 parameters parameter NOUN iajs-3553 12 20 ) ) PUNCT iajs-3553 12 21 causes cause VERB iajs-3553 12 22 a a DET iajs-3553 12 23 sudden sudden ADJ iajs-3553 12 24 shift shift NOUN iajs-3553 12 25 in in ADP iajs-3553 12 26 its its PRON iajs-3553 12 27 " " PUNCT iajs-3553 12 28 specific specific ADJ iajs-3553 12 29 " " PUNCT iajs-3553 12 30 or or CCONJ iajs-3553 12 31 " " PUNCT iajs-3553 12 32 topological topological ADJ iajs-3553 12 33 " " PUNCT iajs-3553 12 34 behavior behavior NOUN iajs-3553 12 35 . . PUNCT iajs-3553 13 1 this this DET iajs-3553 13 2 term term NOUN iajs-3553 13 3 is be AUX iajs-3553 13 4 used use VERB iajs-3553 13 5 a a DET iajs-3553 13 6 lot lot NOUN iajs-3553 13 7 when when SCONJ iajs-3553 13 8 mathematicians mathematician NOUN iajs-3553 13 9 study study VERB iajs-3553 13 10 systems system NOUN iajs-3553 13 11 that that PRON iajs-3553 13 12 change change VERB iajs-3553 13 13 over over ADP iajs-3553 13 14 time time NOUN iajs-3553 13 15 ( ( PUNCT iajs-3553 13 16 1 1 NUM iajs-3553 13 17 - - SYM iajs-3553 13 18 5 5 NUM iajs-3553 13 19 ) ) PUNCT iajs-3553 13 20 . . PUNCT iajs-3553 14 1 there there PRON iajs-3553 14 2 are be VERB iajs-3553 14 3 two two NUM iajs-3553 14 4 major major ADJ iajs-3553 14 5 categories category NOUN iajs-3553 14 6 of of ADP iajs-3553 14 7 bifurcations bifurcation NOUN iajs-3553 14 8 : : PUNCT iajs-3553 14 9 local local ADJ iajs-3553 14 10 and and CCONJ iajs-3553 14 11 global global ADJ iajs-3553 14 12 . . PUNCT iajs-3553 15 1 local local ADJ iajs-3553 15 2 bifurcations bifurcation NOUN iajs-3553 15 3 like like ADP iajs-3553 15 4 the the DET iajs-3553 15 5 saddle saddle NOUN iajs-3553 15 6 node node NOUN iajs-3553 15 7 , , PUNCT iajs-3553 15 8 transcritical transcritical ADJ iajs-3553 15 9 pitchfork pitchfork NOUN iajs-3553 15 10 , , PUNCT iajs-3553 15 11 period period NOUN iajs-3553 15 12 - - PUNCT iajs-3553 15 13 doubling double VERB iajs-3553 15 14 flip flip NOUN iajs-3553 15 15 , , PUNCT iajs-3553 15 16 hopf hopf ADJ iajs-3553 15 17 , , PUNCT iajs-3553 15 18 and and CCONJ iajs-3553 15 19 neimark neimark ADJ iajs-3553 15 20 secondary secondary ADJ iajs-3553 15 21 hopf hopf ADJ iajs-3553 15 22 bifurcation bifurcation NOUN iajs-3553 15 23 , , PUNCT iajs-3553 15 24 which which PRON iajs-3553 15 25 can can AUX iajs-3553 15 26 all all PRON iajs-3553 15 27 be be AUX iajs-3553 15 28 studied study VERB iajs-3553 15 29 purely purely ADV iajs-3553 15 30 by by ADP iajs-3553 15 31 analyzing analyze VERB iajs-3553 15 32 how how SCONJ iajs-3553 15 33 their their PRON iajs-3553 15 34 local local ADJ iajs-3553 15 35 stability stability NOUN iajs-3553 15 36 properties property NOUN iajs-3553 15 37 change change VERB iajs-3553 15 38 , , PUNCT iajs-3553 15 39 happen happen VERB iajs-3553 15 40 as as SCONJ iajs-3553 15 41 parameters parameter NOUN iajs-3553 15 42 cross cross VERB iajs-3553 15 43 critical critical ADJ iajs-3553 15 44 thresholds threshold NOUN iajs-3553 15 45 . . PUNCT iajs-3553 16 1 global global ADJ iajs-3553 16 2 bifurcation bifurcation NOUN iajs-3553 16 3 happens happen VERB iajs-3553 16 4 when when SCONJ iajs-3553 16 5 stability stability NOUN iajs-3553 16 6 and and CCONJ iajs-3553 16 7 bigger big ADJ iajs-3553 16 8 invariant invariant ADJ iajs-3553 16 9 sets set NOUN iajs-3553 16 10 , , PUNCT iajs-3553 16 11 like like ADP iajs-3553 16 12 periodic periodic ADJ iajs-3553 16 13 orbits orbit NOUN iajs-3553 16 14 , , PUNCT iajs-3553 16 15 collide collide NOUN iajs-3553 16 16 . . PUNCT iajs-3553 17 1 contrary contrary ADJ iajs-3553 17 2 to to ADP iajs-3553 17 3 local local ADJ iajs-3553 17 4 bifurcations bifurcation NOUN iajs-3553 17 5 , , PUNCT iajs-3553 17 6 this this PRON iajs-3553 17 7 alters alter VERB iajs-3553 17 8 the the DET iajs-3553 17 9 structure structure NOUN iajs-3553 17 10 of of ADP iajs-3553 17 11 the the DET iajs-3553 17 12 pathways pathway NOUN iajs-3553 17 13 in in ADP iajs-3553 17 14 phase phase NOUN iajs-3553 17 15 space space NOUN iajs-3553 17 16 in in ADP iajs-3553 17 17 a a DET iajs-3553 17 18 nonlocal nonlocal ADJ iajs-3553 17 19 manner manner NOUN iajs-3553 17 20 . . PUNCT iajs-3553 18 1 topological topological ADJ iajs-3553 18 2 shifts shift NOUN iajs-3553 18 3 are be AUX iajs-3553 18 4 widespread widespread ADJ iajs-3553 18 5 because because SCONJ iajs-3553 18 6 they they PRON iajs-3553 18 7 can can AUX iajs-3553 18 8 have have VERB iajs-3553 18 9 an an DET iajs-3553 18 10 impact impact NOUN iajs-3553 18 11 on on ADP iajs-3553 18 12 areas area NOUN iajs-3553 18 13 over over ADP iajs-3553 18 14 incredibly incredibly ADV iajs-3553 18 15 large large ADJ iajs-3553 18 16 distances distance NOUN iajs-3553 18 17 ; ; PUNCT iajs-3553 18 18 for for ADP iajs-3553 18 19 instance instance NOUN iajs-3553 18 20 , , PUNCT iajs-3553 18 21 the the DET iajs-3553 18 22 homoclinic homoclinic ADJ iajs-3553 18 23 ( ( PUNCT iajs-3553 18 24 6 6 NUM iajs-3553 18 25 - - PUNCT iajs-3553 18 26 8).pamuk 8).pamuk NUM iajs-3553 18 27 and and CCONJ iajs-3553 18 28 cay cay PROPN iajs-3553 18 29 ( ( PUNCT iajs-3553 18 30 9 9 NUM iajs-3553 18 31 ) ) PUNCT iajs-3553 18 32 looked look VERB iajs-3553 18 33 at at ADP iajs-3553 18 34 the the DET iajs-3553 18 35 steady steady ADJ iajs-3553 18 36 state state NOUN iajs-3553 18 37 with with ADP iajs-3553 18 38 respect respect NOUN iajs-3553 18 39 to to ADP iajs-3553 18 40 hopf hopf ADJ iajs-3553 18 41 bifurcation bifurcation NOUN iajs-3553 18 42 of of ADP iajs-3553 18 43 a a DET iajs-3553 18 44 feedback feedback NOUN iajs-3553 18 45 diffusion diffusion NOUN iajs-3553 18 46 method method NOUN iajs-3553 18 47 , , PUNCT iajs-3553 18 48 which which PRON iajs-3553 18 49 is be AUX iajs-3553 18 50 used use VERB iajs-3553 18 51 to to PART iajs-3553 18 52 control control VERB iajs-3553 18 53 how how SCONJ iajs-3553 18 54 vascular vascular ADJ iajs-3553 18 55 cells cell NOUN iajs-3553 18 56 and and CCONJ iajs-3553 18 57 inhibitors inhibitor NOUN iajs-3553 18 58 talk talk VERB iajs-3553 18 59 to to ADP iajs-3553 18 60 each each DET iajs-3553 18 61 other other ADJ iajs-3553 18 62 . . PUNCT iajs-3553 19 1 mukherjee mukherjee PROPN iajs-3553 19 2 and and CCONJ iajs-3553 19 3 maji maji PROPN iajs-3553 19 4 ( ( PUNCT iajs-3553 19 5 10 10 NUM iajs-3553 19 6 ) ) PUNCT iajs-3553 19 7 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3553 19 8 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3553 19 9 https://orcid.org/0009-0007-7885-1840 https://orcid.org/0009-0007-7885-1840 PROPN iajs-3553 19 10 mailto:asmaaaziz1998@gmail.com mailto:asmaaaziz1998@gmail.com PROPN iajs-3553 19 11 https://orcid.org/0000-0002-9652-6132 https://orcid.org/0000-0002-9652-6132 PROPN iajs-3553 19 12 mailto:azhar.majeed@sc.uobaghdad.edu.iq mailto:azhar.majeed@sc.uobaghdad.edu.iq ADJ iajs-3553 19 13 ihjpas ihjpas PROPN iajs-3553 19 14 . . PUNCT iajs-3553 20 1 2025 2025 NUM iajs-3553 20 2 , , PUNCT iajs-3553 20 3 38(2 38(2 NUM iajs-3553 20 4 ) ) PUNCT iajs-3553 20 5 317 317 NUM iajs-3553 20 6 established establish VERB iajs-3553 20 7 the the DET iajs-3553 20 8 criteria criterion NOUN iajs-3553 20 9 for for ADP iajs-3553 20 10 local local ADJ iajs-3553 20 11 bifurcations bifurcation NOUN iajs-3553 20 12 for for ADP iajs-3553 20 13 all all DET iajs-3553 20 14 equilibrium equilibrium NOUN iajs-3553 20 15 points point NOUN iajs-3553 20 16 and and CCONJ iajs-3553 20 17 hopf hopf ADJ iajs-3553 20 18 for for ADP iajs-3553 20 19 positive positive ADJ iajs-3553 20 20 equilibrium equilibrium NOUN iajs-3553 20 21 points point NOUN iajs-3553 20 22 in in ADP iajs-3553 20 23 a a DET iajs-3553 20 24 prey prey NOUN iajs-3553 20 25 - - PUNCT iajs-3553 20 26 predator predator NOUN iajs-3553 20 27 model model NOUN iajs-3553 20 28 with with ADP iajs-3553 20 29 prey prey PROPN iajs-3553 20 30 haven haven PROPN iajs-3553 20 31 . . PUNCT iajs-3553 21 1 while while SCONJ iajs-3553 21 2 majeed majeed PROPN iajs-3553 21 3 and and CCONJ iajs-3553 21 4 ali ali PROPN iajs-3553 21 5 ( ( PUNCT iajs-3553 21 6 11 11 NUM iajs-3553 21 7 ) ) PUNCT iajs-3553 21 8 presented present VERB iajs-3553 21 9 a a DET iajs-3553 21 10 model model NOUN iajs-3553 21 11 of of ADP iajs-3553 21 12 bifurcation bifurcation NOUN iajs-3553 21 13 with with ADP iajs-3553 21 14 refuge refuge NOUN iajs-3553 21 15 that that PRON iajs-3553 21 16 included include VERB iajs-3553 21 17 a a DET iajs-3553 21 18 predator predator NOUN iajs-3553 21 19 - - PUNCT iajs-3553 21 20 structured structure VERB iajs-3553 21 21 stage stage NOUN iajs-3553 21 22 food food NOUN iajs-3553 21 23 chain chain NOUN iajs-3553 21 24 . . PUNCT iajs-3553 22 1 in in ADP iajs-3553 22 2 recent recent ADJ iajs-3553 22 3 years year NOUN iajs-3553 22 4 , , PUNCT iajs-3553 22 5 many many ADJ iajs-3553 22 6 researchers researcher NOUN iajs-3553 22 7 have have AUX iajs-3553 22 8 studied study VERB iajs-3553 22 9 the the DET iajs-3553 22 10 importance importance NOUN iajs-3553 22 11 of of ADP iajs-3553 22 12 bifurcation bifurcation NOUN iajs-3553 22 13 . . PUNCT iajs-3553 23 1 majeed majeed PROPN iajs-3553 23 2 and and CCONJ iajs-3553 23 3 alabacy alabacy NOUN iajs-3553 23 4 ( ( PUNCT iajs-3553 23 5 12 12 NUM iajs-3553 23 6 ) ) PUNCT iajs-3553 23 7 found find VERB iajs-3553 23 8 the the DET iajs-3553 23 9 conditions condition NOUN iajs-3553 23 10 for for ADP iajs-3553 23 11 local local ADJ iajs-3553 23 12 bifurcation bifurcation NOUN iajs-3553 23 13 at at ADP iajs-3553 23 14 all all DET iajs-3553 23 15 equilibrium equilibrium NOUN iajs-3553 23 16 points point NOUN iajs-3553 23 17 a a DET iajs-3553 23 18 preypredator preypredator NOUN iajs-3553 23 19 - - PUNCT iajs-3553 23 20 refuge refuge NOUN iajs-3553 23 21 - - PUNCT iajs-3553 23 22 harvesting harvesting NOUN iajs-3553 23 23 model model NOUN iajs-3553 23 24 . . PUNCT iajs-3553 24 1 since since SCONJ iajs-3553 24 2 then then ADV iajs-3553 24 3 , , PUNCT iajs-3553 24 4 lots lot NOUN iajs-3553 24 5 of of ADP iajs-3553 24 6 researchers researcher NOUN iajs-3553 24 7 have have AUX iajs-3553 24 8 considered consider VERB iajs-3553 24 9 the the DET iajs-3553 24 10 fear fear ADJ iajs-3553 24 11 effect effect NOUN iajs-3553 24 12 of of ADP iajs-3553 24 13 predators predator NOUN iajs-3553 24 14 in in ADP iajs-3553 24 15 the the DET iajs-3553 24 16 dynamical dynamical ADJ iajs-3553 24 17 study study NOUN iajs-3553 24 18 on on ADP iajs-3553 24 19 prey prey NOUN iajs-3553 24 20 - - PUNCT iajs-3553 24 21 predator predator NOUN iajs-3553 24 22 models model NOUN iajs-3553 24 23 ( ( PUNCT iajs-3553 24 24 13 13 NUM iajs-3553 24 25 - - SYM iajs-3553 24 26 14 14 NUM iajs-3553 24 27 ) ) PUNCT iajs-3553 24 28 . . PUNCT iajs-3553 25 1 many many ADJ iajs-3553 25 2 other other ADJ iajs-3553 25 3 academics academic NOUN iajs-3553 25 4 have have AUX iajs-3553 25 5 studied study VERB iajs-3553 25 6 the the DET iajs-3553 25 7 local local ADJ iajs-3553 25 8 bifurcation bifurcation NOUN iajs-3553 25 9 in in ADP iajs-3553 25 10 recent recent ADJ iajs-3553 25 11 years year NOUN iajs-3553 25 12 , , PUNCT iajs-3553 25 13 including include VERB iajs-3553 25 14 ( ( PUNCT iajs-3553 25 15 15 15 NUM iajs-3553 25 16 - - SYM iajs-3553 25 17 19 19 NUM iajs-3553 25 18 ) ) PUNCT iajs-3553 25 19 . . PUNCT iajs-3553 26 1 finally finally ADV iajs-3553 26 2 , , PUNCT iajs-3553 26 3 in in ADP iajs-3553 26 4 this this DET iajs-3553 26 5 work work NOUN iajs-3553 26 6 , , PUNCT iajs-3553 26 7 the the DET iajs-3553 26 8 occurrence occurrence NOUN iajs-3553 26 9 of of ADP iajs-3553 26 10 the the DET iajs-3553 26 11 local local ADJ iajs-3553 26 12 bifurcation bifurcation NOUN iajs-3553 26 13 ( ( PUNCT iajs-3553 26 14 lb lb NOUN iajs-3553 26 15 ) ) PUNCT iajs-3553 26 16 of of ADP iajs-3553 26 17 the the DET iajs-3553 26 18 proposed propose VERB iajs-3553 26 19 system system NOUN iajs-3553 26 20 have have AUX iajs-3553 26 21 been be AUX iajs-3553 26 22 discussed discuss VERB iajs-3553 26 23 . . PUNCT iajs-3553 27 1 2 2 X iajs-3553 27 2 . . X iajs-3553 27 3 materials material NOUN iajs-3553 27 4 and and CCONJ iajs-3553 27 5 methods method NOUN iajs-3553 27 6 2.1 2.1 NUM iajs-3553 27 7 . . PUNCT iajs-3553 28 1 model model NOUN iajs-3553 28 2 formulation formulation NOUN iajs-3553 28 3 consider consider VERB iajs-3553 28 4 the the DET iajs-3553 28 5 following follow VERB iajs-3553 28 6 system system NOUN iajs-3553 28 7 that that PRON iajs-3553 28 8 given give VERB iajs-3553 28 9 in in ADP iajs-3553 28 10 ( ( PUNCT iajs-3553 28 11 20 20 NUM iajs-3553 28 12 ) ) PUNCT iajs-3553 28 13 . . PUNCT iajs-3553 29 1 𝑑𝑅1 𝑑𝑅1 ADV iajs-3553 29 2 𝑑𝑑 𝑑𝑑 ADP iajs-3553 29 3 = = SYM iajs-3553 29 4 π‘Ÿ1𝑅1 π‘Ÿ1𝑅1 NUM iajs-3553 29 5 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 29 6 βˆ’ βˆ’ PROPN iajs-3553 29 7 π‘š1𝑅1 π‘š1𝑅1 NUM iajs-3553 29 8 2 2 NUM iajs-3553 29 9 βˆ’ βˆ’ PROPN iajs-3553 29 10 𝛽1𝑅1𝑅2 𝛽1𝑅1𝑅2 PROPN iajs-3553 29 11 ( ( PUNCT iajs-3553 29 12 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 29 13 ) ) PUNCT iajs-3553 29 14 = = PUNCT iajs-3553 29 15 𝑓1(𝑅1 𝑓1(𝑅1 VERB iajs-3553 29 16 . . PUNCT iajs-3553 30 1 𝑅2 𝑅2 PROPN iajs-3553 30 2 . . PUNCT iajs-3553 31 1 𝑅3 𝑅3 PROPN iajs-3553 31 2 ) ) PUNCT iajs-3553 31 3 . . PUNCT iajs-3553 32 1 𝑑𝑅2 𝑑𝑅2 VERB iajs-3553 32 2 𝑑𝑑 𝑑𝑑 ADP iajs-3553 32 3 = = SYM iajs-3553 32 4 π‘Ÿ2𝑅2 π‘Ÿ2𝑅2 PROPN iajs-3553 32 5 1+𝐾2𝑅3 1+𝐾2𝑅3 NUM iajs-3553 32 6 βˆ’ βˆ’ PROPN iajs-3553 32 7 π‘š2𝑅2 π‘š2𝑅2 SYM iajs-3553 32 8 2 2 NUM iajs-3553 33 1 + + CCONJ iajs-3553 33 2 𝑙1𝑅1𝑅2 𝑙1𝑅1𝑅2 ADJ iajs-3553 33 3 ( ( PUNCT iajs-3553 33 4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 33 5 ) ) PUNCT iajs-3553 33 6 βˆ’ βˆ’ PROPN iajs-3553 34 1 𝛽2𝑅2𝑅3 𝛽2𝑅2𝑅3 PROPN iajs-3553 34 2 ( ( PUNCT iajs-3553 34 3 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3 NUM iajs-3553 34 4 ) ) PUNCT iajs-3553 34 5 βˆ’ βˆ’ PROPN iajs-3553 34 6 𝑑1𝑅2 𝑑1𝑅2 PROPN iajs-3553 34 7 = = SYM iajs-3553 34 8 𝑓2 𝑓2 PROPN iajs-3553 34 9 ( ( PUNCT iajs-3553 34 10 𝑅1 𝑅1 PROPN iajs-3553 34 11 . . PUNCT iajs-3553 35 1 𝑅2 𝑅2 PROPN iajs-3553 35 2 . . PUNCT iajs-3553 36 1 𝑅3 𝑅3 PROPN iajs-3553 36 2 ) ) PUNCT iajs-3553 36 3 ( ( PUNCT iajs-3553 36 4 1 1 X iajs-3553 36 5 ) ) PUNCT iajs-3553 36 6 𝑑𝑅3 𝑑𝑅3 PROPN iajs-3553 36 7 𝑑𝑑 𝑑𝑑 ADP iajs-3553 36 8 = = SYM iajs-3553 36 9 𝑙2𝑅2𝑅3 𝑙2𝑅2𝑅3 PROPN iajs-3553 36 10 ( ( PUNCT iajs-3553 36 11 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3 NOUN iajs-3553 36 12 ) ) PUNCT iajs-3553 36 13 βˆ’ βˆ’ PROPN iajs-3553 36 14 𝑑2𝑅3 𝑑2𝑅3 NUM iajs-3553 36 15 = = SYM iajs-3553 36 16 𝑓3 𝑓3 PROPN iajs-3553 36 17 ( ( PUNCT iajs-3553 36 18 𝑅1 𝑅1 PROPN iajs-3553 36 19 . . PUNCT iajs-3553 37 1 𝑅2 𝑅2 PROPN iajs-3553 37 2 . . PUNCT iajs-3553 38 1 𝑅3 𝑅3 PROPN iajs-3553 38 2 ) ) PUNCT iajs-3553 38 3 . . PUNCT iajs-3553 39 1 following follow VERB iajs-3553 39 2 is be AUX iajs-3553 39 3 a a DET iajs-3553 39 4 table table NOUN iajs-3553 39 5 describing describe VERB iajs-3553 39 6 the the DET iajs-3553 39 7 positive positive ADJ iajs-3553 39 8 parameters parameter NOUN iajs-3553 39 9 of of ADP iajs-3553 39 10 system system NOUN iajs-3553 39 11 ( ( PUNCT iajs-3553 39 12 1 1 NUM iajs-3553 39 13 ) ) PUNCT iajs-3553 39 14 . . PUNCT iajs-3553 40 1 table table NOUN iajs-3553 40 2 1 1 NUM iajs-3553 40 3 . . PUNCT iajs-3553 41 1 the the DET iajs-3553 41 2 system system NOUN iajs-3553 41 3 's 's PART iajs-3553 41 4 parameters parameter NOUN iajs-3553 41 5 ( ( PUNCT iajs-3553 41 6 1 1 NUM iajs-3553 41 7 ): ): PUNCT iajs-3553 41 8 parameters parameter NOUN iajs-3553 41 9 description description NOUN iajs-3553 41 10 parameters parameter VERB iajs-3553 41 11 the the DET iajs-3553 41 12 density density NOUN iajs-3553 41 13 of of ADP iajs-3553 41 14 prey prey NOUN iajs-3553 41 15 . . PUNCT iajs-3553 42 1 intermediate intermediate ADJ iajs-3553 42 2 predator predator NOUN iajs-3553 42 3 and and CCONJ iajs-3553 42 4 top top ADJ iajs-3553 42 5 βˆ’ βˆ’ PROPN iajs-3553 42 6 predator predator NOUN iajs-3553 42 7 at at ADP iajs-3553 42 8 time time NOUN iajs-3553 42 9 t. t. PROPN iajs-3553 42 10 respectively respectively ADV iajs-3553 42 11 rii rii VERB iajs-3553 42 12 . . PUNCT iajs-3553 43 1 = = NOUN iajs-3553 44 1 1 1 NUM iajs-3553 44 2 . . NOUN iajs-3553 44 3 2 2 NUM iajs-3553 44 4 . . SYM iajs-3553 44 5 3 3 NUM iajs-3553 44 6 the the DET iajs-3553 44 7 intrinsic intrinsic ADJ iajs-3553 44 8 growth growth NOUN iajs-3553 44 9 rate rate NOUN iajs-3553 44 10 ofthe ofthe NOUN iajs-3553 44 11 prey prey NOUN iajs-3553 44 12 and and CCONJ iajs-3553 44 13 the the DET iajs-3553 44 14 intermediate intermediate ADJ iajs-3553 44 15 predator predator NOUN iajs-3553 44 16 respectively respectively ADV iajs-3553 44 17 rii rii VERB iajs-3553 44 18 . . PUNCT iajs-3553 45 1 = = NOUN iajs-3553 45 2 1 1 NUM iajs-3553 45 3 . . NUM iajs-3553 45 4 2 2 NUM iajs-3553 45 5 the the DET iajs-3553 45 6 rate rate NOUN iajs-3553 45 7 of of ADP iajs-3553 45 8 internal internal ADJ iajs-3553 45 9 competition competition NOUN iajs-3553 45 10 for for ADP iajs-3553 45 11 the the DET iajs-3553 45 12 prey prey NOUN iajs-3553 45 13 and and CCONJ iajs-3553 45 14 the the DET iajs-3553 45 15 intermediate intermediate ADJ iajs-3553 45 16 predator predator NOUN iajs-3553 45 17 respectively respectively ADV iajs-3553 45 18 mi.i mi.i NOUN iajs-3553 45 19 = = SYM iajs-3553 45 20 1 1 NUM iajs-3553 45 21 . . NOUN iajs-3553 45 22 2 2 NUM iajs-3553 45 23 the the DET iajs-3553 45 24 rate rate NOUN iajs-3553 45 25 of of ADP iajs-3553 45 26 fear fear NOUN iajs-3553 45 27 of of ADP iajs-3553 45 28 prey prey NOUN iajs-3553 45 29 and and CCONJ iajs-3553 45 30 intermediate intermediate ADJ iajs-3553 45 31 predator predator NOUN iajs-3553 45 32 respectively respectively ADV iajs-3553 45 33 ki ki PROPN iajs-3553 45 34 . . PUNCT iajs-3553 46 1 i i PRON iajs-3553 46 2 = = NOUN iajs-3553 47 1 1 1 NUM iajs-3553 47 2 . . SYM iajs-3553 47 3 2 2 NUM iajs-3553 47 4 attack attack NOUN iajs-3553 47 5 rate rate NOUN iajs-3553 47 6 for for ADP iajs-3553 47 7 prey prey NOUN iajs-3553 47 8 and and CCONJ iajs-3553 47 9 intermediate intermediate ADJ iajs-3553 47 10 predator predator NOUN iajs-3553 47 11 respectively respectively ADV iajs-3553 47 12 Ξ²i Ξ²i PRON iajs-3553 47 13 . . PUNCT iajs-3553 48 1 i i PRON iajs-3553 48 2 = = NOUN iajs-3553 49 1 1 1 NUM iajs-3553 49 2 . . SYM iajs-3553 49 3 2 2 NUM iajs-3553 49 4 food food NOUN iajs-3553 49 5 transfer transfer NOUN iajs-3553 49 6 rate rate NOUN iajs-3553 49 7 for for ADP iajs-3553 49 8 intermediate intermediate ADJ iajs-3553 49 9 predator predator NOUN iajs-3553 49 10 and and CCONJ iajs-3553 49 11 top top ADJ iajs-3553 49 12 predator predator NOUN iajs-3553 49 13 respectively respectively ADV iajs-3553 49 14 li li PROPN iajs-3553 49 15 . . PUNCT iajs-3553 50 1 i i PRON iajs-3553 50 2 = = NOUN iajs-3553 51 1 1 1 X iajs-3553 51 2 . . SYM iajs-3553 51 3 2 2 NUM iajs-3553 51 4 handling handling NOUN iajs-3553 51 5 time time NOUN iajs-3553 51 6 of of ADP iajs-3553 51 7 prey prey NOUN iajs-3553 51 8 and and CCONJ iajs-3553 51 9 intermediate intermediate ADJ iajs-3553 51 10 predator predator NOUN iajs-3553 51 11 respectively respectively ADV iajs-3553 51 12 ai ai VERB iajs-3553 51 13 . . PUNCT iajs-3553 52 1 i i PRON iajs-3553 52 2 = = NOUN iajs-3553 52 3 1 1 NUM iajs-3553 52 4 . . SYM iajs-3553 52 5 3 3 NUM iajs-3553 52 6 magnitude magnitude NOUN iajs-3553 52 7 of of ADP iajs-3553 52 8 distrubance distrubance NOUN iajs-3553 52 9 among among ADP iajs-3553 52 10 intermediate intermediate ADJ iajs-3553 52 11 predator predator NOUN iajs-3553 52 12 and and CCONJ iajs-3553 52 13 top top ADJ iajs-3553 52 14 predator predator NOUN iajs-3553 52 15 respectively respectively ADV iajs-3553 52 16 ai ai VERB iajs-3553 52 17 . . PUNCT iajs-3553 53 1 i i PRON iajs-3553 53 2 = = NOUN iajs-3553 53 3 2 2 NUM iajs-3553 53 4 . . NOUN iajs-3553 53 5 4 4 NUM iajs-3553 53 6 the the DET iajs-3553 53 7 natural natural ADJ iajs-3553 53 8 death death NOUN iajs-3553 53 9 rate rate NOUN iajs-3553 53 10 of of ADP iajs-3553 53 11 intermediate intermediate ADJ iajs-3553 53 12 predator predator NOUN iajs-3553 53 13 and and CCONJ iajs-3553 53 14 top top ADJ iajs-3553 53 15 predator predator NOUN iajs-3553 53 16 respectively respectively ADV iajs-3553 53 17 di di NOUN iajs-3553 53 18 . . PUNCT iajs-3553 54 1 i i PRON iajs-3553 54 2 = = NOUN iajs-3553 55 1 1 1 NUM iajs-3553 55 2 . . NUM iajs-3553 55 3 2 2 NUM iajs-3553 55 4 2.2 2.2 NUM iajs-3553 55 5 . . PUNCT iajs-3553 56 1 local local ADJ iajs-3553 56 2 bifurcation bifurcation NOUN iajs-3553 56 3 analysis analysis NOUN iajs-3553 56 4 in in ADP iajs-3553 56 5 this this DET iajs-3553 56 6 subsection subsection NOUN iajs-3553 56 7 , , PUNCT iajs-3553 56 8 the the DET iajs-3553 56 9 local local ADJ iajs-3553 56 10 bifurcation bifurcation NOUN iajs-3553 56 11 of of ADP iajs-3553 56 12 the the DET iajs-3553 56 13 model model NOUN iajs-3553 56 14 ( ( PUNCT iajs-3553 56 15 1 1 X iajs-3553 56 16 ) ) PUNCT iajs-3553 56 17 has have AUX iajs-3553 56 18 been be AUX iajs-3553 56 19 examined examine VERB iajs-3553 56 20 , , PUNCT iajs-3553 56 21 with with ADP iajs-3553 56 22 a a DET iajs-3553 56 23 particular particular ADJ iajs-3553 56 24 emphasis emphasis NOUN iajs-3553 56 25 on on ADP iajs-3553 56 26 the the DET iajs-3553 56 27 changes change NOUN iajs-3553 56 28 that that PRON iajs-3553 56 29 occur occur VERB iajs-3553 56 30 around around ADP iajs-3553 56 31 each each DET iajs-3553 56 32 equilibrium equilibrium NOUN iajs-3553 56 33 point point NOUN iajs-3553 56 34 as as ADP iajs-3553 56 35 a a DET iajs-3553 56 36 result result NOUN iajs-3553 56 37 of of ADP iajs-3553 56 38 changes change NOUN iajs-3553 56 39 in in ADP iajs-3553 56 40 the the DET iajs-3553 56 41 parameter parameter NOUN iajs-3553 56 42 values value NOUN iajs-3553 56 43 governing govern VERB iajs-3553 56 44 the the DET iajs-3553 56 45 dynamic dynamic ADJ iajs-3553 56 46 behavior behavior NOUN iajs-3553 56 47 . . PUNCT iajs-3553 57 1 with with ADP iajs-3553 57 2 the the DET iajs-3553 57 3 assistance assistance NOUN iajs-3553 57 4 of of ADP iajs-3553 57 5 sotomayor sotomayor NOUN iajs-3553 57 6 's 's PART iajs-3553 57 7 theorem theorem NOUN iajs-3553 57 8 , , PUNCT iajs-3553 57 9 our our PRON iajs-3553 57 10 mission mission NOUN iajs-3553 57 11 is be AUX iajs-3553 57 12 to to PART iajs-3553 57 13 develop develop VERB iajs-3553 57 14 higher high ADJ iajs-3553 57 15 order order NOUN iajs-3553 57 16 conditions condition NOUN iajs-3553 57 17 with with ADP iajs-3553 57 18 the the DET iajs-3553 57 19 intention intention NOUN iajs-3553 57 20 of of ADP iajs-3553 57 21 ensuring ensure VERB iajs-3553 57 22 the the DET iajs-3553 57 23 appearance appearance NOUN iajs-3553 57 24 of of ADP iajs-3553 57 25 the the DET iajs-3553 57 26 most most ADV iajs-3553 57 27 typical typical ADJ iajs-3553 57 28 of of ADP iajs-3553 57 29 the the DET iajs-3553 57 30 area area NOUN iajs-3553 57 31 's 's PART iajs-3553 57 32 local local ADJ iajs-3553 57 33 bifurcations bifurcation NOUN iajs-3553 57 34 . . PUNCT iajs-3553 58 1 now now ADV iajs-3553 58 2 , , PUNCT iajs-3553 58 3 according accord VERB iajs-3553 58 4 to to ADP iajs-3553 58 5 jacobean jacobean PROPN iajs-3553 58 6 matrix matrix NOUN iajs-3553 58 7 𝐽(𝑅1 𝐽(𝑅1 NOUN iajs-3553 58 8 . . PUNCT iajs-3553 58 9 𝑅2 𝑅2 NOUN iajs-3553 58 10 . . PUNCT iajs-3553 58 11 𝑅3 𝑅3 PROPN iajs-3553 58 12 ) ) PUNCT iajs-3553 58 13 of of ADP iajs-3553 58 14 the the DET iajs-3553 58 15 system system NOUN iajs-3553 58 16 ( ( PUNCT iajs-3553 58 17 1 1 X iajs-3553 58 18 ) ) PUNCT iajs-3553 58 19 which which PRON iajs-3553 58 20 is be AUX iajs-3553 58 21 given give VERB iajs-3553 58 22 in in ADP iajs-3553 58 23 ( ( PUNCT iajs-3553 58 24 16 16 NUM iajs-3553 58 25 ) ) PUNCT iajs-3553 58 26 as as SCONJ iajs-3553 58 27 follows follow VERB iajs-3553 58 28 : : PUNCT iajs-3553 58 29 𝐽 𝐽 PROPN iajs-3553 58 30 = = PUNCT iajs-3553 59 1 [ [ X iajs-3553 59 2 𝑒𝑖𝑗]3Γ—3 𝑒𝑖𝑗]3Γ—3 NOUN iajs-3553 59 3 ( ( PUNCT iajs-3553 59 4 2 2 NUM iajs-3553 59 5 ) ) PUNCT iajs-3553 59 6 where where SCONJ iajs-3553 59 7 : : PUNCT iajs-3553 59 8 u11 u11 PROPN iajs-3553 59 9 = = PROPN iajs-3553 59 10 r1 r1 PROPN iajs-3553 59 11 1+k1r2 1+k1r2 NUM iajs-3553 60 1 βˆ’ βˆ’ PROPN iajs-3553 60 2 2m1r1 2m1r1 NUM iajs-3553 60 3 βˆ’ βˆ’ NOUN iajs-3553 60 4 Ξ²1r2 Ξ²1r2 PUNCT iajs-3553 60 5 ( ( PUNCT iajs-3553 60 6 1+a1r1)2(1+a2r2 1+a1r1)2(1+a2r2 NUM iajs-3553 60 7 ) ) PUNCT iajs-3553 60 8 ، ، NOUN iajs-3553 60 9 𝑒12 𝑒12 PROPN iajs-3553 60 10 βˆ’ βˆ’ PROPN iajs-3553 60 11 ( ( PUNCT iajs-3553 60 12 r1r1k1 r1r1k1 NOUN iajs-3553 60 13 ( ( PUNCT iajs-3553 60 14 1+k1r2)2 1+k1r2)2 NUM iajs-3553 60 15 + + CCONJ iajs-3553 60 16 Ξ²1r1 Ξ²1r1 PUNCT iajs-3553 60 17 ( ( PUNCT iajs-3553 60 18 1+a1r1)(1+a2r2)2 1+a1r1)(1+a2r2)2 NUM iajs-3553 60 19 ) ) PUNCT iajs-3553 60 20 ، ، NOUN iajs-3553 60 21 𝑒13 𝑒13 NOUN iajs-3553 60 22 = = SYM iajs-3553 60 23 0 0 NUM iajs-3553 60 24 ، ، NOUN iajs-3553 60 25 u21 u21 NOUN iajs-3553 60 26 = = SYM iajs-3553 60 27 𝑙1r2 𝑙1r2 X iajs-3553 60 28 ( ( PUNCT iajs-3553 60 29 1+a1r1)2(1+a2r2 1+a1r1)2(1+a2r2 NUM iajs-3553 60 30 ) ) PUNCT iajs-3553 60 31 ، ، NOUN iajs-3553 60 32 𝑒22 𝑒22 PROPN iajs-3553 60 33 = = SYM iajs-3553 60 34 r2 r2 PROPN iajs-3553 60 35 1+k2r3 1+k2r3 NUM iajs-3553 60 36 βˆ’ βˆ’ NUM iajs-3553 60 37 2m2r2 2m2r2 NUM iajs-3553 60 38 + + CCONJ iajs-3553 60 39 𝑙1r1 𝑙1r1 PROPN iajs-3553 60 40 ( ( PUNCT iajs-3553 60 41 1+a1r1)(1+a2r2)2 1+a1r1)(1+a2r2)2 NUM iajs-3553 60 42 βˆ’ βˆ’ NOUN iajs-3553 60 43 Ξ²2r3 Ξ²2r3 PUNCT iajs-3553 60 44 ( ( PUNCT iajs-3553 60 45 1+a3r2)2(1+a4r3 1+a3r2)2(1+a4r3 NUM iajs-3553 60 46 ) ) PUNCT iajs-3553 60 47 d1 d1 PROPN iajs-3553 60 48 ، ، X iajs-3553 60 49 u23 u23 PROPN iajs-3553 60 50 = = SYM iajs-3553 60 51 βˆ’ βˆ’ PROPN iajs-3553 60 52 r2r2k2 r2r2k2 NOUN iajs-3553 60 53 ( ( PUNCT iajs-3553 60 54 1 1 NUM iajs-3553 60 55 + + X iajs-3553 60 56 k2r3)2 k2r3)2 NOUN iajs-3553 60 57 βˆ’ βˆ’ NOUN iajs-3553 60 58 Ξ²2r2 Ξ²2r2 PUNCT iajs-3553 60 59 ( ( PUNCT iajs-3553 60 60 1 1 NUM iajs-3553 60 61 + + NUM iajs-3553 60 62 a3r2)(1 a3r2)(1 PUNCT iajs-3553 60 63 + + CCONJ iajs-3553 60 64 a4r3)2 a4r3)2 ADJ iajs-3553 60 65 ، ، NOUN iajs-3553 60 66 u31 u31 NOUN iajs-3553 60 67 = = SYM iajs-3553 60 68 0،u32 0،u32 NUM iajs-3553 60 69 = = SYM iajs-3553 60 70 𝑙2r3 𝑙2r3 PROPN iajs-3553 60 71 ( ( PUNCT iajs-3553 60 72 1 1 NUM iajs-3553 60 73 + + NUM iajs-3553 60 74 a3r2)2(1 a3r2)2(1 CCONJ iajs-3553 60 75 + + CCONJ iajs-3553 60 76 a4r3 a4r3 NOUN iajs-3553 60 77 ) ) PUNCT iajs-3553 60 78 ، ، NOUN iajs-3553 60 79 ihjpas ihjpas PROPN iajs-3553 60 80 . . PUNCT iajs-3553 61 1 2025 2025 NUM iajs-3553 61 2 , , PUNCT iajs-3553 61 3 38(2 38(2 NUM iajs-3553 61 4 ) ) PUNCT iajs-3553 61 5 318 318 NUM iajs-3553 61 6 u33 u33 NOUN iajs-3553 61 7 = = SYM iajs-3553 61 8 𝑙2r2 𝑙2r2 X iajs-3553 61 9 ( ( PUNCT iajs-3553 61 10 1+a3r2)(1+a4r3)2 1+a3r2)(1+a4r3)2 NUM iajs-3553 61 11 βˆ’ βˆ’ PROPN iajs-3553 61 12 d2 d2 PROPN iajs-3553 61 13 . . PUNCT iajs-3553 62 1 for for ADP iajs-3553 62 2 any any DET iajs-3553 62 3 nonzero nonzero PROPN iajs-3553 62 4 vector vector PROPN iajs-3553 62 5 t t PROPN iajs-3553 62 6 =( =( PROPN iajs-3553 62 7 𝑑1 𝑑1 PROPN iajs-3553 62 8 . . PUNCT iajs-3553 63 1 𝑑2 𝑑2 PROPN iajs-3553 63 2 . . PUNCT iajs-3553 64 1 𝑑3 𝑑3 PROPN iajs-3553 64 2 ) ) PUNCT iajs-3553 64 3 𝑇 𝑇 PROPN iajs-3553 64 4 : : PUNCT iajs-3553 64 5 𝐷2πΉπœ‡(𝑋. 𝐷2πΉπœ‡(𝑋. PROPN iajs-3553 64 6 πœ‡)(𝑇. πœ‡)(𝑇. PROPN iajs-3553 64 7 𝑇 𝑇 PROPN iajs-3553 64 8 ) ) PUNCT iajs-3553 64 9 = = PUNCT iajs-3553 65 1 [ [ X iajs-3553 65 2 π‘Œπ‘–1]3Γ—1 π‘Œπ‘–1]3Γ—1 X iajs-3553 65 3 . . PUNCT iajs-3553 66 1 ( ( PUNCT iajs-3553 66 2 3 3 X iajs-3553 66 3 ) ) PUNCT iajs-3553 66 4 π‘Œ11 π‘Œ11 PROPN iajs-3553 66 5 = = SYM iajs-3553 66 6 2 2 NUM iajs-3553 66 7 ( ( PUNCT iajs-3553 66 8 ( ( PUNCT iajs-3553 66 9 𝛽1𝑅2π‘Ž1 𝛽1𝑅2π‘Ž1 X iajs-3553 66 10 ( ( PUNCT iajs-3553 66 11 1+π‘Ž1𝑅1)3(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)3(1+π‘Ž2𝑅2 NUM iajs-3553 66 12 ) ) PUNCT iajs-3553 66 13 βˆ’ βˆ’ PROPN iajs-3553 66 14 π‘š1 π‘š1 NOUN iajs-3553 66 15 ) ) PUNCT iajs-3553 66 16 𝑑1 𝑑1 NOUN iajs-3553 66 17 2 2 NUM iajs-3553 66 18 βˆ’ βˆ’ PROPN iajs-3553 66 19 ( ( PUNCT iajs-3553 66 20 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 66 21 ( ( PUNCT iajs-3553 66 22 1+𝐾1𝑅2)2 1+𝐾1𝑅2)2 NUM iajs-3553 66 23 + + NUM iajs-3553 66 24 𝛽1 𝛽1 NOUN iajs-3553 66 25 ( ( PUNCT iajs-3553 66 26 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)2 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)2 NUM iajs-3553 66 27 ) ) PUNCT iajs-3553 66 28 𝑑1𝑑2 𝑑1𝑑2 X iajs-3553 66 29 + + CCONJ iajs-3553 66 30 ( ( PUNCT iajs-3553 66 31 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 66 32 2 2 NUM iajs-3553 66 33 ( ( PUNCT iajs-3553 66 34 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 66 35 + + CCONJ iajs-3553 66 36 𝛽1π‘Ž2 𝛽1π‘Ž2 X iajs-3553 66 37 ( ( PUNCT iajs-3553 66 38 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 NUM iajs-3553 66 39 ) ) PUNCT iajs-3553 66 40 𝑑2 𝑑2 NOUN iajs-3553 66 41 2𝑅1 2𝑅1 NUM iajs-3553 66 42 , , PUNCT iajs-3553 66 43 π‘Œ21 π‘Œ21 PROPN iajs-3553 66 44 = = SYM iajs-3553 66 45 βˆ’2 βˆ’2 NOUN iajs-3553 66 46 ( ( PUNCT iajs-3553 66 47 𝑙1r2π‘Ž1 𝑙1r2π‘Ž1 PROPN iajs-3553 66 48 ( ( PUNCT iajs-3553 66 49 1+a1r1)3(1+a2r2 1+a1r1)3(1+a2r2 NUM iajs-3553 66 50 ) ) PUNCT iajs-3553 66 51 𝑑1 𝑑1 NOUN iajs-3553 66 52 2 2 NUM iajs-3553 66 53 βˆ’ βˆ’ NOUN iajs-3553 66 54 𝑙1 𝑙1 NOUN iajs-3553 66 55 ( ( PUNCT iajs-3553 66 56 1+a1r1)2(1+a2r2)2 1+a1r1)2(1+a2r2)2 NUM iajs-3553 66 57 𝑑1𝑑2 𝑑1𝑑2 NOUN iajs-3553 66 58 βˆ’ βˆ’ PROPN iajs-3553 66 59 ( ( PUNCT iajs-3553 66 60 r2k2 r2k2 X iajs-3553 66 61 2r2 2r2 NUM iajs-3553 66 62 ( ( PUNCT iajs-3553 66 63 1+k2r3)3 1+k2r3)3 NUM iajs-3553 66 64 + + X iajs-3553 66 65 Ξ²2 Ξ²2 NOUN iajs-3553 66 66 a4r2 a4r2 NOUN iajs-3553 66 67 ( ( PUNCT iajs-3553 66 68 1+a3r2)(1+a4r3)3 1+a3r2)(1+a4r3)3 NUM iajs-3553 66 69 ) ) PUNCT iajs-3553 66 70 𝑑3 𝑑3 PROPN iajs-3553 66 71 2 2 NUM iajs-3553 66 72 + + CCONJ iajs-3553 66 73 ( ( PUNCT iajs-3553 66 74 π‘š2 π‘š2 NOUN iajs-3553 66 75 + + CCONJ iajs-3553 67 1 𝑙1r1π‘Ž2 𝑙1r1π‘Ž2 PROPN iajs-3553 68 1 ( ( PUNCT iajs-3553 68 2 1+a1r1)(1+a2r2)3 1+a1r1)(1+a2r2)3 NUM iajs-3553 68 3 βˆ’ βˆ’ NOUN iajs-3553 68 4 Ξ²2r3π‘Ž3 Ξ²2r3π‘Ž3 PROPN iajs-3553 68 5 ( ( PUNCT iajs-3553 68 6 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 68 7 ) ) PUNCT iajs-3553 68 8 ) ) PUNCT iajs-3553 69 1 𝑑2 𝑑2 NOUN iajs-3553 69 2 2 2 NUM iajs-3553 69 3 + + CCONJ iajs-3553 69 4 ( ( PUNCT iajs-3553 69 5 r2k2 r2k2 X iajs-3553 69 6 ( ( PUNCT iajs-3553 69 7 1+k2r3)2 1+k2r3)2 NOUN iajs-3553 69 8 + + CCONJ iajs-3553 69 9 Ξ²2 Ξ²2 NOUN iajs-3553 69 10 ( ( PUNCT iajs-3553 69 11 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 NUM iajs-3553 69 12 ) ) PUNCT iajs-3553 69 13 𝑑2𝑑3 𝑑2𝑑3 NOUN iajs-3553 69 14 ) ) PUNCT iajs-3553 69 15 , , PUNCT iajs-3553 69 16 π‘Œ31 π‘Œ31 PROPN iajs-3553 69 17 = = SYM iajs-3553 70 1 βˆ’2 βˆ’2 NOUN iajs-3553 70 2 ( ( PUNCT iajs-3553 70 3 𝑙2𝑅3π‘Ž3 𝑙2𝑅3π‘Ž3 NOUN iajs-3553 70 4 ( ( PUNCT iajs-3553 70 5 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 70 6 ) ) PUNCT iajs-3553 70 7 𝑑2 𝑑2 NOUN iajs-3553 70 8 2 2 NUM iajs-3553 70 9 βˆ’ βˆ’ PROPN iajs-3553 70 10 𝑙2 𝑙2 NOUN iajs-3553 70 11 ( ( PUNCT iajs-3553 70 12 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 NUM iajs-3553 70 13 𝑑2𝑑3 𝑑2𝑑3 X iajs-3553 70 14 + + ADJ iajs-3553 70 15 𝑙2𝑅2π‘Ž4 𝑙2𝑅2π‘Ž4 NOUN iajs-3553 70 16 ( ( PUNCT iajs-3553 70 17 1+a3r2)(1+a4r3)3 1+a3r2)(1+a4r3)3 NUM iajs-3553 70 18 𝑑3 𝑑3 PROPN iajs-3553 70 19 2 2 NUM iajs-3553 70 20 ) ) PUNCT iajs-3553 70 21 , , PUNCT iajs-3553 70 22 and and CCONJ iajs-3553 70 23 𝐷3πΉπœ‡(𝑋. 𝐷3πΉπœ‡(𝑋. PROPN iajs-3553 70 24 πœ‡)(𝑇. πœ‡)(𝑇. PROPN iajs-3553 70 25 𝑇. 𝑇. PROPN iajs-3553 70 26 𝑇 𝑇 PROPN iajs-3553 70 27 ) ) PUNCT iajs-3553 70 28 = = PUNCT iajs-3553 71 1 [ [ X iajs-3553 71 2 π‘Šπ‘–1]3Γ—1 π‘Šπ‘–1]3Γ—1 ADV iajs-3553 71 3 , , PUNCT iajs-3553 71 4 ( ( PUNCT iajs-3553 71 5 4 4 X iajs-3553 71 6 ) ) PUNCT iajs-3553 71 7 π‘Š11 π‘Š11 NOUN iajs-3553 71 8 = = PUNCT iajs-3553 71 9 6 6 NUM iajs-3553 71 10 [ [ X iajs-3553 71 11 ( ( PUNCT iajs-3553 71 12 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 71 13 2 2 NUM iajs-3553 71 14 ( ( PUNCT iajs-3553 71 15 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 71 16 + + CCONJ iajs-3553 71 17 𝛽1π‘Ž2 𝛽1π‘Ž2 X iajs-3553 71 18 ( ( PUNCT iajs-3553 71 19 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)3 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)3 NUM iajs-3553 71 20 ) ) PUNCT iajs-3553 71 21 𝑑1𝑑2 𝑑1𝑑2 X iajs-3553 71 22 2 2 NUM iajs-3553 71 23 + + CCONJ iajs-3553 71 24 ( ( PUNCT iajs-3553 71 25 𝑑2 𝑑2 ADJ iajs-3553 71 26 1+π‘Ž2𝑅2 1+π‘Ž2𝑅2 PROPN iajs-3553 71 27 βˆ’ βˆ’ PROPN iajs-3553 71 28 r2π‘Ž1𝑑1 r2π‘Ž1𝑑1 PROPN iajs-3553 71 29 1+π‘Ž1𝑅1 1+π‘Ž1𝑅1 NUM iajs-3553 71 30 ) ) PUNCT iajs-3553 71 31 𝛽1π‘Ž1𝑑1 𝛽1π‘Ž1𝑑1 ADJ iajs-3553 71 32 2 2 NUM iajs-3553 71 33 ( ( PUNCT iajs-3553 71 34 1+π‘Ž1𝑅1)3(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)3(1+π‘Ž2𝑅2 NUM iajs-3553 71 35 ) ) PUNCT iajs-3553 71 36 βˆ’ βˆ’ PROPN iajs-3553 71 37 ( ( PUNCT iajs-3553 71 38 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 71 39 3𝑅1 3𝑅1 NUM iajs-3553 71 40 ( ( PUNCT iajs-3553 71 41 1+𝐾1𝑅2)4 1+𝐾1𝑅2)4 NUM iajs-3553 71 42 + + CCONJ iajs-3553 71 43 𝛽1π‘Ž2 𝛽1π‘Ž2 ADP iajs-3553 71 44 2𝑅1 2𝑅1 NUM iajs-3553 71 45 ( ( PUNCT iajs-3553 71 46 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 NUM iajs-3553 71 47 ) ) PUNCT iajs-3553 71 48 𝑑2 𝑑2 NOUN iajs-3553 71 49 3 3 NUM iajs-3553 71 50 ] ] PUNCT iajs-3553 71 51 . . PUNCT iajs-3553 72 1 π‘Š21 π‘Š21 NOUN iajs-3553 72 2 = = PUNCT iajs-3553 73 1 6 6 NUM iajs-3553 73 2 [ [ X iajs-3553 73 3 ( ( PUNCT iajs-3553 73 4 r2k2 r2k2 X iajs-3553 73 5 2 2 NUM iajs-3553 73 6 ( ( PUNCT iajs-3553 73 7 1+k2r3)3 1+k2r3)3 NUM iajs-3553 73 8 + + CCONJ iajs-3553 73 9 Ξ²2 Ξ²2 ADJ iajs-3553 73 10 a4 a4 NOUN iajs-3553 73 11 ( ( PUNCT iajs-3553 73 12 1+a3r2)2(1+a4r3)3 1+a3r2)2(1+a4r3)3 NUM iajs-3553 73 13 ) ) PUNCT iajs-3553 73 14 𝑑2𝑑3 𝑑2𝑑3 X iajs-3553 73 15 2 2 NUM iajs-3553 73 16 + + CCONJ iajs-3553 73 17 ( ( PUNCT iajs-3553 73 18 r2π‘Ž1𝑑1 r2π‘Ž1𝑑1 PROPN iajs-3553 73 19 1+a1r1 1+a1r1 NUM iajs-3553 73 20 βˆ’ βˆ’ PROPN iajs-3553 73 21 𝑑2 𝑑2 PROPN iajs-3553 73 22 1+π‘Ž2𝑅2 1+π‘Ž2𝑅2 PROPN iajs-3553 73 23 ) ) PUNCT iajs-3553 73 24 𝑙1π‘Ž1𝑑1 𝑙1π‘Ž1𝑑1 VERB iajs-3553 73 25 2 2 NUM iajs-3553 73 26 ( ( PUNCT iajs-3553 73 27 1+a1r1)3(1+π‘Ž2𝑅2 1+a1r1)3(1+π‘Ž2𝑅2 NOUN iajs-3553 73 28 ) ) PUNCT iajs-3553 73 29 + + CCONJ iajs-3553 73 30 ( ( PUNCT iajs-3553 73 31 𝑙1r1π‘Ž2 𝑙1r1π‘Ž2 ADP iajs-3553 73 32 2 2 NUM iajs-3553 73 33 ( ( PUNCT iajs-3553 73 34 1+a1r1)(1+a2r2)4 1+a1r1)(1+a2r2)4 NUM iajs-3553 73 35 βˆ’ βˆ’ NOUN iajs-3553 73 36 Ξ²2r3π‘Ž3 Ξ²2r3π‘Ž3 PROPN iajs-3553 73 37 2 2 NUM iajs-3553 73 38 ( ( PUNCT iajs-3553 73 39 1+a3r2)4(1+a4r3 1+a3r2)4(1+a4r3 NUM iajs-3553 73 40 ) ) PUNCT iajs-3553 73 41 ) ) PUNCT iajs-3553 74 1 𝑑2 𝑑2 NOUN iajs-3553 74 2 3 3 NUM iajs-3553 75 1 + + CCONJ iajs-3553 75 2 ( ( PUNCT iajs-3553 75 3 Ξ²2 Ξ²2 VERB iajs-3553 75 4 a4𝑑3 a4𝑑3 PROPN iajs-3553 75 5 ( ( PUNCT iajs-3553 75 6 1+a3r2)3(1+a4r3)2 1+a3r2)3(1+a4r3)2 NUM iajs-3553 75 7 βˆ’ βˆ’ NOUN iajs-3553 75 8 𝑙1π‘Ž2𝑑1 𝑙1π‘Ž2𝑑1 NOUN iajs-3553 75 9 ( ( PUNCT iajs-3553 75 10 1+a1r1)2(1+a2r2)3 1+a1r1)2(1+a2r2)3 NUM iajs-3553 75 11 ) ) PUNCT iajs-3553 75 12 𝑑1 𝑑1 NOUN iajs-3553 75 13 2 2 NUM iajs-3553 75 14 βˆ’ βˆ’ NOUN iajs-3553 75 15 ( ( PUNCT iajs-3553 75 16 r2k2 r2k2 X iajs-3553 75 17 3𝑅2 3𝑅2 NUM iajs-3553 75 18 ( ( PUNCT iajs-3553 75 19 1+k2r3)4 1+k2r3)4 NUM iajs-3553 75 20 + + CCONJ iajs-3553 75 21 𝛽1π‘Ž2 𝛽1π‘Ž2 PROPN iajs-3553 75 22 2𝑅1 2𝑅1 NUM iajs-3553 75 23 ( ( PUNCT iajs-3553 75 24 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 NUM iajs-3553 75 25 ) ) PUNCT iajs-3553 75 26 𝑑3 𝑑3 PROPN iajs-3553 75 27 3 3 NUM iajs-3553 75 28 ] ] PUNCT iajs-3553 75 29 , , PUNCT iajs-3553 75 30 π‘Š31 π‘Š31 PROPN iajs-3553 75 31 = = PUNCT iajs-3553 75 32 6 6 NUM iajs-3553 75 33 [ [ X iajs-3553 75 34 ( ( PUNCT iajs-3553 75 35 π‘Ž3𝑅3𝑑2 π‘Ž3𝑅3𝑑2 NOUN iajs-3553 75 36 ( ( PUNCT iajs-3553 75 37 1+a3r2 1+a3r2 NUM iajs-3553 75 38 ) ) PUNCT iajs-3553 75 39 βˆ’ βˆ’ PROPN iajs-3553 75 40 𝑑3 𝑑3 PROPN iajs-3553 75 41 ( ( PUNCT iajs-3553 75 42 1+a4r3 1+a4r3 NUM iajs-3553 75 43 ) ) PUNCT iajs-3553 75 44 ) ) PUNCT iajs-3553 75 45 𝑙2π‘Ž3𝑑2 𝑙2π‘Ž3𝑑2 PUNCT iajs-3553 75 46 2 2 NUM iajs-3553 75 47 ( ( PUNCT iajs-3553 75 48 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 75 49 ) ) PUNCT iajs-3553 75 50 + + CCONJ iajs-3553 75 51 ( ( PUNCT iajs-3553 75 52 π‘Ž4𝑅2𝑑3 π‘Ž4𝑅2𝑑3 INTJ iajs-3553 75 53 ( ( PUNCT iajs-3553 75 54 1+a4r3 1+a4r3 NUM iajs-3553 75 55 ) ) PUNCT iajs-3553 75 56 βˆ’ βˆ’ PROPN iajs-3553 75 57 𝑑2 𝑑2 NOUN iajs-3553 75 58 ( ( PUNCT iajs-3553 75 59 1+a3r2 1+a3r2 NUM iajs-3553 75 60 ) ) PUNCT iajs-3553 75 61 ) ) PUNCT iajs-3553 76 1 𝑙2π‘Ž4𝑑3 𝑙2π‘Ž4𝑑3 PROPN iajs-3553 76 2 2 2 NUM iajs-3553 76 3 ( ( PUNCT iajs-3553 76 4 1+a3r2)(1+a4r3)3 1+a3r2)(1+a4r3)3 NUM iajs-3553 76 5 ] ] PUNCT iajs-3553 76 6 , , PUNCT iajs-3553 76 7 where where SCONJ iajs-3553 76 8 𝑋 𝑋 PROPN iajs-3553 76 9 = = SYM iajs-3553 76 10 ( ( PUNCT iajs-3553 76 11 𝑅1 𝑅1 PROPN iajs-3553 76 12 . . PUNCT iajs-3553 76 13 𝑅2 𝑅2 PROPN iajs-3553 76 14 . . PUNCT iajs-3553 76 15 𝑅3 𝑅3 PROPN iajs-3553 76 16 ) ) PUNCT iajs-3553 76 17 and and CCONJ iajs-3553 76 18 ( ( PUNCT iajs-3553 76 19 πœ‡ πœ‡ X iajs-3553 76 20 ) ) PUNCT iajs-3553 76 21 any any DET iajs-3553 76 22 parameter parameter NOUN iajs-3553 76 23 . . PUNCT iajs-3553 77 1 theorem theorem NOUN iajs-3553 77 2 ( ( PUNCT iajs-3553 77 3 1 1 NUM iajs-3553 77 4 ): ): PUNCT iajs-3553 77 5 system system NOUN iajs-3553 77 6 ( ( PUNCT iajs-3553 77 7 1 1 NUM iajs-3553 77 8 ) ) PUNCT iajs-3553 77 9 with with ADP iajs-3553 77 10 the the DET iajs-3553 77 11 parameter parameter NOUN iajs-3553 77 12 value value NOUN iajs-3553 77 13 𝑑`1 𝑑`1 NOUN iajs-3553 77 14 = = NOUN iajs-3553 77 15 𝑑1 𝑑1 NOUN iajs-3553 77 16 = = NOUN iajs-3553 77 17 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3553 77 18 + + CCONJ iajs-3553 77 19 𝑙1π‘Ÿ1 𝑙1π‘Ÿ1 NOUN iajs-3553 77 20 π‘š1+π‘Ž1π‘Ÿ1 π‘š1+π‘Ž1π‘Ÿ1 NUM iajs-3553 77 21 , , PUNCT iajs-3553 77 22 has have VERB iajs-3553 77 23 a a DET iajs-3553 77 24 transcritical transcritical ADJ iajs-3553 77 25 and and CCONJ iajs-3553 77 26 pitchfork pitchfork NOUN iajs-3553 77 27 bifurcation bifurcation NOUN iajs-3553 77 28 at at ADP iajs-3553 77 29 𝐴1 𝐴1 PROPN iajs-3553 77 30 = = PUNCT iajs-3553 78 1 ( ( PUNCT iajs-3553 78 2 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 78 3 π‘š1 π‘š1 NOUN iajs-3553 78 4 . . PUNCT iajs-3553 79 1 0 0 X iajs-3553 79 2 . . NOUN iajs-3553 79 3 0 0 NUM iajs-3553 79 4 ) ) PUNCT iajs-3553 79 5 , , PUNCT iajs-3553 79 6 if if SCONJ iajs-3553 79 7 the the DET iajs-3553 79 8 next next ADJ iajs-3553 79 9 conditions condition NOUN iajs-3553 79 10 hold hold VERB iajs-3553 79 11 : : PUNCT iajs-3553 79 12 𝑍`1 𝑍`1 PROPN iajs-3553 79 13 β‰  β‰  PROPN iajs-3553 79 14 𝑍`2 𝑍`2 NOUN iajs-3553 79 15 , , PUNCT iajs-3553 79 16 ( ( PUNCT iajs-3553 79 17 5 5 X iajs-3553 79 18 ) ) PUNCT iajs-3553 79 19 𝑍`3 𝑍`3 PROPN iajs-3553 79 20 β‰  β‰  PROPN iajs-3553 79 21 𝑍`4 𝑍`4 PROPN iajs-3553 79 22 . . PUNCT iajs-3553 80 1 ( ( PUNCT iajs-3553 80 2 6 6 NUM iajs-3553 80 3 ) ) PUNCT iajs-3553 81 1 where where SCONJ iajs-3553 81 2 : : PUNCT iajs-3553 81 3 𝑍`1 𝑍`1 NOUN iajs-3553 81 4 = = SYM iajs-3553 81 5 ( ( PUNCT iajs-3553 81 6 𝑙1𝑁 𝑙1𝑁 ADV iajs-3553 81 7 ( ( PUNCT iajs-3553 81 8 1+a1r1)2(1+a2r2)2 1+a1r1)2(1+a2r2)2 NUM iajs-3553 81 9 + + CCONJ iajs-3553 81 10 Ξ²2r3π‘Ž3 Ξ²2r3π‘Ž3 PROPN iajs-3553 81 11 ( ( PUNCT iajs-3553 81 12 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 81 13 ) ) PUNCT iajs-3553 81 14 ) ) PUNCT iajs-3553 81 15 , , PUNCT iajs-3553 81 16 𝑍`2 𝑍`2 NOUN iajs-3553 81 17 = = SYM iajs-3553 81 18 ( ( PUNCT iajs-3553 81 19 𝑙1r2π‘Ž1𝑁2 𝑙1r2π‘Ž1𝑁2 PROPN iajs-3553 81 20 ( ( PUNCT iajs-3553 81 21 1+a1r1)3(1+a2r2 1+a1r1)3(1+a2r2 NUM iajs-3553 81 22 ) ) PUNCT iajs-3553 81 23 + + NUM iajs-3553 81 24 π‘š2 π‘š2 NOUN iajs-3553 81 25 + + CCONJ iajs-3553 81 26 𝑙1r1π‘Ž2 𝑙1r1π‘Ž2 PROPN iajs-3553 81 27 ( ( PUNCT iajs-3553 81 28 1+a1r1)(1+a2r2)3 1+a1r1)(1+a2r2)3 NUM iajs-3553 81 29 ) ) PUNCT iajs-3553 81 30 , , PUNCT iajs-3553 81 31 𝑍`3 𝑍`3 NOUN iajs-3553 81 32 = = PUNCT iajs-3553 81 33 ( ( PUNCT iajs-3553 81 34 𝑙1r2π‘Ž1 𝑙1r2π‘Ž1 PROPN iajs-3553 81 35 2𝑁3 2𝑁3 NUM iajs-3553 81 36 ( ( PUNCT iajs-3553 81 37 1+a1r1)4(1+a2r2 1+a1r1)4(1+a2r2 NUM iajs-3553 81 38 ) ) PUNCT iajs-3553 81 39 + + CCONJ iajs-3553 81 40 𝑙1r1π‘Ž2 𝑙1r1π‘Ž2 ADP iajs-3553 81 41 2 2 NUM iajs-3553 81 42 ( ( PUNCT iajs-3553 81 43 1+a1r1)(1+a2r2)4 1+a1r1)(1+a2r2)4 NUM iajs-3553 81 44 ) ) PUNCT iajs-3553 81 45 , , PUNCT iajs-3553 81 46 𝑍`4 𝑍`4 NOUN iajs-3553 81 47 = = PRON iajs-3553 81 48 ( ( PUNCT iajs-3553 81 49 𝑙1π‘Ž1𝑁2 𝑙1π‘Ž1𝑁2 NOUN iajs-3553 81 50 ( ( PUNCT iajs-3553 81 51 1+a1r1)3(1+a2r2)2 1+a1r1)3(1+a2r2)2 NUM iajs-3553 81 52 + + NUM iajs-3553 81 53 𝑙1π‘Ž2𝑁 𝑙1π‘Ž2𝑁 PROPN iajs-3553 81 54 ( ( PUNCT iajs-3553 81 55 1+a1r1)2(1+a2r2)3 1+a1r1)2(1+a2r2)3 NUM iajs-3553 81 56 + + CCONJ iajs-3553 81 57 Ξ²2r3π‘Ž3 Ξ²2r3π‘Ž3 PROPN iajs-3553 81 58 2 2 NUM iajs-3553 81 59 ( ( PUNCT iajs-3553 81 60 1+a3r2)4(1+a4r3 1+a3r2)4(1+a4r3 NUM iajs-3553 81 61 ) ) PUNCT iajs-3553 81 62 ) ) PUNCT iajs-3553 81 63 . . PUNCT iajs-3553 82 1 proof proof NOUN iajs-3553 82 2 : : PUNCT iajs-3553 82 3 by by ADP iajs-3553 82 4 using use VERB iajs-3553 82 5 the the DET iajs-3553 82 6 jacobian jacobian ADJ iajs-3553 82 7 matrix matrix NOUN iajs-3553 82 8 in in ADP iajs-3553 82 9 equation equation NOUN iajs-3553 82 10 ( ( PUNCT iajs-3553 82 11 1.7 1.7 NUM iajs-3553 82 12 ) ) PUNCT iajs-3553 82 13 in in ADP iajs-3553 82 14 ( ( PUNCT iajs-3553 82 15 20 20 NUM iajs-3553 82 16 ) ) PUNCT iajs-3553 82 17 𝐽`1 𝐽`1 NOUN iajs-3553 82 18 = = PUNCT iajs-3553 82 19 𝐽1(𝐴1 𝐽1(𝐴1 PROPN iajs-3553 82 20 . . PUNCT iajs-3553 82 21 𝑑`1 𝑑`1 ADJ iajs-3553 82 22 ) ) PUNCT iajs-3553 83 1 = = PUNCT iajs-3553 84 1 [ [ X iajs-3553 84 2 π‘ž`𝑖𝑗]3Γ—3 π‘ž`𝑖𝑗]3Γ—3 X iajs-3553 84 3 , , PUNCT iajs-3553 84 4 where where SCONJ iajs-3553 84 5 π‘ž`𝑖𝑗 π‘ž`𝑖𝑗 NOUN iajs-3553 84 6 = = SYM iajs-3553 84 7 π‘žπ‘–π‘— π‘žπ‘–π‘— NOUN iajs-3553 84 8 , , PUNCT iajs-3553 84 9 except except SCONJ iajs-3553 84 10 π‘ž`22 π‘ž`22 NOUN iajs-3553 84 11 = = SYM iajs-3553 84 12 0 0 NUM iajs-3553 84 13 .then .then NOUN iajs-3553 84 14 , , PUNCT iajs-3553 84 15 the the DET iajs-3553 84 16 characterizing characterize VERB iajs-3553 84 17 equation equation NOUN iajs-3553 84 18 for𝐽`1 for𝐽`1 ADV iajs-3553 84 19 has have VERB iajs-3553 84 20 an an DET iajs-3553 84 21 eigenvalue eigenvalue NOUN iajs-3553 84 22 of of ADP iajs-3553 84 23 zero zero NUM iajs-3553 84 24 , , PUNCT iajs-3553 84 25 which which PRON iajs-3553 84 26 is be AUX iajs-3553 84 27 Ξ»1r2 Ξ»1r2 PUNCT iajs-3553 84 28 at at ADP iajs-3553 84 29 𝑑`1 𝑑`1 NOUN iajs-3553 84 30 = = NOUN iajs-3553 84 31 𝑑1 𝑑1 NOUN iajs-3553 84 32 , , PUNCT iajs-3553 84 33 now now ADV iajs-3553 84 34 , , PUNCT iajs-3553 84 35 let let VERB iajs-3553 84 36 𝑇[1 𝑇[1 VERB iajs-3553 84 37 ] ] X iajs-3553 84 38 = = SYM iajs-3553 84 39 ( ( PUNCT iajs-3553 84 40 𝑑1 𝑑1 NOUN iajs-3553 84 41 [ [ X iajs-3553 84 42 1 1 NUM iajs-3553 84 43 ] ] PUNCT iajs-3553 84 44 . . PUNCT iajs-3553 85 1 𝑑2 𝑑2 PROPN iajs-3553 85 2 [ [ X iajs-3553 85 3 1 1 NUM iajs-3553 85 4 ] ] PUNCT iajs-3553 85 5 . . PUNCT iajs-3553 86 1 𝑑3 𝑑3 PROPN iajs-3553 86 2 [ [ X iajs-3553 86 3 1 1 NUM iajs-3553 86 4 ] ] PUNCT iajs-3553 86 5 ) ) PUNCT iajs-3553 86 6 𝑇 𝑇 PROPN iajs-3553 86 7 be be VERB iajs-3553 86 8 the the DET iajs-3553 86 9 eigenvector eigenvector NOUN iajs-3553 86 10 associated associate VERB iajs-3553 86 11 with with ADP iajs-3553 86 12 an an DET iajs-3553 86 13 eigenvalue eigenvalue ADJ iajs-3553 86 14 Ξ»1r2 Ξ»1r2 X iajs-3553 86 15 = = SYM iajs-3553 86 16 0 0 PROPN iajs-3553 86 17 . . NOUN iajs-3553 86 18 thus,(𝐽`1 thus,(𝐽`1 PROPN iajs-3553 86 19 βˆ’ βˆ’ X iajs-3553 86 20 Ξ»1r2 Ξ»1r2 PUNCT iajs-3553 86 21 𝐼)𝑇[1 𝐼)𝑇[1 NOUN iajs-3553 86 22 ] ] X iajs-3553 86 23 = = SYM iajs-3553 86 24 0 0 X iajs-3553 86 25 . . PUNCT iajs-3553 87 1 this this PRON iajs-3553 87 2 gives give VERB iajs-3553 87 3 : : PUNCT iajs-3553 87 4 𝑑1 𝑑1 NOUN iajs-3553 87 5 [ [ X iajs-3553 87 6 1 1 X iajs-3553 87 7 ] ] PUNCT iajs-3553 87 8 = = X iajs-3553 87 9 𝑁𝑑2 𝑁𝑑2 NOUN iajs-3553 88 1 [ [ X iajs-3553 88 2 1 1 NUM iajs-3553 88 3 ] ] PUNCT iajs-3553 88 4 . . PUNCT iajs-3553 89 1 𝑑3 𝑑3 PROPN iajs-3553 89 2 [ [ X iajs-3553 89 3 1 1 NUM iajs-3553 89 4 ] ] X iajs-3553 89 5 = = SYM iajs-3553 89 6 0.π‘€β„Žπ‘’π‘Ÿπ‘’ 0.π‘€β„Žπ‘’π‘Ÿπ‘’ NUM iajs-3553 89 7 𝑁 𝑁 PROPN iajs-3553 89 8 = = PUNCT iajs-3553 89 9 βˆ’ βˆ’ PROPN iajs-3553 89 10 ( ( PUNCT iajs-3553 89 11 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 89 12 π‘š1 π‘š1 X iajs-3553 89 13 + + CCONJ iajs-3553 89 14 𝛽1 𝛽1 PROPN iajs-3553 89 15 π‘š1+π‘Ž1π‘Ÿ1 π‘š1+π‘Ž1π‘Ÿ1 NUM iajs-3553 89 16 ) ) PUNCT iajs-3553 89 17 and and CCONJ iajs-3553 89 18 𝑑2 𝑑2 NOUN iajs-3553 90 1 [ [ X iajs-3553 90 2 1 1 X iajs-3553 90 3 ] ] PUNCT iajs-3553 90 4 any any DET iajs-3553 90 5 real real ADJ iajs-3553 90 6 number number NOUN iajs-3553 90 7 that that PRON iajs-3553 90 8 is be AUX iajs-3553 90 9 not not PART iajs-3553 90 10 zero zero NUM iajs-3553 90 11 . . PUNCT iajs-3553 91 1 ihjpas ihjpas PROPN iajs-3553 91 2 . . PUNCT iajs-3553 92 1 2025 2025 NUM iajs-3553 92 2 , , PUNCT iajs-3553 92 3 38(2 38(2 NUM iajs-3553 92 4 ) ) PUNCT iajs-3553 92 5 319 319 NUM iajs-3553 92 6 let let VERB iajs-3553 92 7 𝑀[1 𝑀[1 NUM iajs-3553 92 8 ] ] X iajs-3553 92 9 = = SYM iajs-3553 92 10 ( ( PUNCT iajs-3553 92 11 π‘š1 π‘š1 X iajs-3553 92 12 [ [ X iajs-3553 92 13 1 1 NUM iajs-3553 92 14 ] ] PUNCT iajs-3553 92 15 . . PUNCT iajs-3553 93 1 π‘š2 π‘š2 NOUN iajs-3553 94 1 [ [ X iajs-3553 94 2 1 1 NUM iajs-3553 94 3 ] ] PUNCT iajs-3553 94 4 . . PUNCT iajs-3553 95 1 π‘š3 π‘š3 NOUN iajs-3553 96 1 [ [ X iajs-3553 96 2 1 1 NUM iajs-3553 96 3 ] ] PUNCT iajs-3553 96 4 ) ) PUNCT iajs-3553 96 5 𝑇 𝑇 PROPN iajs-3553 96 6 be be VERB iajs-3553 96 7 the the DET iajs-3553 96 8 eigenvector eigenvector NOUN iajs-3553 96 9 associated associate VERB iajs-3553 96 10 with with ADP iajs-3553 96 11 an an DET iajs-3553 96 12 eigenvalue eigenvalue ADJ iajs-3553 96 13 Ξ»1r2 Ξ»1r2 X iajs-3553 96 14 = = NOUN iajs-3553 96 15 0 0 NUM iajs-3553 96 16 of of ADP iajs-3553 96 17 the the DET iajs-3553 96 18 matrix matrix NOUN iajs-3553 96 19 𝐽`1 𝐽`1 NOUN iajs-3553 96 20 𝑇.then 𝑇.then ADV iajs-3553 96 21 . . PUNCT iajs-3553 97 1 ( ( PUNCT iajs-3553 97 2 𝐽`1 𝐽`1 NOUN iajs-3553 97 3 𝑇 𝑇 PROPN iajs-3553 97 4 βˆ’ βˆ’ PROPN iajs-3553 97 5 Ξ»1r2 Ξ»1r2 PROPN iajs-3553 97 6 𝐼 𝐼 PROPN iajs-3553 97 7 ) ) PUNCT iajs-3553 97 8 𝑀[1 𝑀[1 PROPN iajs-3553 97 9 ] ] X iajs-3553 97 10 = = SYM iajs-3553 97 11 0 0 NUM iajs-3553 97 12 . . PUNCT iajs-3553 98 1 by by ADP iajs-3553 98 2 solving solve VERB iajs-3553 98 3 this this DET iajs-3553 98 4 equation equation NOUN iajs-3553 98 5 for for ADP iajs-3553 98 6 . . PUNCT iajs-3553 98 7 𝑀[1 𝑀[1 PROPN iajs-3553 98 8 ] ] X iajs-3553 98 9 = = SYM iajs-3553 98 10 ( ( PUNCT iajs-3553 98 11 0.π‘š2 0.π‘š2 X iajs-3553 98 12 [ [ X iajs-3553 98 13 1 1 NUM iajs-3553 98 14 ] ] PUNCT iajs-3553 98 15 . . PUNCT iajs-3553 98 16 0 0 X iajs-3553 98 17 ) ) PUNCT iajs-3553 98 18 𝑇 𝑇 PROPN iajs-3553 98 19 .where .where ADP iajs-3553 98 20 π‘š2 π‘š2 NOUN iajs-3553 98 21 [ [ X iajs-3553 98 22 1 1 X iajs-3553 98 23 ] ] PUNCT iajs-3553 98 24 any any DET iajs-3553 98 25 real real ADJ iajs-3553 98 26 number number NOUN iajs-3553 98 27 that that PRON iajs-3553 98 28 is be AUX iajs-3553 98 29 not not PART iajs-3553 98 30 zero zero NUM iajs-3553 98 31 . . PUNCT iajs-3553 98 32 now now ADV iajs-3553 98 33 consider consider VERB iajs-3553 98 34 this this PRON iajs-3553 98 35 : : PUNCT iajs-3553 98 36 πœ•π‘“ πœ•π‘“ ADJ iajs-3553 98 37 πœ•π‘‘1 πœ•π‘‘1 NOUN iajs-3553 98 38 = = SYM iajs-3553 98 39 𝑓𝑑1 𝑓𝑑1 NOUN iajs-3553 98 40 ( ( PUNCT iajs-3553 98 41 𝑋. 𝑋. PROPN iajs-3553 98 42 𝑑1 𝑑1 PROPN iajs-3553 98 43 ) ) PUNCT iajs-3553 98 44 = = PUNCT iajs-3553 98 45 ( ( PUNCT iajs-3553 98 46 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3553 98 47 πœ•π‘‘1 πœ•π‘‘1 NOUN iajs-3553 98 48 . . PUNCT iajs-3553 99 1 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3553 99 2 πœ•π‘‘1 πœ•π‘‘1 NOUN iajs-3553 99 3 . . PUNCT iajs-3553 100 1 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3553 100 2 πœ•π‘‘1 πœ•π‘‘1 NOUN iajs-3553 100 3 ) ) PUNCT iajs-3553 100 4 𝑇 𝑇 PROPN iajs-3553 100 5 = = SYM iajs-3553 100 6 ( ( PUNCT iajs-3553 100 7 0.βˆ’π‘…2 0.βˆ’π‘…2 NOUN iajs-3553 100 8 . . PUNCT iajs-3553 100 9 0)𝑇 0)𝑇 NOUN iajs-3553 100 10 . . PUNCT iajs-3553 101 1 so so ADV iajs-3553 101 2 , , PUNCT iajs-3553 101 3 𝑓𝑑1 𝑓𝑑1 NOUN iajs-3553 101 4 ( ( PUNCT iajs-3553 101 5 𝐴1 𝐴1 PROPN iajs-3553 101 6 . . PUNCT iajs-3553 102 1 𝑑`1 𝑑`1 ADJ iajs-3553 102 2 ) ) PUNCT iajs-3553 103 1 = = PRON iajs-3553 103 2 ( ( PUNCT iajs-3553 103 3 0 0 NUM iajs-3553 103 4 . . NOUN iajs-3553 103 5 0 0 NUM iajs-3553 103 6 . . X iajs-3553 103 7 0)𝑇 0)𝑇 NOUN iajs-3553 103 8 and and CCONJ iajs-3553 103 9 hence hence ADV iajs-3553 103 10 ( ( PUNCT iajs-3553 103 11 𝑀[1 𝑀[1 PROPN iajs-3553 103 12 ] ] PUNCT iajs-3553 103 13 ) ) PUNCT iajs-3553 103 14 𝑇 𝑇 PROPN iajs-3553 103 15 𝑓𝑑1 𝑓𝑑1 NOUN iajs-3553 103 16 ( ( PUNCT iajs-3553 103 17 𝐴1 𝐴1 PROPN iajs-3553 103 18 . . PUNCT iajs-3553 104 1 𝑑`1 𝑑`1 ADJ iajs-3553 104 2 ) ) PUNCT iajs-3553 105 1 = = SYM iajs-3553 105 2 0 0 NUM iajs-3553 105 3 using use VERB iajs-3553 105 4 sotomayor sotomayor NOUN iajs-3553 105 5 's 's PART iajs-3553 105 6 theorem theorem NOUN iajs-3553 105 7 , , PUNCT iajs-3553 105 8 it it PRON iajs-3553 105 9 is be AUX iajs-3553 105 10 impossible impossible ADJ iajs-3553 105 11 to to PART iajs-3553 105 12 satisfy satisfy VERB iajs-3553 105 13 the the DET iajs-3553 105 14 saddle saddle NOUN iajs-3553 105 15 - - PUNCT iajs-3553 105 16 node node NOUN iajs-3553 105 17 bifurcation bifurcation NOUN iajs-3553 105 18 condition condition NOUN iajs-3553 105 19 . . PUNCT iajs-3553 106 1 the the DET iajs-3553 106 2 first first ADJ iajs-3553 106 3 condition condition NOUN iajs-3553 106 4 for for ADP iajs-3553 106 5 transcritical transcritical ADJ iajs-3553 106 6 bifurcation bifurcation NOUN iajs-3553 106 7 is be AUX iajs-3553 106 8 therefore therefore ADV iajs-3553 106 9 satisfied satisfied ADJ iajs-3553 106 10 . . PUNCT iajs-3553 107 1 now now ADV iajs-3553 107 2 𝐷𝑓𝑑1 𝐷𝑓𝑑1 PROPN iajs-3553 107 3 ( ( PUNCT iajs-3553 107 4 𝑋. 𝑋. PROPN iajs-3553 107 5 𝑑1 𝑑1 PROPN iajs-3553 107 6 ) ) PUNCT iajs-3553 108 1 = = PUNCT iajs-3553 109 1 [ [ PUNCT iajs-3553 109 2 0 0 NUM iajs-3553 109 3 0 0 NUM iajs-3553 109 4 0 0 NUM iajs-3553 109 5 0 0 NUM iajs-3553 110 1 βˆ’ βˆ’ NUM iajs-3553 110 2 1 1 NUM iajs-3553 110 3 0 0 NUM iajs-3553 110 4 0 0 NUM iajs-3553 110 5 0 0 NUM iajs-3553 110 6 0 0 NUM iajs-3553 110 7 ] ] PUNCT iajs-3553 110 8 , , PUNCT iajs-3553 110 9 where where SCONJ iajs-3553 110 10 , , PUNCT iajs-3553 110 11 𝐷𝑓𝑑1 𝐷𝑓𝑑1 PROPN iajs-3553 110 12 ( ( PUNCT iajs-3553 110 13 𝑋. 𝑋. PROPN iajs-3553 110 14 𝑑1 𝑑1 PROPN iajs-3553 110 15 ) ) PUNCT iajs-3553 110 16 represents represent VERB iajs-3553 110 17 the the DET iajs-3553 110 18 derivative derivative NOUN iajs-3553 110 19 of of ADP iajs-3553 110 20 𝑓𝑑1 𝑓𝑑1 NOUN iajs-3553 110 21 ( ( PUNCT iajs-3553 110 22 𝑋. 𝑋. PROPN iajs-3553 110 23 𝑑1 𝑑1 PROPN iajs-3553 110 24 ) ) PUNCT iajs-3553 110 25 with with ADP iajs-3553 110 26 respect respect NOUN iajs-3553 110 27 to to ADP iajs-3553 110 28 𝑋 𝑋 PROPN iajs-3553 110 29 = = SYM iajs-3553 110 30 ( ( PUNCT iajs-3553 110 31 𝑅1 𝑅1 PROPN iajs-3553 110 32 . . PUNCT iajs-3553 110 33 𝑅2 𝑅2 PROPN iajs-3553 110 34 . . PUNCT iajs-3553 111 1 𝑅3 𝑅3 PROPN iajs-3553 111 2 ) ) PUNCT iajs-3553 111 3 𝑇 𝑇 PROPN iajs-3553 111 4 . . PUNCT iajs-3553 112 1 furthermore furthermore ADV iajs-3553 112 2 , , PUNCT iajs-3553 112 3 it it PRON iajs-3553 112 4 is be AUX iajs-3553 112 5 observed observe VERB iajs-3553 112 6 that that SCONJ iajs-3553 112 7 : : PUNCT iajs-3553 112 8 𝐷𝑓𝑑1 𝐷𝑓𝑑1 PROPN iajs-3553 112 9 ( ( PUNCT iajs-3553 112 10 𝐴1 𝐴1 PROPN iajs-3553 112 11 . . PUNCT iajs-3553 112 12 𝑑1)𝑇 𝑑1)𝑇 X iajs-3553 113 1 [ [ X iajs-3553 113 2 1 1 X iajs-3553 113 3 ] ] X iajs-3553 113 4 = = PUNCT iajs-3553 113 5 [ [ PUNCT iajs-3553 113 6 0 0 NUM iajs-3553 113 7 0 0 NUM iajs-3553 113 8 0 0 NUM iajs-3553 113 9 0 0 NUM iajs-3553 114 1 βˆ’ βˆ’ NUM iajs-3553 114 2 1 1 NUM iajs-3553 114 3 0 0 NUM iajs-3553 114 4 0 0 NUM iajs-3553 114 5 0 0 NUM iajs-3553 114 6 0 0 NUM iajs-3553 114 7 ] ] PUNCT iajs-3553 114 8 [ [ PUNCT iajs-3553 114 9 𝑁𝑑2 𝑁𝑑2 X iajs-3553 114 10 [ [ X iajs-3553 114 11 1 1 NUM iajs-3553 114 12 ] ] X iajs-3553 114 13 𝑑2 𝑑2 NOUN iajs-3553 115 1 [ [ X iajs-3553 115 2 1 1 NUM iajs-3553 115 3 ] ] PUNCT iajs-3553 115 4 0 0 NUM iajs-3553 115 5 ] ] PUNCT iajs-3553 116 1 = = PUNCT iajs-3553 116 2 [ [ PUNCT iajs-3553 116 3 0 0 X iajs-3553 116 4 βˆ’π‘‘2 βˆ’π‘‘2 ADJ iajs-3553 116 5 [ [ X iajs-3553 116 6 1 1 NUM iajs-3553 116 7 ] ] PUNCT iajs-3553 116 8 0 0 NUM iajs-3553 116 9 ] ] PUNCT iajs-3553 116 10 ( ( PUNCT iajs-3553 116 11 𝑀[1 𝑀[1 PROPN iajs-3553 116 12 ] ] PUNCT iajs-3553 116 13 ) ) PUNCT iajs-3553 116 14 𝑇 𝑇 PROPN iajs-3553 117 1 [ [ X iajs-3553 117 2 𝐷𝑓𝑑1 𝐷𝑓𝑑1 PROPN iajs-3553 117 3 ( ( PUNCT iajs-3553 117 4 𝐴1 𝐴1 PROPN iajs-3553 117 5 . . PUNCT iajs-3553 117 6 𝑑`1)𝑇 𝑑`1)𝑇 X iajs-3553 118 1 [ [ X iajs-3553 118 2 1 1 NUM iajs-3553 118 3 ] ] X iajs-3553 118 4 ] ] X iajs-3553 118 5 = = SYM iajs-3553 118 6 ( ( PUNCT iajs-3553 118 7 0.π‘š2 0.π‘š2 X iajs-3553 119 1 [ [ X iajs-3553 119 2 1 1 NUM iajs-3553 119 3 ] ] PUNCT iajs-3553 119 4 . . PUNCT iajs-3553 119 5 0 0 X iajs-3553 119 6 ) ) PUNCT iajs-3553 119 7 𝑇 𝑇 PROPN iajs-3553 119 8 ( ( PUNCT iajs-3553 119 9 0.βˆ’π‘‘2 0.βˆ’π‘‘2 X iajs-3553 119 10 [ [ X iajs-3553 119 11 1 1 NUM iajs-3553 119 12 ] ] PUNCT iajs-3553 119 13 . . PUNCT iajs-3553 119 14 0 0 X iajs-3553 119 15 ) ) PUNCT iajs-3553 119 16 = = VERB iajs-3553 120 1 βˆ’π‘‘2 βˆ’π‘‘2 ADJ iajs-3553 120 2 [ [ X iajs-3553 120 3 1 1 NUM iajs-3553 120 4 ] ] X iajs-3553 120 5 π‘š2 π‘š2 NOUN iajs-3553 121 1 [ [ X iajs-3553 121 2 1 1 NUM iajs-3553 121 3 ] ] PUNCT iajs-3553 121 4 β‰  β‰  PROPN iajs-3553 121 5 0 0 X iajs-3553 121 6 . . PUNCT iajs-3553 121 7 by by ADP iajs-3553 121 8 substituting substitute VERB iajs-3553 121 9 𝑇[1 𝑇[1 PROPN iajs-3553 121 10 ] ] PUNCT iajs-3553 121 11 in in ADP iajs-3553 121 12 equation equation NOUN iajs-3553 121 13 ( ( PUNCT iajs-3553 121 14 3 3 X iajs-3553 121 15 ) ) PUNCT iajs-3553 121 16 we we PRON iajs-3553 121 17 get get VERB iajs-3553 121 18 : : PUNCT iajs-3553 121 19 𝐷2πΉπœ‡(𝐴1 𝐷2πΉπœ‡(𝐴1 NOUN iajs-3553 121 20 . . PUNCT iajs-3553 121 21 𝑑`1)(𝑇 𝑑`1)(𝑇 X iajs-3553 122 1 [ [ X iajs-3553 122 2 1 1 NUM iajs-3553 122 3 ] ] PUNCT iajs-3553 122 4 . . PUNCT iajs-3553 123 1 𝑇[1 𝑇[1 PROPN iajs-3553 123 2 ] ] X iajs-3553 123 3 ) ) PUNCT iajs-3553 123 4 = = SYM iajs-3553 123 5 ( ( PUNCT iajs-3553 123 6 π‘Œ`𝑖𝑗 π‘Œ`𝑖𝑗 NOUN iajs-3553 123 7 ) ) PUNCT iajs-3553 123 8 . . PUNCT iajs-3553 124 1 ( ( PUNCT iajs-3553 124 2 7 7 X iajs-3553 124 3 ) ) PUNCT iajs-3553 124 4 π‘Œ`11 π‘Œ`11 NOUN iajs-3553 124 5 = = SYM iajs-3553 124 6 2𝑑2 2𝑑2 NUM iajs-3553 124 7 2 2 NUM iajs-3553 124 8 [ [ X iajs-3553 124 9 ( ( PUNCT iajs-3553 124 10 r2π‘Ž1𝑁2 r2π‘Ž1𝑁2 NOUN iajs-3553 124 11 ( ( PUNCT iajs-3553 124 12 1+π‘Ž1𝑅1)2 1+π‘Ž1𝑅1)2 NUM iajs-3553 124 13 βˆ’ βˆ’ NOUN iajs-3553 124 14 𝑁 𝑁 PROPN iajs-3553 124 15 ( ( PUNCT iajs-3553 124 16 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 124 17 ) ) PUNCT iajs-3553 124 18 + + CCONJ iajs-3553 124 19 r1π‘Ž2 r1π‘Ž2 PROPN iajs-3553 124 20 ( ( PUNCT iajs-3553 124 21 1+a2r2)2 1+a2r2)2 NUM iajs-3553 124 22 ) ) PUNCT iajs-3553 124 23 𝛽1 𝛽1 NOUN iajs-3553 124 24 ( ( PUNCT iajs-3553 124 25 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 124 26 ) ) PUNCT iajs-3553 125 1 + + CCONJ iajs-3553 125 2 ( ( PUNCT iajs-3553 125 3 𝐾1𝑅1 𝐾1𝑅1 PROPN iajs-3553 125 4 ( ( PUNCT iajs-3553 125 5 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 125 6 ) ) PUNCT iajs-3553 125 7 βˆ’ βˆ’ PROPN iajs-3553 126 1 𝑁 𝑁 PROPN iajs-3553 126 2 ) ) PUNCT iajs-3553 126 3 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 126 4 ( ( PUNCT iajs-3553 126 5 1+𝐾1𝑅2)2 1+𝐾1𝑅2)2 NUM iajs-3553 126 6 βˆ’ βˆ’ NOUN iajs-3553 126 7 π‘š1𝑁 π‘š1𝑁 ADJ iajs-3553 126 8 2 2 NUM iajs-3553 126 9 ] ] PUNCT iajs-3553 126 10 , , PUNCT iajs-3553 126 11 π‘Œ`21 π‘Œ`21 NOUN iajs-3553 127 1 = = SYM iajs-3553 127 2 2𝑑2 2𝑑2 NUM iajs-3553 127 3 2 2 NUM iajs-3553 127 4 [ [ X iajs-3553 127 5 ( ( PUNCT iajs-3553 127 6 𝑁 𝑁 PROPN iajs-3553 127 7 ( ( PUNCT iajs-3553 127 8 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 127 9 ) ) PUNCT iajs-3553 127 10 βˆ’ βˆ’ NOUN iajs-3553 127 11 r1π‘Ž2 r1π‘Ž2 PROPN iajs-3553 127 12 ( ( PUNCT iajs-3553 127 13 1+a2r2)2 1+a2r2)2 NUM iajs-3553 127 14 βˆ’ βˆ’ PROPN iajs-3553 127 15 𝑙1r2π‘Ž1𝑁2 𝑙1r2π‘Ž1𝑁2 PROPN iajs-3553 127 16 ( ( PUNCT iajs-3553 127 17 1+a1r1)2 1+a1r1)2 NUM iajs-3553 127 18 ) ) PUNCT iajs-3553 127 19 𝑙1 𝑙1 NOUN iajs-3553 127 20 ( ( PUNCT iajs-3553 127 21 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 127 22 ) ) PUNCT iajs-3553 127 23 βˆ’ βˆ’ NOUN iajs-3553 127 24 π‘š2 π‘š2 NOUN iajs-3553 127 25 + + CCONJ iajs-3553 127 26 Ξ²2r3π‘Ž3 Ξ²2r3π‘Ž3 PROPN iajs-3553 127 27 ( ( PUNCT iajs-3553 127 28 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 127 29 ) ) PUNCT iajs-3553 127 30 ] ] PUNCT iajs-3553 127 31 , , PUNCT iajs-3553 127 32 π‘Œ`31= π‘Œ`31= PROPN iajs-3553 127 33 βˆ’2𝑑2 βˆ’2𝑑2 NUM iajs-3553 127 34 2𝑙2𝑅3π‘Ž3 2𝑙2𝑅3π‘Ž3 NUM iajs-3553 127 35 ( ( PUNCT iajs-3553 127 36 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 127 37 ) ) PUNCT iajs-3553 127 38 , , PUNCT iajs-3553 127 39 hence hence ADV iajs-3553 127 40 , , PUNCT iajs-3553 127 41 it it PRON iajs-3553 127 42 was be AUX iajs-3553 127 43 obtained obtain VERB iajs-3553 127 44 ( ( PUNCT iajs-3553 127 45 𝑀[1 𝑀[1 PROPN iajs-3553 127 46 ] ] PUNCT iajs-3553 127 47 ) ) PUNCT iajs-3553 127 48 𝑇 𝑇 PROPN iajs-3553 128 1 [ [ X iajs-3553 128 2 𝐷2πΉπœ‡(𝐴1 𝐷2πΉπœ‡(𝐴1 NOUN iajs-3553 128 3 . . PUNCT iajs-3553 128 4 𝑑`1)(𝑇 𝑑`1)(𝑇 X iajs-3553 129 1 [ [ X iajs-3553 129 2 1 1 NUM iajs-3553 129 3 ] ] PUNCT iajs-3553 129 4 . . PUNCT iajs-3553 130 1 𝑇[1 𝑇[1 PROPN iajs-3553 130 2 ] ] PUNCT iajs-3553 130 3 ) ) PUNCT iajs-3553 130 4 ] ] PUNCT iajs-3553 131 1 = = SYM iajs-3553 131 2 2 2 X iajs-3553 131 3 ( ( PUNCT iajs-3553 131 4 𝑑2 𝑑2 NOUN iajs-3553 131 5 [ [ X iajs-3553 131 6 1 1 NUM iajs-3553 131 7 ] ] PUNCT iajs-3553 131 8 ) ) PUNCT iajs-3553 131 9 2 2 NUM iajs-3553 131 10 π‘š2 π‘š2 NOUN iajs-3553 131 11 [ [ X iajs-3553 131 12 1 1 NUM iajs-3553 131 13 ] ] PUNCT iajs-3553 131 14 ( ( PUNCT iajs-3553 131 15 z`1 z`1 NOUN iajs-3553 131 16 βˆ’ βˆ’ ADP iajs-3553 131 17 𝑍`2 𝑍`2 NOUN iajs-3553 131 18 ) ) PUNCT iajs-3553 131 19 β‰  β‰  PROPN iajs-3553 131 20 0 0 X iajs-3553 131 21 . . PUNCT iajs-3553 132 1 this this PRON iajs-3553 132 2 indicates indicate VERB iajs-3553 132 3 that that SCONJ iajs-3553 132 4 the the DET iajs-3553 132 5 system system NOUN iajs-3553 132 6 ( ( PUNCT iajs-3553 132 7 1 1 X iajs-3553 132 8 ) ) PUNCT iajs-3553 132 9 exhibits exhibit VERB iajs-3553 132 10 a a DET iajs-3553 132 11 transcritical transcritical ADJ iajs-3553 132 12 bifurcation bifurcation NOUN iajs-3553 132 13 at at ADP iajs-3553 132 14 𝐴1 𝐴1 PROPN iajs-3553 132 15 with with ADP iajs-3553 132 16 a a DET iajs-3553 132 17 parameter parameter NOUN iajs-3553 132 18 𝑑`1 𝑑`1 NOUN iajs-3553 132 19 = = NOUN iajs-3553 132 20 𝑑1 𝑑1 NOUN iajs-3553 132 21 , , PUNCT iajs-3553 132 22 if if SCONJ iajs-3553 132 23 condition condition NOUN iajs-3553 132 24 ( ( PUNCT iajs-3553 132 25 5 5 NUM iajs-3553 132 26 ) ) PUNCT iajs-3553 132 27 not not PART iajs-3553 132 28 satisfied satisfied ADJ iajs-3553 132 29 then then ADV iajs-3553 132 30 . . PUNCT iajs-3553 133 1 by by ADP iajs-3553 133 2 substituting substitute VERB iajs-3553 133 3 𝑇[1 𝑇[1 PROPN iajs-3553 133 4 ] ] PUNCT iajs-3553 133 5 in in ADP iajs-3553 133 6 equation equation NOUN iajs-3553 133 7 ( ( PUNCT iajs-3553 133 8 4 4 X iajs-3553 133 9 ) ) PUNCT iajs-3553 133 10 we we PRON iajs-3553 133 11 get get VERB iajs-3553 133 12 : : PUNCT iajs-3553 133 13 𝐷3πΉπœ‡(𝐴2 𝐷3πΉπœ‡(𝐴2 NOUN iajs-3553 133 14 . . PUNCT iajs-3553 133 15 𝑑`1)(𝑇 𝑑`1)(𝑇 X iajs-3553 134 1 [ [ X iajs-3553 134 2 1 1 NUM iajs-3553 134 3 ] ] PUNCT iajs-3553 134 4 . . PUNCT iajs-3553 135 1 𝑇[1 𝑇[1 PROPN iajs-3553 135 2 ] ] PUNCT iajs-3553 135 3 . . PUNCT iajs-3553 136 1 𝑇[1 𝑇[1 PROPN iajs-3553 136 2 ] ] X iajs-3553 136 3 ) ) PUNCT iajs-3553 136 4 = = SYM iajs-3553 136 5 ( ( PUNCT iajs-3553 136 6 π‘Š`𝑖𝑗 π‘Š`𝑖𝑗 NOUN iajs-3553 136 7 ) ) PUNCT iajs-3553 136 8 , , PUNCT iajs-3553 136 9 ( ( PUNCT iajs-3553 136 10 8) 8) NUM iajs-3553 136 11 π‘Š`11 π‘Š`11 NOUN iajs-3553 136 12 = = SYM iajs-3553 137 1 6𝑑2 6𝑑2 NUM iajs-3553 137 2 3[(𝑁 3[(𝑁 NUM iajs-3553 137 3 βˆ’ βˆ’ PROPN iajs-3553 137 4 𝐾1𝑅1 𝐾1𝑅1 PROPN iajs-3553 137 5 ( ( PUNCT iajs-3553 137 6 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 137 7 ) ) PUNCT iajs-3553 137 8 ) ) PUNCT iajs-3553 137 9 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 137 10 2 2 NUM iajs-3553 137 11 ( ( PUNCT iajs-3553 137 12 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 137 13 + + CCONJ iajs-3553 137 14 ( ( PUNCT iajs-3553 137 15 π‘Ž1𝑁2 π‘Ž1𝑁2 PROPN iajs-3553 137 16 ( ( PUNCT iajs-3553 137 17 1+π‘Ž1𝑅1)2(1+π‘Ž1𝑅1 1+π‘Ž1𝑅1)2(1+π‘Ž1𝑅1 NUM iajs-3553 137 18 ) ) PUNCT iajs-3553 137 19 + + CCONJ iajs-3553 137 20 π‘Ž2𝑁 π‘Ž2𝑁 NUM iajs-3553 137 21 ( ( PUNCT iajs-3553 137 22 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)2 NUM iajs-3553 137 23 βˆ’ βˆ’ NUM iajs-3553 137 24 r2π‘Ž1 r2π‘Ž1 PROPN iajs-3553 137 25 2𝑁3 2𝑁3 NUM iajs-3553 137 26 ( ( PUNCT iajs-3553 137 27 1+π‘Ž1𝑅1)3 1+π‘Ž1𝑅1)3 NUM iajs-3553 137 28 βˆ’ βˆ’ PROPN iajs-3553 137 29 π‘Ž2 π‘Ž2 NOUN iajs-3553 137 30 2𝑅1 2𝑅1 NUM iajs-3553 137 31 ( ( PUNCT iajs-3553 137 32 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 137 33 ) ) PUNCT iajs-3553 137 34 𝛽1 𝛽1 NOUN iajs-3553 137 35 ( ( PUNCT iajs-3553 137 36 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 137 37 ) ) PUNCT iajs-3553 137 38 ] ] PUNCT iajs-3553 137 39 , , PUNCT iajs-3553 137 40 π‘Š`21 π‘Š`21 PROPN iajs-3553 137 41 = = SYM iajs-3553 137 42 6𝑑2 6𝑑2 NUM iajs-3553 137 43 3 3 NUM iajs-3553 137 44 [ [ X iajs-3553 137 45 ( ( PUNCT iajs-3553 137 46 r2π‘Ž1 r2π‘Ž1 X iajs-3553 137 47 2𝑁3 2𝑁3 NUM iajs-3553 137 48 ( ( PUNCT iajs-3553 137 49 1+a1r1)3 1+a1r1)3 NUM iajs-3553 137 50 βˆ’ βˆ’ PROPN iajs-3553 137 51 π‘Ž1𝑁2 π‘Ž1𝑁2 PROPN iajs-3553 137 52 ( ( PUNCT iajs-3553 137 53 1+a1r1)2(1+π‘Ž2𝑅2 1+a1r1)2(1+π‘Ž2𝑅2 NOUN iajs-3553 137 54 ) ) PUNCT iajs-3553 137 55 βˆ’ βˆ’ NOUN iajs-3553 137 56 π‘Ž2𝑁 π‘Ž2𝑁 NUM iajs-3553 137 57 ( ( PUNCT iajs-3553 137 58 1+π‘Ž1𝑅1)(1+a2r2)2 1+π‘Ž1𝑅1)(1+a2r2)2 NUM iajs-3553 137 59 + + CCONJ iajs-3553 137 60 r1π‘Ž2 r1π‘Ž2 PROPN iajs-3553 137 61 2 2 NUM iajs-3553 137 62 ( ( PUNCT iajs-3553 137 63 1+a2r2)3 1+a2r2)3 NUM iajs-3553 137 64 ) ) PUNCT iajs-3553 137 65 𝑙1 𝑙1 NOUN iajs-3553 137 66 ( ( PUNCT iajs-3553 137 67 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 137 68 ) ) PUNCT iajs-3553 137 69 βˆ’ βˆ’ NOUN iajs-3553 137 70 Ξ²2r3π‘Ž3 Ξ²2r3π‘Ž3 PROPN iajs-3553 137 71 2 2 NUM iajs-3553 137 72 ( ( PUNCT iajs-3553 137 73 1+a3r2)4(1+a4r3 1+a3r2)4(1+a4r3 NUM iajs-3553 137 74 ) ) PUNCT iajs-3553 137 75 ] ] PUNCT iajs-3553 137 76 , , PUNCT iajs-3553 137 77 π‘Š`31= π‘Š`31= VERB iajs-3553 137 78 6𝑑2 6𝑑2 NUM iajs-3553 137 79 3𝑙2𝑅3π‘Ž3 3𝑙2𝑅3π‘Ž3 NUM iajs-3553 137 80 2 2 NUM iajs-3553 137 81 ( ( PUNCT iajs-3553 137 82 1+a3r2)4(1+a4r3 1+a3r2)4(1+a4r3 NUM iajs-3553 137 83 ) ) PUNCT iajs-3553 137 84 , , PUNCT iajs-3553 137 85 ( ( PUNCT iajs-3553 137 86 𝑀[1 𝑀[1 PROPN iajs-3553 137 87 ] ] PUNCT iajs-3553 137 88 ) ) PUNCT iajs-3553 137 89 𝑇 𝑇 PROPN iajs-3553 138 1 [ [ X iajs-3553 138 2 𝐷3πΉπœ‡(𝐴2 𝐷3πΉπœ‡(𝐴2 NOUN iajs-3553 138 3 . . PUNCT iajs-3553 139 1 𝑑`1)(𝑇 𝑑`1)(𝑇 X iajs-3553 140 1 [ [ X iajs-3553 140 2 1 1 NUM iajs-3553 140 3 ] ] PUNCT iajs-3553 140 4 . . PUNCT iajs-3553 141 1 𝑇[1 𝑇[1 PROPN iajs-3553 141 2 ] ] PUNCT iajs-3553 141 3 . . PUNCT iajs-3553 142 1 𝑇[1 𝑇[1 PROPN iajs-3553 142 2 ] ] PUNCT iajs-3553 142 3 ) ) PUNCT iajs-3553 142 4 ] ] PUNCT iajs-3553 143 1 = = SYM iajs-3553 143 2 6 6 NUM iajs-3553 143 3 ( ( PUNCT iajs-3553 143 4 𝑑2 𝑑2 NOUN iajs-3553 143 5 [ [ X iajs-3553 143 6 1 1 NUM iajs-3553 143 7 ] ] PUNCT iajs-3553 143 8 ) ) PUNCT iajs-3553 143 9 3 3 NUM iajs-3553 143 10 π‘š2 π‘š2 NOUN iajs-3553 143 11 [ [ X iajs-3553 143 12 1](𝑍`3 1](𝑍`3 NUM iajs-3553 143 13 βˆ’ βˆ’ PROPN iajs-3553 143 14 𝑍`4 𝑍`4 PROPN iajs-3553 143 15 ) ) PUNCT iajs-3553 143 16 β‰  β‰  PROPN iajs-3553 143 17 0 0 NUM iajs-3553 143 18 which which DET iajs-3553 143 19 grantee grantee NOUN iajs-3553 143 20 that that SCONJ iajs-3553 143 21 there there PRON iajs-3553 143 22 is be VERB iajs-3553 143 23 pitch pitch NOUN iajs-3553 143 24 fork fork NOUN iajs-3553 143 25 bifurcation bifurcation NOUN iajs-3553 143 26 at at ADP iajs-3553 143 27 𝐴1 𝐴1 PROPN iajs-3553 143 28 where where SCONJ iajs-3553 143 29 𝑑`1 𝑑`1 NOUN iajs-3553 143 30 = = PROPN iajs-3553 143 31 𝑑1 𝑑1 NOUN iajs-3553 143 32 . . PUNCT iajs-3553 144 1 ihjpas ihjpas PROPN iajs-3553 144 2 . . PUNCT iajs-3553 145 1 2025 2025 NUM iajs-3553 145 2 , , PUNCT iajs-3553 145 3 38(2 38(2 NUM iajs-3553 145 4 ) ) PUNCT iajs-3553 145 5 320 320 NUM iajs-3553 145 6 theorem theorem NOUN iajs-3553 145 7 ( ( PUNCT iajs-3553 145 8 2 2 NUM iajs-3553 145 9 ): ): PUNCT iajs-3553 145 10 system system NOUN iajs-3553 145 11 ( ( PUNCT iajs-3553 145 12 1 1 NUM iajs-3553 145 13 ) ) PUNCT iajs-3553 145 14 with with ADP iajs-3553 145 15 the the DET iajs-3553 145 16 parameter parameter NOUN iajs-3553 145 17 value value NOUN iajs-3553 145 18 οΏ½ οΏ½ PROPN iajs-3553 145 19 Μ… Μ… NOUN iajs-3553 145 20 οΏ½ οΏ½ NOUN iajs-3553 145 21 2 2 NUM iajs-3553 145 22 = = SYM iajs-3553 145 23 𝑑2 𝑑2 NOUN iajs-3553 145 24 = = SYM iajs-3553 145 25 𝑙2 𝑙2 PROPN iajs-3553 145 26 οΏ½ οΏ½ PROPN iajs-3553 145 27 Μ… Μ… NOUN iajs-3553 145 28 οΏ½ οΏ½ NOUN iajs-3553 145 29 2 2 NUM iajs-3553 145 30 1+π‘Ž3 1+π‘Ž3 NUM iajs-3553 145 31 οΏ½ οΏ½ NOUN iajs-3553 145 32 Μ… Μ… NOUN iajs-3553 145 33 οΏ½ οΏ½ NOUN iajs-3553 145 34 2 2 NUM iajs-3553 145 35 , , PUNCT iajs-3553 145 36 has have VERB iajs-3553 145 37 a a DET iajs-3553 145 38 transcritical transcritical ADJ iajs-3553 145 39 and and CCONJ iajs-3553 145 40 pitchfork pitchfork NOUN iajs-3553 145 41 bifurcation bifurcation NOUN iajs-3553 145 42 at at ADP iajs-3553 145 43 𝐴2 𝐴2 PROPN iajs-3553 145 44 = = SYM iajs-3553 145 45 ( ( PUNCT iajs-3553 145 46 0 0 NUM iajs-3553 145 47 . . X iajs-3553 145 48 οΏ½ οΏ½ PROPN iajs-3553 145 49 Μ… Μ… NOUN iajs-3553 145 50 οΏ½ οΏ½ NOUN iajs-3553 145 51 2 2 NUM iajs-3553 145 52 . . NOUN iajs-3553 145 53 0 0 NUM iajs-3553 145 54 ) ) PUNCT iajs-3553 145 55 . . PUNCT iajs-3553 146 1 if if SCONJ iajs-3553 146 2 the the DET iajs-3553 146 3 next next ADJ iajs-3553 146 4 conditions condition NOUN iajs-3553 146 5 hold hold VERB iajs-3553 146 6 : : PUNCT iajs-3553 146 7 zΜ…1 zΜ…1 X iajs-3553 146 8 β‰  β‰  PROPN iajs-3553 146 9 οΏ½ οΏ½ PROPN iajs-3553 146 10 Μ… Μ… NOUN iajs-3553 146 11 οΏ½ οΏ½ NOUN iajs-3553 146 12 2 2 NUM iajs-3553 146 13 . . PUNCT iajs-3553 147 1 ( ( PUNCT iajs-3553 147 2 9 9 NUM iajs-3553 147 3 ) ) PUNCT iajs-3553 147 4 zΜ…3 zΜ…3 NUM iajs-3553 147 5 β‰  β‰  PROPN iajs-3553 147 6 οΏ½ οΏ½ PROPN iajs-3553 147 7 Μ… Μ… NOUN iajs-3553 147 8 οΏ½ οΏ½ NOUN iajs-3553 147 9 4 4 NUM iajs-3553 147 10 . . PUNCT iajs-3553 148 1 ( ( PUNCT iajs-3553 148 2 10 10 NUM iajs-3553 148 3 ) ) PUNCT iajs-3553 148 4 where where SCONJ iajs-3553 148 5 : : PUNCT iajs-3553 148 6 zΜ…1 zΜ…1 X iajs-3553 148 7 = = SYM iajs-3553 148 8 𝑙2βˆ† 𝑙2βˆ† PROPN iajs-3553 148 9 ( ( PUNCT iajs-3553 148 10 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 NUM iajs-3553 148 11 , , PUNCT iajs-3553 148 12 οΏ½ οΏ½ NOUN iajs-3553 148 13 Μ… Μ… NOUN iajs-3553 148 14 οΏ½ οΏ½ NOUN iajs-3553 148 15 2 2 NUM iajs-3553 148 16 = = SYM iajs-3553 148 17 ( ( PUNCT iajs-3553 148 18 𝑙2𝑅3π‘Ž3βˆ†2 𝑙2𝑅3π‘Ž3βˆ†2 X iajs-3553 148 19 ( ( PUNCT iajs-3553 148 20 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 148 21 ) ) PUNCT iajs-3553 148 22 + + NUM iajs-3553 148 23 𝑙2𝑅2π‘Ž4 𝑙2𝑅2π‘Ž4 NOUN iajs-3553 148 24 ( ( PUNCT iajs-3553 148 25 1+a3r2)(1+a4r3)3 1+a3r2)(1+a4r3)3 NUM iajs-3553 148 26 ) ) PUNCT iajs-3553 148 27 , , PUNCT iajs-3553 148 28 οΏ½ οΏ½ PROPN iajs-3553 148 29 Μ… Μ… NOUN iajs-3553 148 30 οΏ½ οΏ½ NOUN iajs-3553 148 31 3 3 NUM iajs-3553 148 32 = = SYM iajs-3553 148 33 𝑙2 𝑙2 PROPN iajs-3553 148 34 ( ( PUNCT iajs-3553 148 35 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 148 36 ) ) PUNCT iajs-3553 148 37 ( ( PUNCT iajs-3553 148 38 𝑅3π‘Ž3 𝑅3π‘Ž3 NOUN iajs-3553 148 39 2βˆ†2 2βˆ†2 PROPN iajs-3553 148 40 ( ( PUNCT iajs-3553 148 41 1+a3r2)3 1+a3r2)3 NUM iajs-3553 148 42 + + NUM iajs-3553 148 43 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 148 44 2 2 NUM iajs-3553 148 45 ( ( PUNCT iajs-3553 148 46 1+a4r3)3 1+a4r3)3 NUM iajs-3553 148 47 ) ) PUNCT iajs-3553 148 48 , , PUNCT iajs-3553 148 49 οΏ½ οΏ½ PROPN iajs-3553 148 50 Μ… Μ… NOUN iajs-3553 148 51 οΏ½ οΏ½ NOUN iajs-3553 148 52 4 4 NUM iajs-3553 148 53 = = SYM iajs-3553 148 54 𝑙2βˆ† 𝑙2βˆ† X iajs-3553 148 55 ( ( PUNCT iajs-3553 148 56 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 PROPN iajs-3553 148 57 ( ( PUNCT iajs-3553 148 58 π‘Ž4 π‘Ž4 PROPN iajs-3553 148 59 ( ( PUNCT iajs-3553 148 60 1+π‘Ž4𝑅3 1+π‘Ž4𝑅3 PROPN iajs-3553 148 61 ) ) PUNCT iajs-3553 148 62 + + CCONJ iajs-3553 148 63 π‘Ž3βˆ† π‘Ž3βˆ† ADJ iajs-3553 148 64 ( ( PUNCT iajs-3553 148 65 1+a3r2 1+a3r2 NUM iajs-3553 148 66 ) ) PUNCT iajs-3553 148 67 ) ) PUNCT iajs-3553 148 68 . . PUNCT iajs-3553 149 1 proof proof NOUN iajs-3553 149 2 : : PUNCT iajs-3553 149 3 by by ADP iajs-3553 149 4 using use VERB iajs-3553 149 5 the the DET iajs-3553 149 6 jacobian jacobian ADJ iajs-3553 149 7 matrix matrix NOUN iajs-3553 149 8 in in ADP iajs-3553 149 9 equation equation NOUN iajs-3553 149 10 ( ( PUNCT iajs-3553 149 11 1.8 1.8 NUM iajs-3553 149 12 ) ) PUNCT iajs-3553 149 13 in in ADP iajs-3553 149 14 ( ( PUNCT iajs-3553 149 15 20 20 NUM iajs-3553 149 16 ) ) PUNCT iajs-3553 149 17 𝐽2Μ… 𝐽2Μ… NOUN iajs-3553 149 18 = = SYM iajs-3553 149 19 𝐽2(𝐴2 𝐽2(𝐴2 PROPN iajs-3553 149 20 . . PUNCT iajs-3553 149 21 οΏ½ οΏ½ PROPN iajs-3553 149 22 Μ… Μ… NOUN iajs-3553 149 23 οΏ½ οΏ½ NOUN iajs-3553 149 24 2 2 NUM iajs-3553 149 25 ) ) PUNCT iajs-3553 149 26 = = PUNCT iajs-3553 150 1 [ [ PUNCT iajs-3553 150 2 οΏ½ οΏ½ NOUN iajs-3553 150 3 Μ… Μ… NOUN iajs-3553 150 4 οΏ½ οΏ½ NOUN iajs-3553 150 5 𝑖𝑗]3Γ—3 𝑖𝑗]3Γ—3 NOUN iajs-3553 150 6 , , PUNCT iajs-3553 150 7 where where SCONJ iajs-3553 150 8 οΏ½ οΏ½ NOUN iajs-3553 150 9 Μ… Μ… VERB iajs-3553 150 10 οΏ½ οΏ½ NOUN iajs-3553 150 11 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 150 12 = = SYM iajs-3553 150 13 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN iajs-3553 150 14 , , PUNCT iajs-3553 150 15 except except SCONJ iajs-3553 150 16 οΏ½ οΏ½ NOUN iajs-3553 150 17 Μ… Μ… NOUN iajs-3553 150 18 οΏ½ οΏ½ NOUN iajs-3553 150 19 33 33 NUM iajs-3553 150 20 = = SYM iajs-3553 150 21 0.then 0.then NUM iajs-3553 150 22 , , PUNCT iajs-3553 150 23 the the DET iajs-3553 150 24 characterizing characterize VERB iajs-3553 150 25 equation equation NOUN iajs-3553 150 26 for𝐽2Μ… for𝐽2Μ… PROPN iajs-3553 150 27 has have VERB iajs-3553 150 28 an an DET iajs-3553 150 29 eigenvalue eigenvalue NOUN iajs-3553 150 30 of of ADP iajs-3553 150 31 zero zero NUM iajs-3553 150 32 , , PUNCT iajs-3553 150 33 which which PRON iajs-3553 150 34 is be AUX iajs-3553 150 35 Ξ»2r3 Ξ»2r3 PUNCT iajs-3553 150 36 at at ADP iajs-3553 150 37 οΏ½ οΏ½ PROPN iajs-3553 150 38 Μ… Μ… NOUN iajs-3553 150 39 οΏ½ οΏ½ NOUN iajs-3553 150 40 2 2 NUM iajs-3553 150 41 = = NOUN iajs-3553 150 42 𝑑2 𝑑2 NOUN iajs-3553 150 43 . . PUNCT iajs-3553 151 1 now now ADV iajs-3553 151 2 , , PUNCT iajs-3553 151 3 let let VERB iajs-3553 151 4 𝑇[2 𝑇[2 PROPN iajs-3553 151 5 ] ] X iajs-3553 151 6 = = PUNCT iajs-3553 151 7 ( ( PUNCT iajs-3553 151 8 𝑑1 𝑑1 NOUN iajs-3553 151 9 [ [ X iajs-3553 151 10 2 2 NUM iajs-3553 151 11 ] ] PUNCT iajs-3553 151 12 . . PUNCT iajs-3553 152 1 𝑑2 𝑑2 PROPN iajs-3553 152 2 [ [ X iajs-3553 152 3 2 2 NUM iajs-3553 152 4 ] ] PUNCT iajs-3553 152 5 . . PUNCT iajs-3553 153 1 𝑑3 𝑑3 PROPN iajs-3553 153 2 [ [ X iajs-3553 153 3 2 2 NUM iajs-3553 153 4 ] ] PUNCT iajs-3553 153 5 ) ) PUNCT iajs-3553 153 6 𝑇 𝑇 PROPN iajs-3553 153 7 be be VERB iajs-3553 153 8 the the DET iajs-3553 153 9 eigenvector eigenvector NOUN iajs-3553 153 10 associated associate VERB iajs-3553 153 11 with with ADP iajs-3553 153 12 an an DET iajs-3553 153 13 eigenvalue eigenvalue ADJ iajs-3553 153 14 Ξ»2r2 Ξ»2r2 NOUN iajs-3553 153 15 = = SYM iajs-3553 153 16 0 0 X iajs-3553 153 17 . . PUNCT iajs-3553 153 18 thus,(𝐽2Μ… thus,(𝐽2Μ… ADJ iajs-3553 154 1 βˆ’ βˆ’ NOUN iajs-3553 154 2 Ξ»2r2 Ξ»2r2 PUNCT iajs-3553 154 3 𝐼)𝑇[2 𝐼)𝑇[2 X iajs-3553 154 4 ] ] PUNCT iajs-3553 154 5 = = SYM iajs-3553 155 1 0 0 X iajs-3553 155 2 . . PUNCT iajs-3553 156 1 this this DET iajs-3553 156 2 gives:𝑑1 gives:𝑑1 NOUN iajs-3553 157 1 [ [ X iajs-3553 157 2 2 2 NUM iajs-3553 157 3 ] ] X iajs-3553 157 4 = = SYM iajs-3553 157 5 0 0 X iajs-3553 157 6 . . PUNCT iajs-3553 158 1 𝑑2 𝑑2 NOUN iajs-3553 158 2 [ [ X iajs-3553 158 3 2 2 NUM iajs-3553 158 4 ] ] PUNCT iajs-3553 158 5 = = PUNCT iajs-3553 158 6 βˆ†π‘‘3 βˆ†π‘‘3 PROPN iajs-3553 159 1 [ [ X iajs-3553 159 2 2 2 NUM iajs-3553 159 3 ] ] PUNCT iajs-3553 159 4 . . PUNCT iajs-3553 160 1 where where SCONJ iajs-3553 160 2 ∢ ∢ NOUN iajs-3553 160 3 βˆ†= βˆ†= VERB iajs-3553 160 4 βˆ’ βˆ’ PROPN iajs-3553 160 5 ( ( PUNCT iajs-3553 160 6 π‘Ÿ2𝑅2𝐾2 π‘Ÿ2𝑅2𝐾2 PROPN iajs-3553 160 7 π‘Ÿ2βˆ’π‘‘1 π‘Ÿ2βˆ’π‘‘1 NUM iajs-3553 161 1 + + NUM iajs-3553 161 2 𝛽2𝑅2 𝛽2𝑅2 NOUN iajs-3553 161 3 ( ( PUNCT iajs-3553 161 4 1+π‘Ž3𝑅2)(π‘Ÿ2βˆ’π‘‘1 1+π‘Ž3𝑅2)(π‘Ÿ2βˆ’π‘‘1 NUM iajs-3553 161 5 ) ) PUNCT iajs-3553 161 6 ) ) PUNCT iajs-3553 161 7 , , PUNCT iajs-3553 161 8 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-3553 161 9 𝑑3 𝑑3 PROPN iajs-3553 161 10 [ [ X iajs-3553 161 11 2 2 NUM iajs-3553 161 12 ] ] PUNCT iajs-3553 161 13 any any DET iajs-3553 161 14 real real ADJ iajs-3553 161 15 number number NOUN iajs-3553 161 16 that that PRON iajs-3553 161 17 is be AUX iajs-3553 161 18 not not PART iajs-3553 161 19 zero zero NUM iajs-3553 161 20 . . PUNCT iajs-3553 162 1 let let VERB iajs-3553 162 2 𝑀[2 𝑀[2 PRON iajs-3553 162 3 ] ] X iajs-3553 162 4 = = SYM iajs-3553 162 5 ( ( PUNCT iajs-3553 162 6 π‘š1 π‘š1 X iajs-3553 162 7 [ [ X iajs-3553 162 8 2 2 NUM iajs-3553 162 9 ] ] PUNCT iajs-3553 162 10 . . PUNCT iajs-3553 163 1 π‘š2 π‘š2 NOUN iajs-3553 164 1 [ [ X iajs-3553 164 2 2 2 NUM iajs-3553 164 3 ] ] PUNCT iajs-3553 164 4 . . PUNCT iajs-3553 165 1 π‘š3 π‘š3 NOUN iajs-3553 166 1 [ [ X iajs-3553 166 2 2 2 NUM iajs-3553 166 3 ] ] PUNCT iajs-3553 166 4 ) ) PUNCT iajs-3553 166 5 𝑇 𝑇 PROPN iajs-3553 166 6 be be VERB iajs-3553 166 7 the the DET iajs-3553 166 8 eigenvector eigenvector NOUN iajs-3553 166 9 associated associate VERB iajs-3553 166 10 with with ADP iajs-3553 166 11 an an DET iajs-3553 166 12 eigenvalue eigenvalue ADJ iajs-3553 166 13 Ξ»2r2 Ξ»2r2 NOUN iajs-3553 166 14 = = SYM iajs-3553 166 15 0 0 NUM iajs-3553 166 16 of of ADP iajs-3553 166 17 the the DET iajs-3553 166 18 matrix matrix NOUN iajs-3553 166 19 𝐽2Μ… 𝐽2Μ… NOUN iajs-3553 166 20 𝑇.then 𝑇.then ADV iajs-3553 166 21 . . PUNCT iajs-3553 167 1 ( ( PUNCT iajs-3553 167 2 𝐽2Μ… 𝐽2Μ… NOUN iajs-3553 167 3 𝑇 𝑇 PROPN iajs-3553 167 4 βˆ’ βˆ’ PROPN iajs-3553 167 5 Ξ»2r2 Ξ»2r2 NOUN iajs-3553 167 6 𝐼 𝐼 PROPN iajs-3553 167 7 ) ) PUNCT iajs-3553 167 8 𝑀[2 𝑀[2 PROPN iajs-3553 167 9 ] ] X iajs-3553 167 10 = = SYM iajs-3553 167 11 0 0 NUM iajs-3553 167 12 . . PUNCT iajs-3553 168 1 by by ADP iajs-3553 168 2 solving solve VERB iajs-3553 168 3 this this DET iajs-3553 168 4 equation equation NOUN iajs-3553 168 5 for for ADP iajs-3553 168 6 . . PUNCT iajs-3553 169 1 𝑀[2 𝑀[2 ADV iajs-3553 169 2 ] ] X iajs-3553 169 3 = = SYM iajs-3553 169 4 ( ( PUNCT iajs-3553 169 5 0 0 NUM iajs-3553 169 6 . . NOUN iajs-3553 169 7 0.π‘š3 0.π‘š3 NOUN iajs-3553 170 1 [ [ X iajs-3553 170 2 2 2 NUM iajs-3553 170 3 ] ] PUNCT iajs-3553 170 4 ) ) PUNCT iajs-3553 170 5 𝑇 𝑇 PROPN iajs-3553 170 6 . . PUNCT iajs-3553 171 1 and and CCONJ iajs-3553 171 2 π‘š3 π‘š3 NOUN iajs-3553 172 1 [ [ X iajs-3553 172 2 2 2 X iajs-3553 172 3 ] ] PUNCT iajs-3553 172 4 any any DET iajs-3553 172 5 real real ADJ iajs-3553 172 6 number number NOUN iajs-3553 172 7 that that PRON iajs-3553 172 8 is be AUX iajs-3553 172 9 not not PART iajs-3553 172 10 zero zero NUM iajs-3553 172 11 . . PUNCT iajs-3553 173 1 now now ADV iajs-3553 173 2 consider consider VERB iajs-3553 173 3 this this PRON iajs-3553 173 4 : : PUNCT iajs-3553 173 5 πœ•π‘“ πœ•π‘“ ADJ iajs-3553 173 6 πœ•π‘‘2 πœ•π‘‘2 NOUN iajs-3553 173 7 = = SYM iajs-3553 173 8 𝑓𝑑2 𝑓𝑑2 X iajs-3553 173 9 ( ( PUNCT iajs-3553 173 10 𝑋. 𝑋. PROPN iajs-3553 173 11 𝑑2 𝑑2 PROPN iajs-3553 173 12 ) ) PUNCT iajs-3553 173 13 = = SYM iajs-3553 173 14 ( ( PUNCT iajs-3553 173 15 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3553 173 16 πœ•π‘‘2 πœ•π‘‘2 PROPN iajs-3553 173 17 . . PUNCT iajs-3553 174 1 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3553 174 2 πœ•π‘‘2 πœ•π‘‘2 VERB iajs-3553 174 3 . . PUNCT iajs-3553 175 1 πœ•π‘“3 πœ•π‘“3 PROPN iajs-3553 175 2 πœ•π‘‘2 πœ•π‘‘2 PROPN iajs-3553 175 3 ) ) PUNCT iajs-3553 176 1 𝑇 𝑇 PROPN iajs-3553 176 2 = = SYM iajs-3553 176 3 ( ( PUNCT iajs-3553 176 4 0 0 NUM iajs-3553 176 5 . . NOUN iajs-3553 176 6 0 0 NUM iajs-3553 176 7 . . PUNCT iajs-3553 176 8 βˆ’π‘…3 βˆ’π‘…3 NOUN iajs-3553 176 9 ) ) PUNCT iajs-3553 176 10 𝑇. 𝑇. PROPN iajs-3553 177 1 so so ADV iajs-3553 177 2 , , PUNCT iajs-3553 177 3 𝑓𝑑2 𝑓𝑑2 PROPN iajs-3553 177 4 ( ( PUNCT iajs-3553 177 5 𝐴2 𝐴2 PROPN iajs-3553 177 6 . . PUNCT iajs-3553 177 7 οΏ½ οΏ½ PROPN iajs-3553 177 8 Μ… Μ… NOUN iajs-3553 177 9 οΏ½ οΏ½ NOUN iajs-3553 177 10 2 2 NUM iajs-3553 177 11 ) ) PUNCT iajs-3553 177 12 = = SYM iajs-3553 177 13 ( ( PUNCT iajs-3553 177 14 0 0 NUM iajs-3553 177 15 . . NOUN iajs-3553 177 16 0 0 NUM iajs-3553 177 17 . . X iajs-3553 177 18 0)𝑇 0)𝑇 NOUN iajs-3553 177 19 and and CCONJ iajs-3553 177 20 hence hence ADV iajs-3553 177 21 ( ( PUNCT iajs-3553 177 22 𝑀[2 𝑀[2 PROPN iajs-3553 177 23 ] ] X iajs-3553 177 24 ) ) PUNCT iajs-3553 177 25 𝑇 𝑇 PROPN iajs-3553 177 26 𝑓𝑑2 𝑓𝑑2 PROPN iajs-3553 177 27 ( ( PUNCT iajs-3553 177 28 𝐴2 𝐴2 PROPN iajs-3553 177 29 . . PUNCT iajs-3553 177 30 οΏ½ οΏ½ PROPN iajs-3553 177 31 Μ… Μ… NOUN iajs-3553 177 32 οΏ½ οΏ½ NOUN iajs-3553 177 33 2 2 NUM iajs-3553 177 34 ) ) PUNCT iajs-3553 177 35 = = SYM iajs-3553 177 36 0 0 X iajs-3553 177 37 . . X iajs-3553 178 1 using use VERB iajs-3553 178 2 sotomayor sotomayor NOUN iajs-3553 178 3 's 's PART iajs-3553 178 4 theorem theorem NOUN iajs-3553 178 5 , , PUNCT iajs-3553 178 6 it it PRON iajs-3553 178 7 is be AUX iajs-3553 178 8 impossible impossible ADJ iajs-3553 178 9 to to PART iajs-3553 178 10 satisfy satisfy VERB iajs-3553 178 11 the the DET iajs-3553 178 12 saddle saddle NOUN iajs-3553 178 13 - - PUNCT iajs-3553 178 14 node node NOUN iajs-3553 178 15 bifurcation bifurcation NOUN iajs-3553 178 16 condition condition NOUN iajs-3553 178 17 . . PUNCT iajs-3553 179 1 the the DET iajs-3553 179 2 first first ADJ iajs-3553 179 3 condition condition NOUN iajs-3553 179 4 for for ADP iajs-3553 179 5 transcritical transcritical ADJ iajs-3553 179 6 bifurcation bifurcation NOUN iajs-3553 179 7 is be AUX iajs-3553 179 8 therefore therefore ADV iajs-3553 179 9 satisfied satisfied ADJ iajs-3553 179 10 . . PUNCT iajs-3553 180 1 now now ADV iajs-3553 180 2 𝐷𝑓𝑑(𝑋. 𝐷𝑓𝑑(𝑋. NOUN iajs-3553 180 3 𝑑2 𝑑2 NOUN iajs-3553 180 4 ) ) PUNCT iajs-3553 180 5 = = PUNCT iajs-3553 181 1 [ [ PUNCT iajs-3553 181 2 0 0 NUM iajs-3553 181 3 0 0 NUM iajs-3553 181 4 0 0 NUM iajs-3553 181 5 0 0 NUM iajs-3553 181 6 0 0 NUM iajs-3553 181 7 0 0 NUM iajs-3553 181 8 0 0 NUM iajs-3553 181 9 0 0 NUM iajs-3553 181 10 βˆ’ βˆ’ NOUN iajs-3553 181 11 1 1 NUM iajs-3553 181 12 ] ] PUNCT iajs-3553 181 13 . . PUNCT iajs-3553 182 1 where where SCONJ iajs-3553 182 2 , , PUNCT iajs-3553 182 3 𝐷𝑓𝑑2 𝐷𝑓𝑑2 PROPN iajs-3553 182 4 ( ( PUNCT iajs-3553 182 5 𝑋. 𝑋. PROPN iajs-3553 182 6 𝑑2 𝑑2 NOUN iajs-3553 182 7 ) ) PUNCT iajs-3553 182 8 represents represent VERB iajs-3553 182 9 the the DET iajs-3553 182 10 derivative derivative NOUN iajs-3553 182 11 of of ADP iajs-3553 182 12 𝑓𝑑2 𝑓𝑑2 PROPN iajs-3553 182 13 ( ( PUNCT iajs-3553 182 14 𝑋. 𝑋. PROPN iajs-3553 182 15 𝑑2 𝑑2 PROPN iajs-3553 182 16 ) ) PUNCT iajs-3553 182 17 with with ADP iajs-3553 182 18 respect respect NOUN iajs-3553 182 19 to to ADP iajs-3553 182 20 𝑋 𝑋 PROPN iajs-3553 182 21 = = SYM iajs-3553 182 22 ( ( PUNCT iajs-3553 182 23 𝑅1 𝑅1 PROPN iajs-3553 182 24 . . PUNCT iajs-3553 182 25 𝑅2 𝑅2 PROPN iajs-3553 182 26 . . PUNCT iajs-3553 183 1 𝑅3 𝑅3 PROPN iajs-3553 183 2 ) ) PUNCT iajs-3553 183 3 𝑇 𝑇 PROPN iajs-3553 183 4 . . PUNCT iajs-3553 184 1 furthermore furthermore ADV iajs-3553 184 2 , , PUNCT iajs-3553 184 3 it it PRON iajs-3553 184 4 is be AUX iajs-3553 184 5 observed observe VERB iajs-3553 184 6 that that SCONJ iajs-3553 184 7 : : PUNCT iajs-3553 184 8 𝐷𝑓𝑑2 𝐷𝑓𝑑2 NOUN iajs-3553 184 9 ( ( PUNCT iajs-3553 184 10 𝐴2 𝐴2 PROPN iajs-3553 184 11 . . PUNCT iajs-3553 185 1 𝑑2)𝑇 𝑑2)𝑇 NOUN iajs-3553 186 1 [ [ X iajs-3553 186 2 2 2 X iajs-3553 186 3 ] ] X iajs-3553 186 4 = = PUNCT iajs-3553 186 5 [ [ PUNCT iajs-3553 186 6 0 0 NUM iajs-3553 186 7 0 0 NUM iajs-3553 186 8 0 0 NUM iajs-3553 186 9 0 0 NUM iajs-3553 186 10 0 0 NUM iajs-3553 186 11 0 0 NUM iajs-3553 186 12 0 0 NUM iajs-3553 186 13 0 0 NUM iajs-3553 186 14 βˆ’ βˆ’ NOUN iajs-3553 186 15 1 1 NUM iajs-3553 186 16 ] ] PUNCT iajs-3553 186 17 [ [ PUNCT iajs-3553 186 18 0 0 NUM iajs-3553 186 19 βˆ†π‘‘3 βˆ†π‘‘3 PROPN iajs-3553 187 1 [ [ X iajs-3553 187 2 2 2 NUM iajs-3553 187 3 ] ] X iajs-3553 187 4 𝑑3 𝑑3 NOUN iajs-3553 187 5 [ [ X iajs-3553 187 6 2 2 NUM iajs-3553 187 7 ] ] PUNCT iajs-3553 187 8 ] ] PUNCT iajs-3553 188 1 = = PUNCT iajs-3553 188 2 [ [ PUNCT iajs-3553 188 3 0 0 NUM iajs-3553 188 4 0 0 NUM iajs-3553 188 5 βˆ’π‘‘3 βˆ’π‘‘3 PROPN iajs-3553 189 1 [ [ X iajs-3553 189 2 2 2 NUM iajs-3553 189 3 ] ] PUNCT iajs-3553 189 4 ] ] X iajs-3553 189 5 ( ( PUNCT iajs-3553 189 6 𝑀[2 𝑀[2 PROPN iajs-3553 189 7 ] ] X iajs-3553 189 8 ) ) PUNCT iajs-3553 189 9 𝑇 𝑇 PROPN iajs-3553 190 1 [ [ X iajs-3553 190 2 𝐷𝑓𝑑2 𝐷𝑓𝑑2 X iajs-3553 190 3 ( ( PUNCT iajs-3553 190 4 𝐴2 𝐴2 PROPN iajs-3553 190 5 . . PUNCT iajs-3553 190 6 οΏ½ οΏ½ PROPN iajs-3553 190 7 Μ… Μ… NOUN iajs-3553 190 8 οΏ½ οΏ½ NOUN iajs-3553 190 9 2)𝑇 2)𝑇 NOUN iajs-3553 191 1 [ [ X iajs-3553 191 2 2 2 NUM iajs-3553 191 3 ] ] X iajs-3553 191 4 ] ] X iajs-3553 191 5 = = SYM iajs-3553 191 6 ( ( PUNCT iajs-3553 191 7 0 0 NUM iajs-3553 191 8 . . NOUN iajs-3553 191 9 0.π‘š3 0.π‘š3 NOUN iajs-3553 192 1 [ [ X iajs-3553 192 2 2 2 NUM iajs-3553 192 3 ] ] PUNCT iajs-3553 192 4 ) ) PUNCT iajs-3553 192 5 ( ( PUNCT iajs-3553 192 6 0 0 NUM iajs-3553 192 7 . . NOUN iajs-3553 192 8 0 0 NUM iajs-3553 192 9 . . PUNCT iajs-3553 193 1 βˆ’π‘‘3 βˆ’π‘‘3 PROPN iajs-3553 194 1 [ [ X iajs-3553 194 2 2 2 NUM iajs-3553 194 3 ] ] PUNCT iajs-3553 194 4 ) ) PUNCT iajs-3553 194 5 𝑇 𝑇 PROPN iajs-3553 194 6 = = SYM iajs-3553 194 7 βˆ’π‘š3 βˆ’π‘š3 PROPN iajs-3553 195 1 [ [ X iajs-3553 195 2 2 2 NUM iajs-3553 195 3 ] ] X iajs-3553 195 4 𝑑3 𝑑3 NOUN iajs-3553 195 5 [ [ X iajs-3553 195 6 2 2 NUM iajs-3553 195 7 ] ] PUNCT iajs-3553 195 8 β‰  β‰  X iajs-3553 195 9 0 0 NUM iajs-3553 195 10 by by ADP iajs-3553 195 11 substituting substitute VERB iajs-3553 195 12 𝑇[2]𝑖𝑛 𝑇[2]𝑖𝑛 PROPN iajs-3553 195 13 equation equation NOUN iajs-3553 195 14 ( ( PUNCT iajs-3553 195 15 3 3 X iajs-3553 195 16 ) ) PUNCT iajs-3553 195 17 we we PRON iajs-3553 195 18 get get VERB iajs-3553 195 19 : : PUNCT iajs-3553 195 20 𝐷2πΉπœ‡(𝐴2 𝐷2πΉπœ‡(𝐴2 PROPN iajs-3553 195 21 . . PUNCT iajs-3553 196 1 οΏ½ οΏ½ PROPN iajs-3553 196 2 Μ… Μ… NOUN iajs-3553 196 3 οΏ½ οΏ½ NOUN iajs-3553 196 4 2)(𝑇 2)(𝑇 PRON iajs-3553 196 5 [ [ X iajs-3553 196 6 2 2 NUM iajs-3553 196 7 ] ] PUNCT iajs-3553 196 8 . . PUNCT iajs-3553 197 1 𝑇[2 𝑇[2 NOUN iajs-3553 197 2 ] ] X iajs-3553 197 3 ) ) PUNCT iajs-3553 197 4 = = SYM iajs-3553 197 5 ( ( PUNCT iajs-3553 197 6 οΏ½ οΏ½ NOUN iajs-3553 197 7 Μ… Μ… NOUN iajs-3553 197 8 οΏ½ οΏ½ NOUN iajs-3553 197 9 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 197 10 ) ) PUNCT iajs-3553 197 11 . . PUNCT iajs-3553 198 1 ( ( PUNCT iajs-3553 198 2 11 11 NUM iajs-3553 198 3 ) ) PUNCT iajs-3553 198 4 οΏ½ οΏ½ NOUN iajs-3553 198 5 Μ… Μ… NOUN iajs-3553 198 6 οΏ½ οΏ½ NOUN iajs-3553 198 7 11 11 NUM iajs-3553 198 8 = = SYM iajs-3553 198 9 2βˆ†2𝑅1𝑑3 2βˆ†2𝑅1𝑑3 PROPN iajs-3553 199 1 [ [ X iajs-3553 199 2 2 2 NUM iajs-3553 199 3 ] ] X iajs-3553 199 4 ( ( PUNCT iajs-3553 199 5 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 199 6 2 2 NUM iajs-3553 199 7 ( ( PUNCT iajs-3553 199 8 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 199 9 + + CCONJ iajs-3553 199 10 𝛽1π‘Ž2 𝛽1π‘Ž2 X iajs-3553 199 11 ( ( PUNCT iajs-3553 199 12 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 PROPN iajs-3553 199 13 ) ) PUNCT iajs-3553 199 14 , , PUNCT iajs-3553 199 15 οΏ½ οΏ½ PROPN iajs-3553 199 16 Μ… Μ… NOUN iajs-3553 199 17 οΏ½ οΏ½ NOUN iajs-3553 199 18 21 21 NUM iajs-3553 199 19 = = SYM iajs-3553 199 20 2 2 NUM iajs-3553 199 21 𝑑3 𝑑3 PROPN iajs-3553 199 22 [ [ X iajs-3553 199 23 2 2 NUM iajs-3553 199 24 ] ] PUNCT iajs-3553 199 25 [ [ X iajs-3553 199 26 ( ( PUNCT iajs-3553 199 27 r3π‘Ž3βˆ†2 r3π‘Ž3βˆ†2 X iajs-3553 199 28 ( ( PUNCT iajs-3553 199 29 1+a3r2)2 1+a3r2)2 NUM iajs-3553 199 30 + + CCONJ iajs-3553 199 31 a4𝑅2 a4𝑅2 PRON iajs-3553 199 32 ( ( PUNCT iajs-3553 199 33 1+a4r3)2 1+a4r3)2 NUM iajs-3553 199 34 βˆ’ βˆ’ NOUN iajs-3553 199 35 βˆ† βˆ† PROPN iajs-3553 199 36 ( ( PUNCT iajs-3553 199 37 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 199 38 ) ) PUNCT iajs-3553 199 39 ) ) PUNCT iajs-3553 199 40 𝛽2 𝛽2 NOUN iajs-3553 199 41 ( ( PUNCT iajs-3553 199 42 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 199 43 ) ) PUNCT iajs-3553 199 44 βˆ’ βˆ’ PROPN iajs-3553 200 1 𝑙1r1π‘Ž2βˆ†2 𝑙1r1π‘Ž2βˆ†2 PROPN iajs-3553 200 2 ( ( PUNCT iajs-3553 200 3 1+a1r1)(1+a2r2)3 1+a1r1)(1+a2r2)3 NUM iajs-3553 200 4 + + CCONJ iajs-3553 200 5 ( ( PUNCT iajs-3553 200 6 βˆ† βˆ† PROPN iajs-3553 200 7 βˆ’ βˆ’ PROPN iajs-3553 200 8 𝐾2𝑅2 𝐾2𝑅2 NOUN iajs-3553 200 9 ( ( PUNCT iajs-3553 200 10 1+k2r3 1+k2r3 NUM iajs-3553 200 11 ) ) PUNCT iajs-3553 200 12 ) ) PUNCT iajs-3553 200 13 r2k2 r2k2 X iajs-3553 200 14 ( ( PUNCT iajs-3553 200 15 1+k2r3)2 1+k2r3)2 NOUN iajs-3553 200 16 βˆ’ βˆ’ NOUN iajs-3553 200 17 π‘š2βˆ† π‘š2βˆ† ADJ iajs-3553 200 18 2 2 NUM iajs-3553 200 19 ] ] PUNCT iajs-3553 200 20 , , PUNCT iajs-3553 200 21 οΏ½ οΏ½ NOUN iajs-3553 200 22 Μ… Μ… NOUN iajs-3553 200 23 οΏ½ οΏ½ NOUN iajs-3553 200 24 31 31 NUM iajs-3553 200 25 = = SYM iajs-3553 200 26 2𝑑3 2𝑑3 NUM iajs-3553 201 1 [ [ X iajs-3553 201 2 2 2 NUM iajs-3553 201 3 ] ] X iajs-3553 201 4 𝑙2 𝑙2 NOUN iajs-3553 201 5 ( ( PUNCT iajs-3553 201 6 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 201 7 ) ) PUNCT iajs-3553 201 8 ( ( PUNCT iajs-3553 201 9 1 1 NUM iajs-3553 201 10 ( ( PUNCT iajs-3553 201 11 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 201 12 ) ) PUNCT iajs-3553 201 13 βˆ’ βˆ’ PROPN iajs-3553 201 14 𝑅3π‘Ž3βˆ†2 𝑅3π‘Ž3βˆ†2 NOUN iajs-3553 201 15 ( ( PUNCT iajs-3553 201 16 1+a3r2)2 1+a3r2)2 NUM iajs-3553 201 17 βˆ’ βˆ’ NOUN iajs-3553 201 18 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 201 19 ( ( PUNCT iajs-3553 201 20 1+a4r3)2 1+a4r3)2 NUM iajs-3553 201 21 ) ) PUNCT iajs-3553 201 22 , , PUNCT iajs-3553 201 23 hence hence ADV iajs-3553 201 24 , , PUNCT iajs-3553 201 25 it it PRON iajs-3553 201 26 was be AUX iajs-3553 201 27 obtained obtain VERB iajs-3553 201 28 . . PUNCT iajs-3553 202 1 ihjpas ihjpas PROPN iajs-3553 202 2 . . PUNCT iajs-3553 203 1 2025 2025 NUM iajs-3553 203 2 , , PUNCT iajs-3553 203 3 38(2 38(2 NUM iajs-3553 203 4 ) ) PUNCT iajs-3553 203 5 321 321 NUM iajs-3553 203 6 ( ( PUNCT iajs-3553 203 7 𝑀[2 𝑀[2 PROPN iajs-3553 203 8 ] ] X iajs-3553 203 9 ) ) PUNCT iajs-3553 203 10 𝑇 𝑇 PROPN iajs-3553 204 1 [ [ X iajs-3553 204 2 𝐷2πΉπœ‡(𝐴2 𝐷2πΉπœ‡(𝐴2 PROPN iajs-3553 204 3 . . PUNCT iajs-3553 205 1 οΏ½ οΏ½ PROPN iajs-3553 205 2 Μ… Μ… NOUN iajs-3553 205 3 οΏ½ οΏ½ NOUN iajs-3553 205 4 2)(𝑇 2)(𝑇 PRON iajs-3553 205 5 [ [ X iajs-3553 205 6 2 2 NUM iajs-3553 205 7 ] ] PUNCT iajs-3553 205 8 . . PUNCT iajs-3553 206 1 𝑇[2 𝑇[2 NOUN iajs-3553 206 2 ] ] PUNCT iajs-3553 206 3 ) ) PUNCT iajs-3553 206 4 ] ] PUNCT iajs-3553 207 1 = = SYM iajs-3553 207 2 2 2 X iajs-3553 207 3 ( ( PUNCT iajs-3553 207 4 𝑑3 𝑑3 PROPN iajs-3553 207 5 [ [ X iajs-3553 207 6 2 2 NUM iajs-3553 207 7 ] ] PUNCT iajs-3553 207 8 ) ) PUNCT iajs-3553 207 9 2 2 NUM iajs-3553 207 10 π‘š3 π‘š3 NOUN iajs-3553 207 11 [ [ X iajs-3553 207 12 2 2 NUM iajs-3553 207 13 ] ] PUNCT iajs-3553 207 14 ( ( PUNCT iajs-3553 207 15 zΜ…1 zΜ…1 NOUN iajs-3553 207 16 βˆ’ βˆ’ NOUN iajs-3553 207 17 οΏ½ οΏ½ PROPN iajs-3553 207 18 Μ… Μ… NOUN iajs-3553 207 19 οΏ½ οΏ½ NOUN iajs-3553 207 20 2 2 NUM iajs-3553 207 21 ) ) PUNCT iajs-3553 207 22 β‰  β‰  PROPN iajs-3553 207 23 0 0 X iajs-3553 207 24 . . PUNCT iajs-3553 208 1 this this PRON iajs-3553 208 2 indicates indicate VERB iajs-3553 208 3 that that SCONJ iajs-3553 208 4 the the DET iajs-3553 208 5 system system NOUN iajs-3553 208 6 ( ( PUNCT iajs-3553 208 7 1 1 X iajs-3553 208 8 ) ) PUNCT iajs-3553 208 9 exhibits exhibit VERB iajs-3553 208 10 a a DET iajs-3553 208 11 transcritical transcritical ADJ iajs-3553 208 12 bifurcation bifurcation NOUN iajs-3553 208 13 at at ADP iajs-3553 208 14 𝐴2 𝐴2 NOUN iajs-3553 208 15 with with ADP iajs-3553 208 16 a a DET iajs-3553 208 17 parameter parameter NOUN iajs-3553 208 18 οΏ½ οΏ½ NOUN iajs-3553 208 19 Μ… Μ… NOUN iajs-3553 208 20 οΏ½ οΏ½ NOUN iajs-3553 208 21 2 2 NUM iajs-3553 208 22 = = SYM iajs-3553 208 23 𝑑2 𝑑2 NOUN iajs-3553 208 24 . . PUNCT iajs-3553 209 1 if if SCONJ iajs-3553 209 2 condition condition NOUN iajs-3553 209 3 ( ( PUNCT iajs-3553 209 4 9 9 NUM iajs-3553 209 5 ) ) PUNCT iajs-3553 209 6 not not PART iajs-3553 209 7 satisfied satisfied ADJ iajs-3553 209 8 , , PUNCT iajs-3553 209 9 then then ADV iajs-3553 209 10 . . PUNCT iajs-3553 210 1 by by ADP iajs-3553 210 2 substituting substitute VERB iajs-3553 210 3 𝑇[2 𝑇[2 PROPN iajs-3553 210 4 ] ] PUNCT iajs-3553 210 5 in in ADP iajs-3553 210 6 equation equation NOUN iajs-3553 210 7 ( ( PUNCT iajs-3553 210 8 4 4 X iajs-3553 210 9 ) ) PUNCT iajs-3553 210 10 we we PRON iajs-3553 210 11 obtain obtain VERB iajs-3553 210 12 : : PUNCT iajs-3553 210 13 𝐷3πΉπœ‡(𝐴2 𝐷3πΉπœ‡(𝐴2 NOUN iajs-3553 210 14 . . PUNCT iajs-3553 210 15 οΏ½ οΏ½ PROPN iajs-3553 210 16 Μ… Μ… NOUN iajs-3553 210 17 οΏ½ οΏ½ NOUN iajs-3553 210 18 2)(𝑇 2)(𝑇 PRON iajs-3553 210 19 [ [ X iajs-3553 210 20 2 2 NUM iajs-3553 210 21 ] ] PUNCT iajs-3553 210 22 . . PUNCT iajs-3553 211 1 𝑇[2 𝑇[2 NOUN iajs-3553 211 2 ] ] X iajs-3553 211 3 . . PUNCT iajs-3553 212 1 𝑇[2 𝑇[2 NOUN iajs-3553 212 2 ] ] X iajs-3553 212 3 ) ) PUNCT iajs-3553 212 4 = = SYM iajs-3553 212 5 ( ( PUNCT iajs-3553 212 6 οΏ½ οΏ½ NOUN iajs-3553 212 7 Μ… Μ… NOUN iajs-3553 212 8 οΏ½ οΏ½ NOUN iajs-3553 212 9 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 212 10 ) ) PUNCT iajs-3553 212 11 . . PUNCT iajs-3553 213 1 ( ( PUNCT iajs-3553 213 2 12 12 NUM iajs-3553 213 3 ) ) PUNCT iajs-3553 213 4 οΏ½ οΏ½ NOUN iajs-3553 213 5 Μ… Μ… NOUN iajs-3553 213 6 οΏ½ οΏ½ NOUN iajs-3553 213 7 11 11 NUM iajs-3553 213 8 = = SYM iajs-3553 213 9 βˆ’6βˆ†2 βˆ’6βˆ†2 NOUN iajs-3553 213 10 ( ( PUNCT iajs-3553 213 11 𝑑3 𝑑3 PROPN iajs-3553 213 12 [ [ X iajs-3553 213 13 2 2 NUM iajs-3553 213 14 ] ] PUNCT iajs-3553 213 15 ) ) PUNCT iajs-3553 213 16 3 3 NUM iajs-3553 213 17 ( ( PUNCT iajs-3553 213 18 𝛽1π‘Ž2 𝛽1π‘Ž2 PROPN iajs-3553 213 19 2𝑅1 2𝑅1 NUM iajs-3553 213 20 ( ( PUNCT iajs-3553 213 21 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 NUM iajs-3553 213 22 + + SYM iajs-3553 213 23 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 213 24 3𝑅1 3𝑅1 NUM iajs-3553 213 25 ( ( PUNCT iajs-3553 213 26 1+𝐾1𝑅2)4 1+𝐾1𝑅2)4 NUM iajs-3553 213 27 ) ) PUNCT iajs-3553 213 28 , , PUNCT iajs-3553 213 29 οΏ½ οΏ½ PROPN iajs-3553 213 30 Μ… Μ… NOUN iajs-3553 213 31 οΏ½ οΏ½ NOUN iajs-3553 213 32 21 21 NUM iajs-3553 213 33 = = SYM iajs-3553 213 34 6(𝑑3 6(𝑑3 PROPN iajs-3553 214 1 [ [ X iajs-3553 214 2 2 2 NUM iajs-3553 214 3 ] ] PUNCT iajs-3553 214 4 ) ) PUNCT iajs-3553 214 5 3 3 NUM iajs-3553 215 1 [ [ X iajs-3553 215 2 ( ( PUNCT iajs-3553 215 3 βˆ† βˆ† PROPN iajs-3553 215 4 βˆ’ βˆ’ PROPN iajs-3553 215 5 k2𝑅2 k2𝑅2 PROPN iajs-3553 215 6 ( ( PUNCT iajs-3553 215 7 1+k2r3 1+k2r3 NUM iajs-3553 215 8 ) ) PUNCT iajs-3553 215 9 ) ) PUNCT iajs-3553 215 10 r2k2 r2k2 X iajs-3553 215 11 2 2 NUM iajs-3553 215 12 ( ( PUNCT iajs-3553 215 13 1+k2r3)3 1+k2r3)3 NUM iajs-3553 215 14 + + CCONJ iajs-3553 215 15 ( ( PUNCT iajs-3553 215 16 𝑙1βˆ†3 𝑙1βˆ†3 NOUN iajs-3553 215 17 ( ( PUNCT iajs-3553 215 18 1+a2r2)3 1+a2r2)3 NUM iajs-3553 215 19 βˆ’ βˆ’ PROPN iajs-3553 215 20 𝛽1 𝛽1 NOUN iajs-3553 215 21 ( ( PUNCT iajs-3553 215 22 1+π‘Ž2𝑅2)3 1+π‘Ž2𝑅2)3 NUM iajs-3553 215 23 ) ) PUNCT iajs-3553 215 24 π‘Ž2 π‘Ž2 NOUN iajs-3553 215 25 2𝑅1 2𝑅1 NUM iajs-3553 215 26 ( ( PUNCT iajs-3553 215 27 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 215 28 ) ) PUNCT iajs-3553 216 1 + + CCONJ iajs-3553 216 2 ( ( PUNCT iajs-3553 216 3 π‘Ž4βˆ† π‘Ž4βˆ† X iajs-3553 216 4 ( ( PUNCT iajs-3553 216 5 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3 NOUN iajs-3553 216 6 ) ) PUNCT iajs-3553 216 7 + + NUM iajs-3553 216 8 a4 a4 NOUN iajs-3553 216 9 ( ( PUNCT iajs-3553 216 10 1+a4r3)2 1+a4r3)2 ADV iajs-3553 216 11 βˆ’ βˆ’ NOUN iajs-3553 216 12 r3π‘Ž3 r3π‘Ž3 NOUN iajs-3553 216 13 2βˆ†2 2βˆ†2 NUM iajs-3553 216 14 ( ( PUNCT iajs-3553 216 15 1+a3r2)2 1+a3r2)2 NUM iajs-3553 216 16 ) ) PUNCT iajs-3553 216 17 𝛽2βˆ† 𝛽2βˆ† PROPN iajs-3553 216 18 ( ( PUNCT iajs-3553 216 19 1+π‘Ž3𝑅2)2(1+π‘Ž4𝑅3 1+π‘Ž3𝑅2)2(1+π‘Ž4𝑅3 NUM iajs-3553 216 20 ) ) PUNCT iajs-3553 216 21 ] ] PUNCT iajs-3553 216 22 , , PUNCT iajs-3553 216 23 οΏ½ οΏ½ PROPN iajs-3553 216 24 Μ… Μ… NOUN iajs-3553 216 25 οΏ½ οΏ½ NOUN iajs-3553 216 26 31 31 NUM iajs-3553 216 27 = = SYM iajs-3553 216 28 6(𝑑3 6(𝑑3 PROPN iajs-3553 217 1 [ [ X iajs-3553 217 2 2 2 NUM iajs-3553 217 3 ] ] PUNCT iajs-3553 217 4 ) ) PUNCT iajs-3553 217 5 3 3 NUM iajs-3553 217 6 𝑙2 𝑙2 NOUN iajs-3553 217 7 ( ( PUNCT iajs-3553 217 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 217 9 ) ) PUNCT iajs-3553 217 10 ( ( PUNCT iajs-3553 217 11 𝑅3π‘Ž3 𝑅3π‘Ž3 NOUN iajs-3553 217 12 2βˆ†2 2βˆ†2 PROPN iajs-3553 217 13 ( ( PUNCT iajs-3553 217 14 1+a3r2)3 1+a3r2)3 NUM iajs-3553 217 15 + + NUM iajs-3553 217 16 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 217 17 2 2 NUM iajs-3553 217 18 ( ( PUNCT iajs-3553 217 19 1+a4r3)3 1+a4r3)3 NUM iajs-3553 217 20 βˆ’ βˆ’ PROPN iajs-3553 217 21 π‘Ž4βˆ† π‘Ž4βˆ† X iajs-3553 217 22 ( ( PUNCT iajs-3553 217 23 1+a3r2)(1+a4r3)2 1+a3r2)(1+a4r3)2 NUM iajs-3553 217 24 βˆ’ βˆ’ PROPN iajs-3553 217 25 π‘Ž3βˆ†2 π‘Ž3βˆ†2 PROPN iajs-3553 217 26 ( ( PUNCT iajs-3553 217 27 1+a3r2)2(1+a4r3 1+a3r2)2(1+a4r3 NUM iajs-3553 217 28 ) ) PUNCT iajs-3553 217 29 ) ) PUNCT iajs-3553 217 30 . . PUNCT iajs-3553 218 1 ( ( PUNCT iajs-3553 218 2 𝑀[2 𝑀[2 PROPN iajs-3553 218 3 ] ] X iajs-3553 218 4 ) ) PUNCT iajs-3553 218 5 𝑇 𝑇 PROPN iajs-3553 219 1 [ [ X iajs-3553 219 2 𝐷3πΉπœ‡(𝐴2 𝐷3πΉπœ‡(𝐴2 PROPN iajs-3553 219 3 . . PUNCT iajs-3553 220 1 οΏ½ οΏ½ PROPN iajs-3553 220 2 Μ… Μ… NOUN iajs-3553 220 3 οΏ½ οΏ½ NOUN iajs-3553 220 4 2)(𝑇 2)(𝑇 PRON iajs-3553 220 5 [ [ X iajs-3553 220 6 2 2 NUM iajs-3553 220 7 ] ] PUNCT iajs-3553 220 8 . . PUNCT iajs-3553 221 1 𝑇[2 𝑇[2 NOUN iajs-3553 221 2 ] ] X iajs-3553 221 3 . . PUNCT iajs-3553 222 1 𝑇[2 𝑇[2 NOUN iajs-3553 222 2 ] ] PUNCT iajs-3553 222 3 ) ) PUNCT iajs-3553 222 4 ] ] PUNCT iajs-3553 223 1 = = SYM iajs-3553 223 2 6 6 NUM iajs-3553 223 3 ( ( PUNCT iajs-3553 223 4 𝑑3 𝑑3 PROPN iajs-3553 223 5 [ [ X iajs-3553 223 6 2 2 NUM iajs-3553 223 7 ] ] PUNCT iajs-3553 223 8 ) ) PUNCT iajs-3553 223 9 3 3 NUM iajs-3553 223 10 π‘š3 π‘š3 NOUN iajs-3553 223 11 [ [ X iajs-3553 223 12 2 2 NUM iajs-3553 223 13 ] ] X iajs-3553 223 14 ( ( PUNCT iajs-3553 223 15 οΏ½ οΏ½ NOUN iajs-3553 223 16 Μ… Μ… NOUN iajs-3553 223 17 οΏ½ οΏ½ NOUN iajs-3553 223 18 3 3 NUM iajs-3553 223 19 βˆ’ βˆ’ PROPN iajs-3553 223 20 οΏ½ οΏ½ PROPN iajs-3553 223 21 Μ… Μ… NOUN iajs-3553 223 22 οΏ½ οΏ½ NOUN iajs-3553 223 23 4 4 NUM iajs-3553 223 24 ) ) PUNCT iajs-3553 223 25 β‰  β‰  PROPN iajs-3553 223 26 0 0 NUM iajs-3553 223 27 which which DET iajs-3553 223 28 grantee grantee NOUN iajs-3553 223 29 that that SCONJ iajs-3553 223 30 there there PRON iajs-3553 223 31 is be VERB iajs-3553 223 32 pitch pitch NOUN iajs-3553 223 33 fork fork NOUN iajs-3553 223 34 bifurcation bifurcation NOUN iajs-3553 223 35 at at ADP iajs-3553 223 36 𝐴2 𝐴2 PROPN iajs-3553 223 37 where where SCONJ iajs-3553 223 38 οΏ½ οΏ½ NOUN iajs-3553 223 39 Μ… Μ… NOUN iajs-3553 223 40 οΏ½ οΏ½ NOUN iajs-3553 223 41 2 2 NUM iajs-3553 223 42 = = SYM iajs-3553 223 43 𝑑2 𝑑2 NOUN iajs-3553 223 44 . . PUNCT iajs-3553 224 1 theorem theorem NOUN iajs-3553 224 2 ( ( PUNCT iajs-3553 224 3 3 3 NUM iajs-3553 224 4 ): ): PUNCT iajs-3553 224 5 system system NOUN iajs-3553 224 6 ( ( PUNCT iajs-3553 224 7 1 1 NUM iajs-3553 224 8 ) ) PUNCT iajs-3553 224 9 with with ADP iajs-3553 224 10 the the DET iajs-3553 224 11 parameter parameter NOUN iajs-3553 224 12 value value NOUN iajs-3553 224 13 οΏ½ οΏ½ PROPN iajs-3553 224 14 Μ‚ Μ‚ VERB iajs-3553 224 15 οΏ½ οΏ½ NOUN iajs-3553 224 16 2 2 NUM iajs-3553 224 17 = = SYM iajs-3553 224 18 𝑑2 𝑑2 NOUN iajs-3553 224 19 = = SYM iajs-3553 224 20 𝑙2 𝑙2 PROPN iajs-3553 224 21 οΏ½ οΏ½ PROPN iajs-3553 224 22 Μ‚ Μ‚ VERB iajs-3553 224 23 οΏ½ οΏ½ NOUN iajs-3553 224 24 2 2 NUM iajs-3553 224 25 1+π‘Ž3 1+π‘Ž3 NOUN iajs-3553 224 26 οΏ½ οΏ½ NOUN iajs-3553 224 27 Μ‚ Μ‚ NOUN iajs-3553 224 28 οΏ½ οΏ½ NOUN iajs-3553 224 29 2 2 NUM iajs-3553 224 30 , , PUNCT iajs-3553 224 31 has have VERB iajs-3553 224 32 a a DET iajs-3553 224 33 transcritical transcritical ADJ iajs-3553 224 34 and and CCONJ iajs-3553 224 35 pitchfork pitchfork NOUN iajs-3553 224 36 bifurcation bifurcation NOUN iajs-3553 224 37 at at ADP iajs-3553 224 38 𝐴3 𝐴3 PROPN iajs-3553 224 39 = = SYM iajs-3553 224 40 ( ( PUNCT iajs-3553 224 41 οΏ½ οΏ½ PROPN iajs-3553 224 42 Μ‚ Μ‚ VERB iajs-3553 224 43 οΏ½ οΏ½ NOUN iajs-3553 224 44 1 1 NUM iajs-3553 224 45 . . PUNCT iajs-3553 224 46 οΏ½ οΏ½ PROPN iajs-3553 224 47 Μ‚ Μ‚ VERB iajs-3553 224 48 οΏ½ οΏ½ NOUN iajs-3553 224 49 2 2 NUM iajs-3553 224 50 . . NOUN iajs-3553 224 51 0 0 NUM iajs-3553 224 52 ) ) PUNCT iajs-3553 224 53 , , PUNCT iajs-3553 224 54 if if SCONJ iajs-3553 224 55 the the DET iajs-3553 224 56 next next ADJ iajs-3553 224 57 conditions condition NOUN iajs-3553 224 58 hold hold VERB iajs-3553 224 59 : : PUNCT iajs-3553 224 60 οΏ½ οΏ½ PROPN iajs-3553 224 61 Μ‚ Μ‚ VERB iajs-3553 224 62 οΏ½ οΏ½ PROPN iajs-3553 224 63 1 1 NUM iajs-3553 224 64 β‰  β‰  PROPN iajs-3553 224 65 οΏ½ οΏ½ PROPN iajs-3553 224 66 Μ‚ Μ‚ NOUN iajs-3553 224 67 οΏ½ οΏ½ NOUN iajs-3553 224 68 2 2 NUM iajs-3553 224 69 . . PUNCT iajs-3553 225 1 ( ( PUNCT iajs-3553 225 2 13 13 NUM iajs-3553 225 3 ) ) PUNCT iajs-3553 225 4 οΏ½ οΏ½ NOUN iajs-3553 225 5 Μ‚ Μ‚ VERB iajs-3553 225 6 οΏ½ οΏ½ PROPN iajs-3553 225 7 3 3 NUM iajs-3553 225 8 β‰  β‰  PROPN iajs-3553 225 9 οΏ½ οΏ½ PROPN iajs-3553 225 10 Μ‚ Μ‚ NOUN iajs-3553 225 11 οΏ½ οΏ½ NOUN iajs-3553 225 12 4 4 NUM iajs-3553 225 13 . . PUNCT iajs-3553 226 1 ( ( PUNCT iajs-3553 226 2 14 14 NUM iajs-3553 226 3 ) ) PUNCT iajs-3553 226 4 where where SCONJ iajs-3553 226 5 : : PUNCT iajs-3553 226 6 οΏ½ οΏ½ PROPN iajs-3553 226 7 Μ‚ Μ‚ VERB iajs-3553 226 8 οΏ½ οΏ½ NOUN iajs-3553 226 9 1 1 NUM iajs-3553 226 10 = = SYM iajs-3553 226 11 𝑙2βˆ…2 𝑙2βˆ…2 VERB iajs-3553 226 12 ( ( PUNCT iajs-3553 226 13 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 NUM iajs-3553 226 14 ، ، X iajs-3553 226 15 οΏ½ οΏ½ PROPN iajs-3553 226 16 Μ‚ Μ‚ VERB iajs-3553 226 17 οΏ½ οΏ½ NOUN iajs-3553 226 18 2 2 NUM iajs-3553 226 19 = = SYM iajs-3553 226 20 𝑙2𝑅3π‘Ž3βˆ…2 𝑙2𝑅3π‘Ž3βˆ…2 PROPN iajs-3553 226 21 2 2 NUM iajs-3553 226 22 ( ( PUNCT iajs-3553 226 23 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 226 24 ) ) PUNCT iajs-3553 227 1 + + NUM iajs-3553 227 2 𝑙2𝑅2π‘Ž4 𝑙2𝑅2π‘Ž4 NOUN iajs-3553 227 3 ( ( PUNCT iajs-3553 227 4 1+a3r2)(1+a4r3)3 1+a3r2)(1+a4r3)3 NUM iajs-3553 227 5 ) ) PUNCT iajs-3553 227 6 ، ، X iajs-3553 227 7 οΏ½ οΏ½ PROPN iajs-3553 227 8 Μ‚ Μ‚ VERB iajs-3553 227 9 οΏ½ οΏ½ NOUN iajs-3553 227 10 3 3 NUM iajs-3553 227 11 = = SYM iajs-3553 227 12 𝑙2 𝑙2 PROPN iajs-3553 227 13 ( ( PUNCT iajs-3553 227 14 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 227 15 ) ) PUNCT iajs-3553 227 16 ( ( PUNCT iajs-3553 227 17 𝑅3π‘Ž3 𝑅3π‘Ž3 NOUN iajs-3553 227 18 2βˆ…2 2βˆ…2 PROPN iajs-3553 227 19 3 3 NUM iajs-3553 227 20 ( ( PUNCT iajs-3553 227 21 1+a3r2)3 1+a3r2)3 NUM iajs-3553 227 22 + + NUM iajs-3553 227 23 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 227 24 2 2 NUM iajs-3553 227 25 ( ( PUNCT iajs-3553 227 26 1+a4r3)3 1+a4r3)3 NUM iajs-3553 227 27 ) ) PUNCT iajs-3553 227 28 ، ، PROPN iajs-3553 227 29 οΏ½ οΏ½ PROPN iajs-3553 227 30 Μ‚ Μ‚ VERB iajs-3553 227 31 οΏ½ οΏ½ NOUN iajs-3553 227 32 4 4 NUM iajs-3553 227 33 = = SYM iajs-3553 227 34 𝑙2βˆ…2 𝑙2βˆ…2 VERB iajs-3553 227 35 ( ( PUNCT iajs-3553 227 36 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 NUM iajs-3553 227 37 ( ( PUNCT iajs-3553 227 38 π‘Ž4 π‘Ž4 PROPN iajs-3553 227 39 ( ( PUNCT iajs-3553 227 40 1+π‘Ž4𝑅3 1+π‘Ž4𝑅3 PROPN iajs-3553 227 41 ) ) PUNCT iajs-3553 227 42 + + CCONJ iajs-3553 227 43 π‘Ž3βˆ…2 π‘Ž3βˆ…2 NOUN iajs-3553 227 44 ( ( PUNCT iajs-3553 227 45 1+a3r2 1+a3r2 NUM iajs-3553 227 46 ) ) PUNCT iajs-3553 227 47 ) ) PUNCT iajs-3553 227 48 . . PUNCT iajs-3553 228 1 proof proof NOUN iajs-3553 228 2 : : PUNCT iajs-3553 228 3 by by ADP iajs-3553 228 4 using use VERB iajs-3553 228 5 the the DET iajs-3553 228 6 jacobian jacobian ADJ iajs-3553 228 7 matrix matrix NOUN iajs-3553 228 8 inequation inequation NOUN iajs-3553 228 9 ( ( PUNCT iajs-3553 228 10 1.9 1.9 NUM iajs-3553 228 11 ) ) PUNCT iajs-3553 228 12 in in ADP iajs-3553 228 13 ( ( PUNCT iajs-3553 228 14 20 20 NUM iajs-3553 228 15 ) ) PUNCT iajs-3553 228 16 𝐽3 𝐽3 NOUN iajs-3553 228 17 = = SYM iajs-3553 228 18 𝐽3(𝐴3 𝐽3(𝐴3 PROPN iajs-3553 228 19 . . PROPN iajs-3553 228 20 οΏ½ οΏ½ PROPN iajs-3553 228 21 Μ‚ Μ‚ VERB iajs-3553 228 22 οΏ½ οΏ½ NOUN iajs-3553 228 23 2 2 NUM iajs-3553 228 24 ) ) PUNCT iajs-3553 228 25 = = PUNCT iajs-3553 229 1 [ [ X iajs-3553 229 2 οΏ½ οΏ½ NOUN iajs-3553 229 3 Μ‚ Μ‚ SYM iajs-3553 229 4 οΏ½ οΏ½ NOUN iajs-3553 229 5 𝑖𝑗]3Γ—3 𝑖𝑗]3Γ—3 PROPN iajs-3553 229 6 , , PUNCT iajs-3553 229 7 where where SCONJ iajs-3553 229 8 οΏ½ οΏ½ PROPN iajs-3553 229 9 Μ‚ Μ‚ NOUN iajs-3553 229 10 οΏ½ οΏ½ NOUN iajs-3553 229 11 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 229 12 = = PUNCT iajs-3553 229 13 𝑑𝑖𝑗 𝑑𝑖𝑗 INTJ iajs-3553 229 14 , , PUNCT iajs-3553 229 15 except except SCONJ iajs-3553 229 16 οΏ½ οΏ½ PROPN iajs-3553 229 17 Μ‚ Μ‚ SYM iajs-3553 229 18 οΏ½ οΏ½ NOUN iajs-3553 229 19 33 33 NUM iajs-3553 229 20 = = SYM iajs-3553 229 21 0 0 NUM iajs-3553 230 1 .then .then NOUN iajs-3553 230 2 , , PUNCT iajs-3553 230 3 the the DET iajs-3553 230 4 characterizing characterize VERB iajs-3553 230 5 equation equation NOUN iajs-3553 230 6 for for ADP iajs-3553 230 7 𝐽3 𝐽3 PROPN iajs-3553 230 8 has have VERB iajs-3553 230 9 an an DET iajs-3553 230 10 eigenvalue eigenvalue NOUN iajs-3553 230 11 of of ADP iajs-3553 230 12 zero zero NUM iajs-3553 230 13 , , PUNCT iajs-3553 230 14 which which PRON iajs-3553 230 15 is be AUX iajs-3553 230 16 Ξ»3r3 Ξ»3r3 X iajs-3553 230 17 at at ADP iajs-3553 230 18 οΏ½ οΏ½ PROPN iajs-3553 230 19 Μ‚ Μ‚ X iajs-3553 230 20 οΏ½ οΏ½ NOUN iajs-3553 230 21 2 2 NUM iajs-3553 230 22 = = NOUN iajs-3553 230 23 𝑑2 𝑑2 NOUN iajs-3553 230 24 . . PUNCT iajs-3553 231 1 now now ADV iajs-3553 231 2 , , PUNCT iajs-3553 231 3 let let VERB iajs-3553 231 4 𝑇[3 𝑇[3 PRON iajs-3553 231 5 ] ] X iajs-3553 231 6 = = PUNCT iajs-3553 231 7 ( ( PUNCT iajs-3553 231 8 𝑑1 𝑑1 NOUN iajs-3553 231 9 [ [ X iajs-3553 231 10 3 3 NUM iajs-3553 231 11 ] ] PUNCT iajs-3553 231 12 . . PUNCT iajs-3553 232 1 𝑑2 𝑑2 PROPN iajs-3553 232 2 [ [ X iajs-3553 232 3 3 3 NUM iajs-3553 232 4 ] ] PUNCT iajs-3553 232 5 . . PUNCT iajs-3553 233 1 𝑑3 𝑑3 PROPN iajs-3553 233 2 [ [ X iajs-3553 233 3 3 3 NUM iajs-3553 233 4 ] ] PUNCT iajs-3553 233 5 ) ) PUNCT iajs-3553 233 6 𝑇 𝑇 PROPN iajs-3553 233 7 be be VERB iajs-3553 233 8 the the DET iajs-3553 233 9 eigenvector eigenvector NOUN iajs-3553 233 10 associated associate VERB iajs-3553 233 11 with with ADP iajs-3553 233 12 an an DET iajs-3553 233 13 eigenvalue eigenvalue NOUN iajs-3553 233 14 Ξ»3r3 Ξ»3r3 X iajs-3553 233 15 = = SYM iajs-3553 233 16 0 0 PROPN iajs-3553 233 17 . . PUNCT iajs-3553 234 1 thus,(𝐽3 thus,(𝐽3 ADJ iajs-3553 234 2 βˆ’ βˆ’ PROPN iajs-3553 234 3 Ξ»3r3 Ξ»3r3 SYM iajs-3553 234 4 𝐼)𝑇[3 𝐼)𝑇[3 NOUN iajs-3553 234 5 ] ] X iajs-3553 234 6 = = PUNCT iajs-3553 235 1 0 0 X iajs-3553 235 2 . . PUNCT iajs-3553 236 1 this this PRON iajs-3553 236 2 gives give VERB iajs-3553 236 3 : : PUNCT iajs-3553 236 4 𝑑1 𝑑1 NOUN iajs-3553 236 5 [ [ X iajs-3553 236 6 3 3 NUM iajs-3553 236 7 ] ] PUNCT iajs-3553 236 8 = = SYM iajs-3553 236 9 βˆ…1𝑑3 βˆ…1𝑑3 NOUN iajs-3553 236 10 [ [ X iajs-3553 236 11 3 3 NUM iajs-3553 236 12 ] ] PUNCT iajs-3553 236 13 . . PUNCT iajs-3553 237 1 𝑑2 𝑑2 PROPN iajs-3553 237 2 [ [ X iajs-3553 237 3 3 3 NUM iajs-3553 237 4 ] ] PUNCT iajs-3553 237 5 = = PUNCT iajs-3553 237 6 βˆ…2𝑑3 βˆ…2𝑑3 PROPN iajs-3553 238 1 [ [ X iajs-3553 238 2 3 3 NUM iajs-3553 238 3 ] ] PUNCT iajs-3553 238 4 . . PUNCT iajs-3553 239 1 where where SCONJ iajs-3553 239 2 : : PUNCT iajs-3553 239 3 βˆ…1 βˆ…1 PROPN iajs-3553 239 4 = = SYM iajs-3553 239 5 𝑑23𝑑12 𝑑23𝑑12 PROPN iajs-3553 239 6 𝑑11𝑑22βˆ’π‘‘12𝑑21 𝑑11𝑑22βˆ’π‘‘12𝑑21 DET iajs-3553 239 7 ، ، NOUN iajs-3553 239 8 βˆ…2 βˆ…2 PROPN iajs-3553 239 9 = = SYM iajs-3553 239 10 𝑑23𝑑11 𝑑23𝑑11 NOUN iajs-3553 239 11 𝑑12𝑑21βˆ’π‘‘11𝑑22 𝑑12𝑑21βˆ’π‘‘11𝑑22 NOUN iajs-3553 239 12 . . PUNCT iajs-3553 240 1 and and CCONJ iajs-3553 240 2 𝑑3 𝑑3 PROPN iajs-3553 240 3 [ [ X iajs-3553 240 4 3 3 NUM iajs-3553 240 5 ] ] X iajs-3553 240 6 any any DET iajs-3553 240 7 real real ADJ iajs-3553 240 8 number number NOUN iajs-3553 240 9 that that PRON iajs-3553 240 10 is be AUX iajs-3553 240 11 not not PART iajs-3553 240 12 zero zero NUM iajs-3553 240 13 . . PUNCT iajs-3553 241 1 let let VERB iajs-3553 241 2 𝑀[3 𝑀[3 VERB iajs-3553 241 3 ] ] X iajs-3553 241 4 = = SYM iajs-3553 241 5 ( ( PUNCT iajs-3553 241 6 π‘š1 π‘š1 X iajs-3553 241 7 [ [ X iajs-3553 241 8 3 3 NUM iajs-3553 241 9 ] ] PUNCT iajs-3553 241 10 . . PUNCT iajs-3553 242 1 π‘š2 π‘š2 NOUN iajs-3553 243 1 [ [ X iajs-3553 243 2 3 3 NUM iajs-3553 243 3 ] ] PUNCT iajs-3553 243 4 . . PUNCT iajs-3553 244 1 π‘š3 π‘š3 NOUN iajs-3553 245 1 [ [ X iajs-3553 245 2 3 3 X iajs-3553 245 3 ] ] PUNCT iajs-3553 245 4 ) ) PUNCT iajs-3553 245 5 𝑇 𝑇 PROPN iajs-3553 245 6 be be VERB iajs-3553 245 7 the the DET iajs-3553 245 8 eigenvector eigenvector NOUN iajs-3553 245 9 associated associate VERB iajs-3553 245 10 with with ADP iajs-3553 245 11 an an DET iajs-3553 245 12 eigenvalue eigenvalue NOUN iajs-3553 245 13 Ξ»3r3 Ξ»3r3 X iajs-3553 245 14 = = SYM iajs-3553 245 15 0 0 NUM iajs-3553 245 16 of of ADP iajs-3553 245 17 the the DET iajs-3553 245 18 matrix matrix NOUN iajs-3553 245 19 𝐽3 𝐽3 ADP iajs-3553 245 20 𝑇.then 𝑇.then ADV iajs-3553 245 21 . . PUNCT iajs-3553 246 1 ( ( PUNCT iajs-3553 246 2 𝐽3 𝐽3 PROPN iajs-3553 246 3 𝑇 𝑇 PROPN iajs-3553 246 4 βˆ’ βˆ’ PROPN iajs-3553 246 5 Ξ»3r3 Ξ»3r3 SYM iajs-3553 246 6 𝐼)𝑀[3 𝐼)𝑀[3 NOUN iajs-3553 246 7 ] ] PUNCT iajs-3553 247 1 = = SYM iajs-3553 247 2 0 0 X iajs-3553 247 3 . . PUNCT iajs-3553 248 1 by by ADP iajs-3553 248 2 solving solve VERB iajs-3553 248 3 this this DET iajs-3553 248 4 equation equation NOUN iajs-3553 248 5 for for ADP iajs-3553 248 6 . . PUNCT iajs-3553 249 1 𝑀[3 𝑀[3 PROPN iajs-3553 249 2 ] ] X iajs-3553 249 3 = = SYM iajs-3553 249 4 ( ( PUNCT iajs-3553 249 5 0 0 NUM iajs-3553 249 6 . . NOUN iajs-3553 249 7 0.π‘š3 0.π‘š3 NOUN iajs-3553 250 1 [ [ X iajs-3553 250 2 3 3 NUM iajs-3553 250 3 ] ] PUNCT iajs-3553 250 4 ) ) PUNCT iajs-3553 250 5 𝑇 𝑇 PROPN iajs-3553 250 6 , , PUNCT iajs-3553 250 7 and and CCONJ iajs-3553 250 8 π‘š3 π‘š3 NOUN iajs-3553 251 1 [ [ X iajs-3553 251 2 3 3 X iajs-3553 251 3 ] ] X iajs-3553 251 4 any any DET iajs-3553 251 5 real real ADJ iajs-3553 251 6 number number NOUN iajs-3553 251 7 that that PRON iajs-3553 251 8 is be AUX iajs-3553 251 9 not not PART iajs-3553 251 10 zero zero NUM iajs-3553 251 11 . . PUNCT iajs-3553 252 1 now now ADV iajs-3553 252 2 consider consider VERB iajs-3553 252 3 this this PRON iajs-3553 252 4 : : PUNCT iajs-3553 252 5 πœ•π‘“ πœ•π‘“ ADJ iajs-3553 252 6 πœ•π‘‘2 πœ•π‘‘2 NOUN iajs-3553 252 7 = = SYM iajs-3553 252 8 𝑓𝑑2 𝑓𝑑2 X iajs-3553 252 9 ( ( PUNCT iajs-3553 252 10 𝑋. 𝑋. PROPN iajs-3553 252 11 𝑑2 𝑑2 PROPN iajs-3553 252 12 ) ) PUNCT iajs-3553 252 13 = = SYM iajs-3553 252 14 ( ( PUNCT iajs-3553 252 15 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3553 252 16 πœ•π‘‘2 πœ•π‘‘2 PROPN iajs-3553 252 17 . . PUNCT iajs-3553 253 1 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3553 253 2 πœ•π‘‘2 πœ•π‘‘2 VERB iajs-3553 253 3 . . PUNCT iajs-3553 254 1 πœ•π‘“3 πœ•π‘“3 PROPN iajs-3553 254 2 πœ•π‘‘2 πœ•π‘‘2 PROPN iajs-3553 254 3 ) ) PUNCT iajs-3553 255 1 𝑇 𝑇 PROPN iajs-3553 255 2 = = SYM iajs-3553 255 3 ( ( PUNCT iajs-3553 255 4 0 0 NUM iajs-3553 255 5 . . NOUN iajs-3553 255 6 0 0 NUM iajs-3553 255 7 . . PUNCT iajs-3553 256 1 βˆ’π‘…3 βˆ’π‘…3 NOUN iajs-3553 256 2 ) ) PUNCT iajs-3553 256 3 𝑇 𝑇 NOUN iajs-3553 256 4 . . PUNCT iajs-3553 257 1 so so ADV iajs-3553 257 2 , , PUNCT iajs-3553 257 3 𝑓𝑑2 𝑓𝑑2 PROPN iajs-3553 257 4 ( ( PUNCT iajs-3553 257 5 𝐴3 𝐴3 PROPN iajs-3553 257 6 . . PUNCT iajs-3553 257 7 οΏ½ οΏ½ PROPN iajs-3553 257 8 Μ‚ Μ‚ VERB iajs-3553 257 9 οΏ½ οΏ½ NOUN iajs-3553 257 10 2 2 NUM iajs-3553 257 11 ) ) PUNCT iajs-3553 257 12 = = SYM iajs-3553 257 13 ( ( PUNCT iajs-3553 257 14 0 0 NUM iajs-3553 257 15 . . NOUN iajs-3553 257 16 0 0 NUM iajs-3553 257 17 . . X iajs-3553 257 18 0)𝑇and 0)𝑇and PUNCT iajs-3553 258 1 hence hence ADV iajs-3553 258 2 ( ( PUNCT iajs-3553 258 3 𝑀[3 𝑀[3 PROPN iajs-3553 258 4 ] ] X iajs-3553 258 5 ) ) PUNCT iajs-3553 258 6 𝑇 𝑇 PROPN iajs-3553 258 7 𝑓𝑑2 𝑓𝑑2 X iajs-3553 258 8 ( ( PUNCT iajs-3553 258 9 𝐴3 𝐴3 PROPN iajs-3553 258 10 . . PUNCT iajs-3553 258 11 οΏ½ οΏ½ PROPN iajs-3553 258 12 Μ‚ Μ‚ VERB iajs-3553 258 13 οΏ½ οΏ½ NOUN iajs-3553 258 14 2 2 NUM iajs-3553 258 15 ) ) PUNCT iajs-3553 258 16 = = SYM iajs-3553 258 17 0 0 X iajs-3553 258 18 . . X iajs-3553 259 1 using use VERB iajs-3553 259 2 sotomayor sotomayor NOUN iajs-3553 259 3 's 's PART iajs-3553 259 4 theorem theorem NOUN iajs-3553 259 5 , , PUNCT iajs-3553 259 6 it it PRON iajs-3553 259 7 is be AUX iajs-3553 259 8 impossible impossible ADJ iajs-3553 259 9 to to PART iajs-3553 259 10 satisfy satisfy VERB iajs-3553 259 11 the the DET iajs-3553 259 12 saddle saddle NOUN iajs-3553 259 13 - - PUNCT iajs-3553 259 14 node node NOUN iajs-3553 259 15 bifurcation bifurcation NOUN iajs-3553 259 16 condition condition NOUN iajs-3553 259 17 . . PUNCT iajs-3553 260 1 the the DET iajs-3553 260 2 first first ADJ iajs-3553 260 3 condition condition NOUN iajs-3553 260 4 for for ADP iajs-3553 260 5 transcritical transcritical ADJ iajs-3553 260 6 bifurcation bifurcation NOUN iajs-3553 260 7 is be AUX iajs-3553 260 8 therefore therefore ADV iajs-3553 260 9 satisfied satisfied ADJ iajs-3553 260 10 . . PUNCT iajs-3553 261 1 now now ADV iajs-3553 261 2 𝐷𝑓𝑑2 𝐷𝑓𝑑2 VERB iajs-3553 261 3 ( ( PUNCT iajs-3553 261 4 𝑋. 𝑋. PROPN iajs-3553 261 5 𝑑2 𝑑2 PROPN iajs-3553 261 6 ) ) PUNCT iajs-3553 261 7 = = PUNCT iajs-3553 262 1 [ [ PUNCT iajs-3553 262 2 0 0 NUM iajs-3553 262 3 0 0 NUM iajs-3553 262 4 0 0 NUM iajs-3553 262 5 0 0 NUM iajs-3553 262 6 0 0 NUM iajs-3553 262 7 0 0 NUM iajs-3553 262 8 0 0 NUM iajs-3553 262 9 0 0 NUM iajs-3553 263 1 βˆ’ βˆ’ NOUN iajs-3553 263 2 1 1 NUM iajs-3553 263 3 ] ] PUNCT iajs-3553 263 4 . . PUNCT iajs-3553 264 1 ihjpas ihjpas PROPN iajs-3553 264 2 . . PUNCT iajs-3553 265 1 2025 2025 NUM iajs-3553 265 2 , , PUNCT iajs-3553 265 3 38(2 38(2 NUM iajs-3553 265 4 ) ) PUNCT iajs-3553 265 5 322 322 NUM iajs-3553 265 6 where where SCONJ iajs-3553 265 7 , , PUNCT iajs-3553 265 8 𝐷𝑓𝑑2 𝐷𝑓𝑑2 PROPN iajs-3553 265 9 ( ( PUNCT iajs-3553 265 10 𝑋. 𝑋. PROPN iajs-3553 265 11 𝑑2 𝑑2 NOUN iajs-3553 265 12 ) ) PUNCT iajs-3553 265 13 represents represent VERB iajs-3553 265 14 the the DET iajs-3553 265 15 derivative derivative NOUN iajs-3553 265 16 of of ADP iajs-3553 265 17 𝑓𝑑2 𝑓𝑑2 PROPN iajs-3553 265 18 ( ( PUNCT iajs-3553 265 19 𝑋. 𝑋. PROPN iajs-3553 265 20 𝑑2 𝑑2 PROPN iajs-3553 265 21 ) ) PUNCT iajs-3553 265 22 with with ADP iajs-3553 265 23 respect respect NOUN iajs-3553 265 24 to to ADP iajs-3553 265 25 𝑋 𝑋 PROPN iajs-3553 265 26 = = SYM iajs-3553 265 27 ( ( PUNCT iajs-3553 265 28 𝑅1 𝑅1 PROPN iajs-3553 265 29 . . PUNCT iajs-3553 265 30 𝑅2 𝑅2 PROPN iajs-3553 265 31 . . PUNCT iajs-3553 266 1 𝑅3 𝑅3 PROPN iajs-3553 266 2 ) ) PUNCT iajs-3553 266 3 𝑇 𝑇 PROPN iajs-3553 266 4 . . PUNCT iajs-3553 267 1 furthermore furthermore ADV iajs-3553 267 2 , , PUNCT iajs-3553 267 3 it it PRON iajs-3553 267 4 is be AUX iajs-3553 267 5 observed observe VERB iajs-3553 267 6 that that SCONJ iajs-3553 267 7 : : PUNCT iajs-3553 267 8 𝐷𝑓𝑑2 𝐷𝑓𝑑2 NOUN iajs-3553 267 9 ( ( PUNCT iajs-3553 267 10 𝐴3 𝐴3 PROPN iajs-3553 267 11 . . PUNCT iajs-3553 268 1 𝑑2)𝑇 𝑑2)𝑇 NOUN iajs-3553 269 1 [ [ X iajs-3553 269 2 3 3 X iajs-3553 269 3 ] ] X iajs-3553 269 4 = = PUNCT iajs-3553 270 1 [ [ PUNCT iajs-3553 270 2 0 0 NUM iajs-3553 270 3 0 0 NUM iajs-3553 270 4 0 0 NUM iajs-3553 270 5 0 0 NUM iajs-3553 270 6 0 0 NUM iajs-3553 270 7 0 0 NUM iajs-3553 270 8 0 0 NUM iajs-3553 270 9 0 0 NUM iajs-3553 270 10 βˆ’ βˆ’ NOUN iajs-3553 270 11 1 1 NUM iajs-3553 270 12 ] ] PUNCT iajs-3553 270 13 [ [ PUNCT iajs-3553 270 14 βˆ…1𝑑3 βˆ…1𝑑3 NOUN iajs-3553 270 15 [ [ X iajs-3553 270 16 3 3 NUM iajs-3553 270 17 ] ] X iajs-3553 270 18 βˆ…2𝑑3 βˆ…2𝑑3 NOUN iajs-3553 271 1 [ [ X iajs-3553 271 2 3 3 NUM iajs-3553 271 3 ] ] X iajs-3553 271 4 𝑑3 𝑑3 NOUN iajs-3553 271 5 [ [ X iajs-3553 271 6 3 3 NUM iajs-3553 271 7 ] ] PUNCT iajs-3553 271 8 ] ] PUNCT iajs-3553 272 1 = = PUNCT iajs-3553 272 2 [ [ PUNCT iajs-3553 272 3 0 0 NUM iajs-3553 272 4 0 0 NUM iajs-3553 272 5 βˆ’π‘‘3 βˆ’π‘‘3 PROPN iajs-3553 273 1 [ [ X iajs-3553 273 2 3 3 X iajs-3553 273 3 ] ] PUNCT iajs-3553 273 4 ] ] PUNCT iajs-3553 273 5 ( ( PUNCT iajs-3553 273 6 𝑀[3 𝑀[3 PROPN iajs-3553 273 7 ] ] X iajs-3553 273 8 ) ) PUNCT iajs-3553 273 9 𝑇 𝑇 PROPN iajs-3553 274 1 [ [ X iajs-3553 274 2 𝐷𝑓𝑑2 𝐷𝑓𝑑2 X iajs-3553 274 3 ( ( PUNCT iajs-3553 274 4 𝐴3 𝐴3 PROPN iajs-3553 274 5 . . PUNCT iajs-3553 274 6 οΏ½ οΏ½ PROPN iajs-3553 274 7 Μ‚ Μ‚ NOUN iajs-3553 274 8 οΏ½ οΏ½ NOUN iajs-3553 274 9 2)𝑇 2)𝑇 NOUN iajs-3553 274 10 [ [ X iajs-3553 274 11 3 3 NUM iajs-3553 274 12 ] ] X iajs-3553 274 13 ] ] X iajs-3553 274 14 = = SYM iajs-3553 274 15 ( ( PUNCT iajs-3553 274 16 0 0 NUM iajs-3553 274 17 . . NOUN iajs-3553 274 18 0.π‘š3 0.π‘š3 NOUN iajs-3553 275 1 [ [ X iajs-3553 275 2 3 3 NUM iajs-3553 275 3 ] ] PUNCT iajs-3553 275 4 ) ) PUNCT iajs-3553 275 5 ( ( PUNCT iajs-3553 275 6 0 0 NUM iajs-3553 275 7 . . NOUN iajs-3553 275 8 0 0 NUM iajs-3553 275 9 . . PUNCT iajs-3553 276 1 βˆ’π‘‘3 βˆ’π‘‘3 PROPN iajs-3553 277 1 [ [ X iajs-3553 277 2 3 3 NUM iajs-3553 277 3 ] ] PUNCT iajs-3553 277 4 ) ) PUNCT iajs-3553 277 5 𝑇 𝑇 PROPN iajs-3553 277 6 = = SYM iajs-3553 277 7 βˆ’π‘š3 βˆ’π‘š3 PROPN iajs-3553 278 1 [ [ X iajs-3553 278 2 3 3 X iajs-3553 278 3 ] ] X iajs-3553 278 4 𝑑3 𝑑3 NOUN iajs-3553 278 5 [ [ X iajs-3553 278 6 3 3 NUM iajs-3553 278 7 ] ] PUNCT iajs-3553 278 8 β‰  β‰  X iajs-3553 278 9 0 0 NUM iajs-3553 278 10 by by ADP iajs-3553 278 11 substituting substitute VERB iajs-3553 278 12 𝑇[3 𝑇[3 PROPN iajs-3553 278 13 ] ] PUNCT iajs-3553 278 14 in in ADP iajs-3553 278 15 equation equation NOUN iajs-3553 278 16 ( ( PUNCT iajs-3553 278 17 3 3 X iajs-3553 278 18 ) ) PUNCT iajs-3553 278 19 we we PRON iajs-3553 278 20 get get VERB iajs-3553 278 21 : : PUNCT iajs-3553 279 1 𝐷2πΉπœ‡(𝐴3 𝐷2πΉπœ‡(𝐴3 PROPN iajs-3553 279 2 . . PUNCT iajs-3553 279 3 οΏ½ οΏ½ PROPN iajs-3553 279 4 Μ‚ Μ‚ VERB iajs-3553 279 5 οΏ½ οΏ½ NOUN iajs-3553 279 6 2)(𝑇 2)(𝑇 PRON iajs-3553 279 7 [ [ X iajs-3553 279 8 3 3 NUM iajs-3553 279 9 ] ] PUNCT iajs-3553 279 10 . . PUNCT iajs-3553 280 1 𝑇[3 𝑇[3 PROPN iajs-3553 280 2 ] ] X iajs-3553 280 3 ) ) PUNCT iajs-3553 280 4 = = SYM iajs-3553 280 5 ( ( PUNCT iajs-3553 280 6 οΏ½ οΏ½ PROPN iajs-3553 280 7 Μ‚ Μ‚ NOUN iajs-3553 280 8 οΏ½ οΏ½ NOUN iajs-3553 280 9 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 280 10 ) ) PUNCT iajs-3553 280 11 . . PUNCT iajs-3553 281 1 ( ( PUNCT iajs-3553 281 2 15 15 NUM iajs-3553 281 3 ) ) PUNCT iajs-3553 281 4 οΏ½ οΏ½ NOUN iajs-3553 281 5 Μ‚ Μ‚ NOUN iajs-3553 281 6 οΏ½ οΏ½ NOUN iajs-3553 281 7 11 11 NUM iajs-3553 281 8 = = SYM iajs-3553 281 9 2(𝑑3 2(𝑑3 NUM iajs-3553 282 1 [ [ X iajs-3553 282 2 3 3 NUM iajs-3553 282 3 ] ] PUNCT iajs-3553 282 4 ) ) PUNCT iajs-3553 282 5 2 2 NUM iajs-3553 283 1 [ [ X iajs-3553 283 2 ( ( PUNCT iajs-3553 283 3 𝑅2π‘Ž1βˆ…1 𝑅2π‘Ž1βˆ…1 NOUN iajs-3553 283 4 2 2 NUM iajs-3553 283 5 ( ( PUNCT iajs-3553 283 6 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 283 7 βˆ’ βˆ’ NUM iajs-3553 283 8 βˆ…1βˆ…2 βˆ…1βˆ…2 PROPN iajs-3553 283 9 ( ( PUNCT iajs-3553 283 10 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 283 11 ) ) PUNCT iajs-3553 283 12 + + NUM iajs-3553 283 13 π‘Ž2𝑅2βˆ…2 π‘Ž2𝑅2βˆ…2 SYM iajs-3553 283 14 2 2 NUM iajs-3553 283 15 ( ( PUNCT iajs-3553 283 16 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 283 17 ) ) PUNCT iajs-3553 283 18 𝛽1 𝛽1 NOUN iajs-3553 283 19 ( ( PUNCT iajs-3553 283 20 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 283 21 ) ) PUNCT iajs-3553 283 22 + + CCONJ iajs-3553 283 23 ( ( PUNCT iajs-3553 283 24 𝐾1𝑅1βˆ…1 𝐾1𝑅1βˆ…1 NOUN iajs-3553 283 25 2 2 NUM iajs-3553 283 26 ( ( PUNCT iajs-3553 283 27 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 283 28 ) ) PUNCT iajs-3553 283 29 βˆ’βˆ…1βˆ…2 βˆ’βˆ…1βˆ…2 PROPN iajs-3553 283 30 ) ) PUNCT iajs-3553 283 31 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 283 32 ( ( PUNCT iajs-3553 283 33 1+𝐾1𝑅2)2 1+𝐾1𝑅2)2 NUM iajs-3553 283 34 βˆ’ βˆ’ PROPN iajs-3553 283 35 π‘š1βˆ…1 π‘š1βˆ…1 NOUN iajs-3553 283 36 2 2 NUM iajs-3553 283 37 ] ] PUNCT iajs-3553 283 38 , , PUNCT iajs-3553 283 39 οΏ½ οΏ½ PROPN iajs-3553 283 40 Μ‚ Μ‚ VERB iajs-3553 283 41 οΏ½ οΏ½ NOUN iajs-3553 283 42 21 21 NUM iajs-3553 283 43 = = SYM iajs-3553 283 44 2(𝑑3 2(𝑑3 NUM iajs-3553 284 1 [ [ X iajs-3553 284 2 3 3 NUM iajs-3553 284 3 ] ] PUNCT iajs-3553 284 4 ) ) PUNCT iajs-3553 284 5 2 2 NUM iajs-3553 285 1 [ [ X iajs-3553 285 2 ( ( PUNCT iajs-3553 285 3 βˆ…1βˆ…2 βˆ…1βˆ…2 X iajs-3553 285 4 ( ( PUNCT iajs-3553 285 5 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 285 6 ) ) PUNCT iajs-3553 285 7 βˆ’ βˆ’ PROPN iajs-3553 286 1 r2π‘Ž1βˆ…1 r2π‘Ž1βˆ…1 NOUN iajs-3553 286 2 2 2 NUM iajs-3553 286 3 ( ( PUNCT iajs-3553 286 4 1+a1r1)2 1+a1r1)2 NUM iajs-3553 286 5 βˆ’ βˆ’ NOUN iajs-3553 286 6 r1π‘Ž2βˆ…2 r1π‘Ž2βˆ…2 PROPN iajs-3553 286 7 2 2 NUM iajs-3553 286 8 ( ( PUNCT iajs-3553 286 9 1+a2r2)2 1+a2r2)2 NUM iajs-3553 286 10 ) ) PUNCT iajs-3553 286 11 𝑙1 𝑙1 NOUN iajs-3553 286 12 ( ( PUNCT iajs-3553 286 13 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 286 14 ) ) PUNCT iajs-3553 286 15 + + CCONJ iajs-3553 286 16 ( ( PUNCT iajs-3553 286 17 a3r3βˆ…2 a3r3βˆ…2 NOUN iajs-3553 286 18 2 2 NUM iajs-3553 286 19 ( ( PUNCT iajs-3553 286 20 1+a3r2)2 1+a3r2)2 NUM iajs-3553 286 21 + + CCONJ iajs-3553 286 22 βˆ…2 βˆ…2 NOUN iajs-3553 286 23 ( ( PUNCT iajs-3553 286 24 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 286 25 ) ) PUNCT iajs-3553 287 1 + + CCONJ iajs-3553 287 2 a4𝑅2 a4𝑅2 PRON iajs-3553 287 3 ( ( PUNCT iajs-3553 287 4 1+a4r3)2 1+a4r3)2 X iajs-3553 287 5 ) ) PUNCT iajs-3553 287 6 𝛽2 𝛽2 PROPN iajs-3553 287 7 ( ( PUNCT iajs-3553 287 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 287 9 ) ) PUNCT iajs-3553 287 10 βˆ’π‘š2βˆ…2 βˆ’π‘š2βˆ…2 NOUN iajs-3553 287 11 2 2 NUM iajs-3553 287 12 + + CCONJ iajs-3553 287 13 ( ( PUNCT iajs-3553 287 14 𝐾2𝑅2 𝐾2𝑅2 NOUN iajs-3553 287 15 ( ( PUNCT iajs-3553 287 16 1+k2r3 1+k2r3 NUM iajs-3553 287 17 ) ) PUNCT iajs-3553 287 18 βˆ’ βˆ’ NOUN iajs-3553 287 19 βˆ…2 βˆ…2 NOUN iajs-3553 287 20 ) ) PUNCT iajs-3553 287 21 r2k2 r2k2 NOUN iajs-3553 287 22 ( ( PUNCT iajs-3553 287 23 1+k2r3)2 1+k2r3)2 NOUN iajs-3553 287 24 ] ] X iajs-3553 287 25 , , PUNCT iajs-3553 287 26 οΏ½ οΏ½ PROPN iajs-3553 287 27 Μ‚ Μ‚ NOUN iajs-3553 287 28 οΏ½ οΏ½ NOUN iajs-3553 287 29 31 31 NUM iajs-3553 287 30 = = SYM iajs-3553 287 31 2(𝑑3 2(𝑑3 NUM iajs-3553 288 1 [ [ X iajs-3553 288 2 3 3 NUM iajs-3553 288 3 ] ] PUNCT iajs-3553 288 4 ) ) PUNCT iajs-3553 288 5 2 2 NUM iajs-3553 288 6 𝑙2 𝑙2 NOUN iajs-3553 288 7 ( ( PUNCT iajs-3553 288 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 288 9 ) ) PUNCT iajs-3553 288 10 ( ( PUNCT iajs-3553 288 11 βˆ…2 βˆ…2 NOUN iajs-3553 288 12 ( ( PUNCT iajs-3553 288 13 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 288 14 ) ) PUNCT iajs-3553 288 15 βˆ’ βˆ’ PROPN iajs-3553 288 16 𝑅3π‘Ž3βˆ…2 𝑅3π‘Ž3βˆ…2 ADP iajs-3553 288 17 2 2 NUM iajs-3553 288 18 ( ( PUNCT iajs-3553 288 19 1+a3r2)2 1+a3r2)2 NUM iajs-3553 288 20 βˆ’ βˆ’ NOUN iajs-3553 288 21 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 288 22 ( ( PUNCT iajs-3553 288 23 1+a4r3)2 1+a4r3)2 NUM iajs-3553 288 24 ) ) PUNCT iajs-3553 288 25 . . PUNCT iajs-3553 289 1 hence hence ADV iajs-3553 289 2 , , PUNCT iajs-3553 289 3 it it PRON iajs-3553 289 4 was be AUX iajs-3553 289 5 obtained obtain VERB iajs-3553 289 6 ( ( PUNCT iajs-3553 289 7 𝑀[3 𝑀[3 PROPN iajs-3553 289 8 ] ] NOUN iajs-3553 289 9 ) ) PUNCT iajs-3553 289 10 𝑇 𝑇 PROPN iajs-3553 290 1 [ [ X iajs-3553 290 2 𝐷2πΉπœ‡(𝐴3 𝐷2πΉπœ‡(𝐴3 PROPN iajs-3553 290 3 . . PUNCT iajs-3553 290 4 οΏ½ οΏ½ PROPN iajs-3553 290 5 Μ‚ Μ‚ VERB iajs-3553 290 6 οΏ½ οΏ½ NOUN iajs-3553 290 7 2)(𝑇 2)(𝑇 PRON iajs-3553 290 8 [ [ X iajs-3553 290 9 3 3 NUM iajs-3553 290 10 ] ] PUNCT iajs-3553 290 11 . . PUNCT iajs-3553 291 1 𝑇[3 𝑇[3 PROPN iajs-3553 291 2 ] ] PUNCT iajs-3553 291 3 ) ) PUNCT iajs-3553 291 4 ] ] PUNCT iajs-3553 292 1 = = SYM iajs-3553 292 2 2 2 X iajs-3553 292 3 ( ( PUNCT iajs-3553 292 4 𝑑3 𝑑3 PROPN iajs-3553 292 5 [ [ X iajs-3553 292 6 3 3 NUM iajs-3553 292 7 ] ] PUNCT iajs-3553 292 8 ) ) PUNCT iajs-3553 292 9 2 2 NUM iajs-3553 292 10 π‘š3 π‘š3 NOUN iajs-3553 292 11 [ [ X iajs-3553 292 12 3 3 NUM iajs-3553 292 13 ] ] X iajs-3553 292 14 ( ( PUNCT iajs-3553 292 15 οΏ½ οΏ½ NOUN iajs-3553 292 16 Μ‚ Μ‚ VERB iajs-3553 292 17 οΏ½ οΏ½ NOUN iajs-3553 292 18 1 1 NUM iajs-3553 292 19 βˆ’ βˆ’ PROPN iajs-3553 292 20 οΏ½ οΏ½ PROPN iajs-3553 292 21 Μ‚ Μ‚ VERB iajs-3553 292 22 οΏ½ οΏ½ NOUN iajs-3553 292 23 2 2 NUM iajs-3553 292 24 ) ) PUNCT iajs-3553 292 25 β‰  β‰  PROPN iajs-3553 292 26 0 0 X iajs-3553 292 27 . . PUNCT iajs-3553 293 1 this this PRON iajs-3553 293 2 indicates indicate VERB iajs-3553 293 3 that that SCONJ iajs-3553 293 4 the the DET iajs-3553 293 5 system system NOUN iajs-3553 293 6 ( ( PUNCT iajs-3553 293 7 1 1 X iajs-3553 293 8 ) ) PUNCT iajs-3553 293 9 exhibits exhibit VERB iajs-3553 293 10 a a DET iajs-3553 293 11 transcritical transcritical ADJ iajs-3553 293 12 bifurcation bifurcation NOUN iajs-3553 293 13 at at ADP iajs-3553 293 14 𝐴2 𝐴2 NOUN iajs-3553 293 15 with with ADP iajs-3553 293 16 a a DET iajs-3553 293 17 parameter parameter NOUN iajs-3553 293 18 οΏ½ οΏ½ NOUN iajs-3553 293 19 Μ‚ Μ‚ NOUN iajs-3553 293 20 οΏ½ οΏ½ NOUN iajs-3553 293 21 2 2 NUM iajs-3553 293 22 = = SYM iajs-3553 293 23 𝑑2 𝑑2 NOUN iajs-3553 293 24 . . PUNCT iajs-3553 294 1 if if SCONJ iajs-3553 294 2 condition condition NOUN iajs-3553 294 3 ( ( PUNCT iajs-3553 294 4 13 13 NUM iajs-3553 294 5 ) ) PUNCT iajs-3553 294 6 not not PART iajs-3553 294 7 satisfied satisfied ADJ iajs-3553 294 8 then then ADV iajs-3553 294 9 . . PUNCT iajs-3553 295 1 by by ADP iajs-3553 295 2 substituting substitute VERB iajs-3553 295 3 𝑇[3 𝑇[3 PROPN iajs-3553 295 4 ] ] PUNCT iajs-3553 295 5 in in ADP iajs-3553 295 6 equation equation NOUN iajs-3553 295 7 ( ( PUNCT iajs-3553 295 8 4 4 X iajs-3553 295 9 ) ) PUNCT iajs-3553 295 10 we we PRON iajs-3553 295 11 get get VERB iajs-3553 295 12 : : PUNCT iajs-3553 295 13 𝐷3πΉπœ‡(𝐴3 𝐷3πΉπœ‡(𝐴3 X iajs-3553 295 14 . . PUNCT iajs-3553 295 15 οΏ½ οΏ½ PROPN iajs-3553 295 16 Μ‚ Μ‚ VERB iajs-3553 295 17 οΏ½ οΏ½ NOUN iajs-3553 295 18 2)(𝑇 2)(𝑇 PRON iajs-3553 295 19 [ [ X iajs-3553 295 20 3 3 NUM iajs-3553 295 21 ] ] PUNCT iajs-3553 295 22 . . PUNCT iajs-3553 296 1 𝑇[3 𝑇[3 PROPN iajs-3553 296 2 ] ] PUNCT iajs-3553 296 3 . . PUNCT iajs-3553 297 1 𝑇[3 𝑇[3 PROPN iajs-3553 297 2 ] ] X iajs-3553 297 3 ) ) PUNCT iajs-3553 298 1 = = SYM iajs-3553 298 2 ( ( PUNCT iajs-3553 298 3 οΏ½ οΏ½ PROPN iajs-3553 298 4 Μ‚ Μ‚ NOUN iajs-3553 298 5 οΏ½ οΏ½ NOUN iajs-3553 298 6 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 298 7 ) ) PUNCT iajs-3553 298 8 , , PUNCT iajs-3553 298 9 ( ( PUNCT iajs-3553 298 10 16 16 NUM iajs-3553 298 11 ) ) PUNCT iajs-3553 298 12 οΏ½ οΏ½ PROPN iajs-3553 298 13 Μ‚ Μ‚ VERB iajs-3553 298 14 οΏ½ οΏ½ NOUN iajs-3553 298 15 11 11 NUM iajs-3553 298 16 = = SYM iajs-3553 298 17 6(𝑑3 6(𝑑3 PROPN iajs-3553 299 1 [ [ X iajs-3553 299 2 3 3 NUM iajs-3553 299 3 ] ] PUNCT iajs-3553 299 4 ) ) PUNCT iajs-3553 299 5 3 3 NUM iajs-3553 300 1 [ [ X iajs-3553 300 2 ( ( PUNCT iajs-3553 300 3 π‘Ž2βˆ…1βˆ…2 π‘Ž2βˆ…1βˆ…2 ADJ iajs-3553 300 4 2 2 NUM iajs-3553 300 5 ( ( PUNCT iajs-3553 300 6 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)2 NUM iajs-3553 300 7 βˆ’ βˆ’ NOUN iajs-3553 300 8 βˆ…2 βˆ…2 VERB iajs-3553 300 9 3π‘Ž2 3π‘Ž2 NUM iajs-3553 300 10 2𝑅1 2𝑅1 NUM iajs-3553 300 11 ( ( PUNCT iajs-3553 300 12 1+π‘Ž2𝑅2)3 1+π‘Ž2𝑅2)3 NUM iajs-3553 300 13 βˆ’ βˆ’ NOUN iajs-3553 300 14 r2π‘Ž1 r2π‘Ž1 NUM iajs-3553 300 15 2βˆ…1 2βˆ…1 NUM iajs-3553 300 16 3 3 NUM iajs-3553 300 17 ( ( PUNCT iajs-3553 300 18 1+π‘Ž1𝑅1)3 1+π‘Ž1𝑅1)3 NUM iajs-3553 300 19 π‘Ž1βˆ…1 π‘Ž1βˆ…1 NOUN iajs-3553 300 20 2βˆ…2 2βˆ…2 NUM iajs-3553 300 21 ( ( PUNCT iajs-3553 300 22 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2 NUM iajs-3553 300 23 ) ) PUNCT iajs-3553 300 24 ) ) PUNCT iajs-3553 300 25 𝛽1 𝛽1 NOUN iajs-3553 300 26 ( ( PUNCT iajs-3553 300 27 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 300 28 ) ) PUNCT iajs-3553 300 29 + + CCONJ iajs-3553 300 30 ( ( PUNCT iajs-3553 300 31 βˆ…1 βˆ…1 VERB iajs-3553 300 32 βˆ’ βˆ’ PROPN iajs-3553 300 33 𝐾1𝑅1βˆ…1 𝐾1𝑅1βˆ…1 NOUN iajs-3553 300 34 ( ( PUNCT iajs-3553 300 35 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 300 36 ) ) PUNCT iajs-3553 300 37 ) ) PUNCT iajs-3553 300 38 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 300 39 2βˆ…2 2βˆ…2 NUM iajs-3553 300 40 2 2 NUM iajs-3553 300 41 ( ( PUNCT iajs-3553 300 42 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 300 43 ] ] PUNCT iajs-3553 300 44 , , PUNCT iajs-3553 300 45 οΏ½ οΏ½ PROPN iajs-3553 300 46 Μ‚ Μ‚ VERB iajs-3553 300 47 οΏ½ οΏ½ NOUN iajs-3553 300 48 21 21 NUM iajs-3553 300 49 = = SYM iajs-3553 300 50 6(𝑑3 6(𝑑3 PROPN iajs-3553 301 1 [ [ X iajs-3553 301 2 3 3 NUM iajs-3553 301 3 ] ] PUNCT iajs-3553 301 4 ) ) PUNCT iajs-3553 301 5 3 3 NUM iajs-3553 302 1 [ [ X iajs-3553 302 2 ( ( PUNCT iajs-3553 302 3 βˆ…2 βˆ…2 VERB iajs-3553 302 4 βˆ’ βˆ’ NOUN iajs-3553 302 5 k2𝑅2 k2𝑅2 PROPN iajs-3553 302 6 ( ( PUNCT iajs-3553 302 7 1+k2r3 1+k2r3 NUM iajs-3553 302 8 ) ) PUNCT iajs-3553 302 9 ) ) PUNCT iajs-3553 302 10 r2k2 r2k2 X iajs-3553 302 11 2 2 NUM iajs-3553 302 12 ( ( PUNCT iajs-3553 302 13 1+k2r3)3 1+k2r3)3 VERB iajs-3553 302 14 βˆ’ βˆ’ PROPN iajs-3553 303 1 𝛽1π‘Ž2 𝛽1π‘Ž2 ADP iajs-3553 303 2 2𝑅1 2𝑅1 NUM iajs-3553 303 3 ( ( PUNCT iajs-3553 303 4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 NUM iajs-3553 303 5 + + NUM iajs-3553 303 6 ( ( PUNCT iajs-3553 303 7 r2π‘Ž1 r2π‘Ž1 PROPN iajs-3553 303 8 2βˆ…1 2βˆ…1 NUM iajs-3553 303 9 3 3 NUM iajs-3553 303 10 ( ( PUNCT iajs-3553 303 11 1+a1r1)3 1+a1r1)3 NOUN iajs-3553 303 12 + + CCONJ iajs-3553 303 13 π‘Ž1βˆ…1 π‘Ž1βˆ…1 NOUN iajs-3553 303 14 2βˆ…2 2βˆ…2 NUM iajs-3553 303 15 ( ( PUNCT iajs-3553 303 16 1+a1r1)2(1+a2r2 1+a1r1)2(1+a2r2 NUM iajs-3553 303 17 ) ) PUNCT iajs-3553 303 18 βˆ’ βˆ’ PROPN iajs-3553 304 1 π‘Ž2βˆ…1βˆ…2 π‘Ž2βˆ…1βˆ…2 ADJ iajs-3553 304 2 2 2 NUM iajs-3553 304 3 ( ( PUNCT iajs-3553 304 4 1+a1r1)(1+a2r2)2 1+a1r1)(1+a2r2)2 NUM iajs-3553 304 5 + + CCONJ iajs-3553 304 6 r1π‘Ž2 r1π‘Ž2 PROPN iajs-3553 304 7 2βˆ…2 2βˆ…2 NUM iajs-3553 304 8 3 3 NUM iajs-3553 304 9 ( ( PUNCT iajs-3553 304 10 1+a2r2)3 1+a2r2)3 NUM iajs-3553 304 11 ) ) PUNCT iajs-3553 304 12 𝑙1 𝑙1 NOUN iajs-3553 304 13 ( ( PUNCT iajs-3553 304 14 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 304 15 ) ) PUNCT iajs-3553 304 16 + + CCONJ iajs-3553 304 17 ( ( PUNCT iajs-3553 304 18 π‘Ž4βˆ…2 π‘Ž4βˆ…2 ADV iajs-3553 304 19 ( ( PUNCT iajs-3553 304 20 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3)2 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3)2 NUM iajs-3553 304 21 + + CCONJ iajs-3553 304 22 a4βˆ…2 a4βˆ…2 PROPN iajs-3553 304 23 2 2 NUM iajs-3553 304 24 ( ( PUNCT iajs-3553 304 25 1+a3r2)(1+a4r3)2 1+a3r2)(1+a4r3)2 NUM iajs-3553 304 26 βˆ’ βˆ’ NOUN iajs-3553 304 27 r3π‘Ž3 r3π‘Ž3 NOUN iajs-3553 304 28 2βˆ…2 2βˆ…2 NUM iajs-3553 304 29 3 3 NUM iajs-3553 304 30 ( ( PUNCT iajs-3553 304 31 1+a3r2)3 1+a3r2)3 NUM iajs-3553 304 32 ) ) PUNCT iajs-3553 304 33 𝛽2𝐼 𝛽2𝐼 ADV iajs-3553 304 34 ( ( PUNCT iajs-3553 304 35 1+π‘Ž3𝑅2)2(1+π‘Ž4𝑅3 1+π‘Ž3𝑅2)2(1+π‘Ž4𝑅3 NUM iajs-3553 304 36 ) ) PUNCT iajs-3553 304 37 ] ] PUNCT iajs-3553 304 38 , , PUNCT iajs-3553 304 39 οΏ½ οΏ½ PROPN iajs-3553 304 40 Μ‚ Μ‚ NOUN iajs-3553 304 41 οΏ½ οΏ½ NOUN iajs-3553 304 42 31 31 NUM iajs-3553 304 43 = = SYM iajs-3553 304 44 6(𝑑3 6(𝑑3 PROPN iajs-3553 305 1 [ [ X iajs-3553 305 2 3 3 NUM iajs-3553 305 3 ] ] PUNCT iajs-3553 305 4 ) ) PUNCT iajs-3553 305 5 3 3 NUM iajs-3553 305 6 𝑙2 𝑙2 NOUN iajs-3553 305 7 ( ( PUNCT iajs-3553 305 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 305 9 ) ) PUNCT iajs-3553 305 10 ( ( PUNCT iajs-3553 305 11 𝑅3π‘Ž3 𝑅3π‘Ž3 NOUN iajs-3553 305 12 2βˆ…2 2βˆ…2 PROPN iajs-3553 305 13 3 3 NUM iajs-3553 305 14 ( ( PUNCT iajs-3553 305 15 1+a3r2)3 1+a3r2)3 NUM iajs-3553 305 16 + + NUM iajs-3553 305 17 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 305 18 2 2 NUM iajs-3553 305 19 ( ( PUNCT iajs-3553 305 20 1+a4r3)3 1+a4r3)3 NUM iajs-3553 305 21 βˆ’ βˆ’ NOUN iajs-3553 305 22 π‘Ž4βˆ…2 π‘Ž4βˆ…2 NOUN iajs-3553 305 23 ( ( PUNCT iajs-3553 305 24 1+a3r2)(1+a4r3)2 1+a3r2)(1+a4r3)2 NUM iajs-3553 305 25 βˆ’ βˆ’ NOUN iajs-3553 305 26 π‘Ž3βˆ…2 π‘Ž3βˆ…2 NOUN iajs-3553 305 27 2 2 NUM iajs-3553 305 28 ( ( PUNCT iajs-3553 305 29 1+a3r2)2(1+a4r3 1+a3r2)2(1+a4r3 NUM iajs-3553 305 30 ) ) PUNCT iajs-3553 305 31 ) ) PUNCT iajs-3553 305 32 . . PUNCT iajs-3553 306 1 hence hence ADV iajs-3553 306 2 , , PUNCT iajs-3553 306 3 ( ( PUNCT iajs-3553 306 4 𝑀[3 𝑀[3 PROPN iajs-3553 306 5 ] ] X iajs-3553 306 6 ) ) PUNCT iajs-3553 306 7 𝑇 𝑇 PROPN iajs-3553 306 8 [ [ X iajs-3553 306 9 𝐷3πΉπœ‡(𝐴3 𝐷3πΉπœ‡(𝐴3 PROPN iajs-3553 306 10 . . PUNCT iajs-3553 306 11 οΏ½ οΏ½ PROPN iajs-3553 306 12 Μ‚ Μ‚ NOUN iajs-3553 306 13 οΏ½ οΏ½ NOUN iajs-3553 306 14 2)(𝑇 2)(𝑇 PRON iajs-3553 306 15 [ [ X iajs-3553 306 16 3 3 NUM iajs-3553 306 17 ] ] PUNCT iajs-3553 306 18 . . PUNCT iajs-3553 307 1 𝑇[3 𝑇[3 PROPN iajs-3553 307 2 ] ] PUNCT iajs-3553 307 3 . . PUNCT iajs-3553 308 1 𝑇[3 𝑇[3 PROPN iajs-3553 308 2 ] ] PUNCT iajs-3553 308 3 ) ) PUNCT iajs-3553 308 4 ] ] PUNCT iajs-3553 309 1 = = SYM iajs-3553 309 2 6 6 NUM iajs-3553 309 3 ( ( PUNCT iajs-3553 309 4 𝑑3 𝑑3 PROPN iajs-3553 309 5 [ [ X iajs-3553 309 6 3 3 NUM iajs-3553 309 7 ] ] PUNCT iajs-3553 309 8 ) ) PUNCT iajs-3553 309 9 3 3 NUM iajs-3553 309 10 π‘š3 π‘š3 NOUN iajs-3553 309 11 [ [ X iajs-3553 309 12 3 3 NUM iajs-3553 309 13 ] ] X iajs-3553 309 14 ( ( PUNCT iajs-3553 309 15 οΏ½ οΏ½ NOUN iajs-3553 309 16 Μ‚ Μ‚ NOUN iajs-3553 309 17 οΏ½ οΏ½ NOUN iajs-3553 309 18 3 3 NUM iajs-3553 309 19 βˆ’ βˆ’ PROPN iajs-3553 309 20 οΏ½ οΏ½ PROPN iajs-3553 309 21 Μ‚ Μ‚ NOUN iajs-3553 309 22 οΏ½ οΏ½ NOUN iajs-3553 309 23 4 4 NUM iajs-3553 309 24 ) ) PUNCT iajs-3553 309 25 β‰  β‰  PROPN iajs-3553 309 26 0 0 NUM iajs-3553 309 27 . . PUNCT iajs-3553 310 1 by by ADP iajs-3553 310 2 condition condition NOUN iajs-3553 310 3 ( ( PUNCT iajs-3553 310 4 14 14 NUM iajs-3553 310 5 ) ) PUNCT iajs-3553 310 6 . . PUNCT iajs-3553 311 1 which which DET iajs-3553 311 2 grantee grantee NOUN iajs-3553 311 3 that that SCONJ iajs-3553 311 4 there there PRON iajs-3553 311 5 is be VERB iajs-3553 311 6 pitch pitch NOUN iajs-3553 311 7 fork fork NOUN iajs-3553 311 8 bifurcation bifurcation NOUN iajs-3553 311 9 at at ADP iajs-3553 311 10 𝐴3 𝐴3 PROPN iajs-3553 311 11 where where SCONJ iajs-3553 311 12 οΏ½ οΏ½ PROPN iajs-3553 311 13 Μ… Μ… NOUN iajs-3553 311 14 οΏ½ οΏ½ NOUN iajs-3553 311 15 2 2 NUM iajs-3553 311 16 = = SYM iajs-3553 311 17 𝑑2 𝑑2 NOUN iajs-3553 311 18 . . PUNCT iajs-3553 312 1 theorem theorem NOUN iajs-3553 312 2 ( ( PUNCT iajs-3553 312 3 4 4 NUM iajs-3553 312 4 ): ): PUNCT iajs-3553 312 5 system system NOUN iajs-3553 312 6 ( ( PUNCT iajs-3553 312 7 1 1 NUM iajs-3553 312 8 ) ) PUNCT iajs-3553 312 9 with with ADP iajs-3553 312 10 the the DET iajs-3553 312 11 parameter parameter NOUN iajs-3553 312 12 value value PROPN iajs-3553 312 13 οΏ½ οΏ½ PROPN iajs-3553 312 14 ΜΏ ΜΏ NOUN iajs-3553 312 15 οΏ½ οΏ½ PROPN iajs-3553 312 16 1 1 NUM iajs-3553 312 17 = = SYM iajs-3553 312 18 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 312 19 = = SYM iajs-3553 312 20 𝛽1 𝛽1 PROPN iajs-3553 312 21 οΏ½ οΏ½ PROPN iajs-3553 312 22 ΜΏ ΜΏ NOUN iajs-3553 312 23 οΏ½ οΏ½ PROPN iajs-3553 312 24 2(1+𝐾1 2(1+𝐾1 SYM iajs-3553 312 25 οΏ½ οΏ½ PROPN iajs-3553 312 26 ΜΏ ΜΏ NOUN iajs-3553 312 27 οΏ½ οΏ½ NOUN iajs-3553 312 28 2 2 NUM iajs-3553 312 29 ) ) PUNCT iajs-3553 312 30 ( ( PUNCT iajs-3553 312 31 1+π‘Ž2 1+π‘Ž2 NUM iajs-3553 312 32 οΏ½ οΏ½ PROPN iajs-3553 312 33 ΜΏ ΜΏ NOUN iajs-3553 312 34 οΏ½ οΏ½ NOUN iajs-3553 312 35 2 2 NUM iajs-3553 312 36 ) ) PUNCT iajs-3553 312 37 , , PUNCT iajs-3553 312 38 has have VERB iajs-3553 312 39 a a DET iajs-3553 312 40 transcritical transcritical ADJ iajs-3553 312 41 and and CCONJ iajs-3553 312 42 pitchfork pitchfork NOUN iajs-3553 312 43 bifurcation bifurcation NOUN iajs-3553 312 44 at at ADP iajs-3553 312 45 𝐴4 𝐴4 PROPN iajs-3553 312 46 = = PUNCT iajs-3553 312 47 ( ( PUNCT iajs-3553 312 48 0 0 NUM iajs-3553 312 49 . . X iajs-3553 312 50 οΏ½ οΏ½ PROPN iajs-3553 312 51 ΜΏ ΜΏ NOUN iajs-3553 312 52 οΏ½ οΏ½ PROPN iajs-3553 312 53 2 2 NUM iajs-3553 312 54 . . PUNCT iajs-3553 312 55 οΏ½ οΏ½ PROPN iajs-3553 312 56 ΜΏ ΜΏ NOUN iajs-3553 312 57 οΏ½ οΏ½ PROPN iajs-3553 312 58 3 3 NUM iajs-3553 312 59 ) ) PUNCT iajs-3553 312 60 , , PUNCT iajs-3553 312 61 if if SCONJ iajs-3553 312 62 the the DET iajs-3553 312 63 next next ADJ iajs-3553 312 64 conditions condition NOUN iajs-3553 312 65 hold hold VERB iajs-3553 312 66 : : PUNCT iajs-3553 312 67 οΏ½ οΏ½ PROPN iajs-3553 312 68 ΜΏ ΜΏ NOUN iajs-3553 312 69 οΏ½ οΏ½ PROPN iajs-3553 312 70 1 1 NUM iajs-3553 312 71 β‰  β‰  PROPN iajs-3553 312 72 οΏ½ οΏ½ PROPN iajs-3553 312 73 ΜΏ ΜΏ NOUN iajs-3553 312 74 οΏ½ οΏ½ PROPN iajs-3553 312 75 2 2 NUM iajs-3553 312 76 . . PUNCT iajs-3553 313 1 ( ( PUNCT iajs-3553 313 2 17 17 NUM iajs-3553 313 3 ) ) PUNCT iajs-3553 313 4 οΏ½ οΏ½ PROPN iajs-3553 313 5 ΜΏ ΜΏ NOUN iajs-3553 313 6 οΏ½ οΏ½ PROPN iajs-3553 313 7 3 3 NUM iajs-3553 313 8 β‰  β‰  PROPN iajs-3553 313 9 οΏ½ οΏ½ PROPN iajs-3553 313 10 ΜΏ ΜΏ NOUN iajs-3553 313 11 οΏ½ οΏ½ PROPN iajs-3553 313 12 4 4 NUM iajs-3553 313 13 . . PUNCT iajs-3553 314 1 ( ( PUNCT iajs-3553 314 2 18 18 NUM iajs-3553 314 3 ) ) PUNCT iajs-3553 314 4 where where SCONJ iajs-3553 314 5 : : PUNCT iajs-3553 314 6 οΏ½ οΏ½ PROPN iajs-3553 314 7 ΜΏ ΜΏ NOUN iajs-3553 314 8 οΏ½ οΏ½ PROPN iajs-3553 314 9 1 1 NUM iajs-3553 314 10 = = SYM iajs-3553 314 11 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 314 12 2𝐼1 2𝐼1 NUM iajs-3553 314 13 2 2 NUM iajs-3553 314 14 ( ( PUNCT iajs-3553 314 15 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 314 16 + + CCONJ iajs-3553 314 17 𝛽1π‘Ž2𝐼1 𝛽1π‘Ž2𝐼1 ADJ iajs-3553 314 18 2 2 NUM iajs-3553 314 19 ( ( PUNCT iajs-3553 314 20 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 PROPN iajs-3553 314 21 + + NUM iajs-3553 314 22 𝛽1𝑅2π‘Ž1 𝛽1𝑅2π‘Ž1 PROPN iajs-3553 314 23 ( ( PUNCT iajs-3553 314 24 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)3 NUM iajs-3553 314 25 , , PUNCT iajs-3553 314 26 ihjpas ihjpas PROPN iajs-3553 314 27 . . PUNCT iajs-3553 315 1 2025 2025 NUM iajs-3553 315 2 , , PUNCT iajs-3553 315 3 38(2 38(2 NUM iajs-3553 315 4 ) ) PUNCT iajs-3553 315 5 323 323 NUM iajs-3553 315 6 οΏ½ οΏ½ PROPN iajs-3553 315 7 ΜΏ ΜΏ NOUN iajs-3553 315 8 οΏ½ οΏ½ PROPN iajs-3553 315 9 2 2 NUM iajs-3553 315 10 = = SYM iajs-3553 315 11 ( ( PUNCT iajs-3553 315 12 π‘š1 π‘š1 NOUN iajs-3553 315 13 + + CCONJ iajs-3553 315 14 π‘Ÿ1𝐾1𝐼1 π‘Ÿ1𝐾1𝐼1 NUM iajs-3553 315 15 ( ( PUNCT iajs-3553 315 16 1+𝐾1𝑅2)2 1+𝐾1𝑅2)2 NUM iajs-3553 315 17 + + NUM iajs-3553 315 18 𝛽1𝐼1 𝛽1𝐼1 NUM iajs-3553 315 19 ( ( PUNCT iajs-3553 315 20 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)2 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)2 NUM iajs-3553 315 21 ) ) PUNCT iajs-3553 315 22 , , PUNCT iajs-3553 315 23 οΏ½ οΏ½ PROPN iajs-3553 315 24 ΜΏ ΜΏ NOUN iajs-3553 315 25 οΏ½ οΏ½ PROPN iajs-3553 315 26 3 3 NUM iajs-3553 315 27 = = SYM iajs-3553 315 28 ( ( PUNCT iajs-3553 315 29 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 315 30 2 2 NUM iajs-3553 315 31 ( ( PUNCT iajs-3553 315 32 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 315 33 𝐼1 𝐼1 NOUN iajs-3553 315 34 2 2 NUM iajs-3553 315 35 + + CCONJ iajs-3553 315 36 𝛽1π‘Ž1 𝛽1π‘Ž1 X iajs-3553 315 37 ( ( PUNCT iajs-3553 315 38 1+π‘Ž1𝑅1)3(1+π‘Ž2𝑅2)2 1+π‘Ž1𝑅1)3(1+π‘Ž2𝑅2)2 NUM iajs-3553 315 39 𝐼1 𝐼1 NOUN iajs-3553 315 40 + + X iajs-3553 315 41 𝛽1π‘Ž2 𝛽1π‘Ž2 X iajs-3553 315 42 ( ( PUNCT iajs-3553 315 43 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)3 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2)3 NUM iajs-3553 315 44 𝐼1 𝐼1 NOUN iajs-3553 315 45 2 2 NUM iajs-3553 315 46 βˆ’ βˆ’ NOUN iajs-3553 315 47 Ξ²1r2π‘Ž1 Ξ²1r2π‘Ž1 PROPN iajs-3553 315 48 2 2 NUM iajs-3553 315 49 ( ( PUNCT iajs-3553 315 50 1+π‘Ž1𝑅1)4(1+a2r2 1+π‘Ž1𝑅1)4(1+a2r2 NUM iajs-3553 315 51 ) ) PUNCT iajs-3553 315 52 ) ) PUNCT iajs-3553 315 53 , , PUNCT iajs-3553 315 54 οΏ½ οΏ½ PROPN iajs-3553 315 55 ΜΏ ΜΏ NOUN iajs-3553 315 56 οΏ½ οΏ½ PROPN iajs-3553 315 57 4 4 NUM iajs-3553 315 58 = = SYM iajs-3553 315 59 ( ( PUNCT iajs-3553 315 60 Ξ²1r2π‘Ž1 Ξ²1r2π‘Ž1 PROPN iajs-3553 315 61 2 2 NUM iajs-3553 315 62 ( ( PUNCT iajs-3553 315 63 1+π‘Ž1𝑅1)4(1+a2r2 1+π‘Ž1𝑅1)4(1+a2r2 NUM iajs-3553 315 64 ) ) PUNCT iajs-3553 316 1 + + SYM iajs-3553 316 2 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 316 3 3𝑅1 3𝑅1 NUM iajs-3553 316 4 ( ( PUNCT iajs-3553 316 5 1+𝐾1𝑅2)4 1+𝐾1𝑅2)4 NUM iajs-3553 316 6 𝐼1 𝐼1 NOUN iajs-3553 316 7 3 3 NUM iajs-3553 316 8 + + CCONJ iajs-3553 316 9 𝛽1π‘Ž2 𝛽1π‘Ž2 PRON iajs-3553 316 10 2𝑅1 2𝑅1 NUM iajs-3553 316 11 ( ( PUNCT iajs-3553 316 12 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 NUM iajs-3553 316 13 𝐼1 𝐼1 NOUN iajs-3553 316 14 3 3 NUM iajs-3553 316 15 ) ) PUNCT iajs-3553 316 16 . . PUNCT iajs-3553 317 1 proof proof NOUN iajs-3553 317 2 : : PUNCT iajs-3553 317 3 by by ADP iajs-3553 317 4 using use VERB iajs-3553 317 5 the the DET iajs-3553 317 6 jacobian jacobian ADJ iajs-3553 317 7 matrix matrix NOUN iajs-3553 317 8 from from ADP iajs-3553 317 9 equation equation NOUN iajs-3553 317 10 ( ( PUNCT iajs-3553 317 11 1.10 1.10 NUM iajs-3553 317 12 ) ) PUNCT iajs-3553 317 13 in in ADP iajs-3553 317 14 ( ( PUNCT iajs-3553 317 15 20 20 NUM iajs-3553 317 16 ) ) PUNCT iajs-3553 317 17 𝐽4ΜΏ 𝐽4ΜΏ NOUN iajs-3553 318 1 = = SYM iajs-3553 318 2 𝐽4(𝐴4 𝐽4(𝐴4 PROPN iajs-3553 318 3 . . PUNCT iajs-3553 318 4 οΏ½ οΏ½ PROPN iajs-3553 318 5 ΜΏ ΜΏ NOUN iajs-3553 318 6 οΏ½ οΏ½ PROPN iajs-3553 318 7 1 1 NUM iajs-3553 318 8 ) ) PUNCT iajs-3553 318 9 = = PUNCT iajs-3553 319 1 [ [ PUNCT iajs-3553 319 2 οΏ½ οΏ½ PROPN iajs-3553 319 3 ΜΏ ΜΏ NOUN iajs-3553 319 4 οΏ½ οΏ½ NOUN iajs-3553 319 5 𝑖𝑗]3Γ—3 𝑖𝑗]3Γ—3 PROPN iajs-3553 319 6 , , PUNCT iajs-3553 319 7 where where SCONJ iajs-3553 319 8 οΏ½ οΏ½ PROPN iajs-3553 319 9 ΜΏ ΜΏ NOUN iajs-3553 319 10 οΏ½ οΏ½ PROPN iajs-3553 319 11 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 319 12 = = NOUN iajs-3553 319 13 𝑒𝑖𝑗 𝑒𝑖𝑗 PROPN iajs-3553 319 14 . . PUNCT iajs-3553 320 1 except except SCONJ iajs-3553 320 2 οΏ½ οΏ½ PROPN iajs-3553 320 3 ΜΏ ΜΏ NOUN iajs-3553 320 4 οΏ½ οΏ½ PROPN iajs-3553 320 5 11 11 NUM iajs-3553 320 6 = = SYM iajs-3553 320 7 0 0 NUM iajs-3553 320 8 .then .then NOUN iajs-3553 320 9 , , PUNCT iajs-3553 320 10 the the DET iajs-3553 320 11 characterising characterise VERB iajs-3553 320 12 equation equation NOUN iajs-3553 320 13 for for ADP iajs-3553 320 14 𝐽4ΜΏ 𝐽4ΜΏ PROPN iajs-3553 320 15 has have VERB iajs-3553 320 16 an an DET iajs-3553 320 17 eigenvalue eigenvalue NOUN iajs-3553 320 18 of of ADP iajs-3553 320 19 zero zero NUM iajs-3553 320 20 , , PUNCT iajs-3553 320 21 which which PRON iajs-3553 320 22 is be AUX iajs-3553 320 23 Ξ»4r1 Ξ»4r1 PRON iajs-3553 320 24 = = SYM iajs-3553 320 25 0 0 NUM iajs-3553 320 26 at at ADP iajs-3553 320 27 οΏ½ οΏ½ PROPN iajs-3553 320 28 ΜΏ ΜΏ NOUN iajs-3553 320 29 οΏ½ οΏ½ PROPN iajs-3553 320 30 1 1 NUM iajs-3553 320 31 = = SYM iajs-3553 320 32 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 320 33 . . PUNCT iajs-3553 321 1 now now ADV iajs-3553 321 2 , , PUNCT iajs-3553 321 3 let let VERB iajs-3553 321 4 𝑇[4 𝑇[4 PRON iajs-3553 321 5 ] ] X iajs-3553 322 1 = = PUNCT iajs-3553 322 2 ( ( PUNCT iajs-3553 322 3 𝑑1 𝑑1 NOUN iajs-3553 322 4 [ [ X iajs-3553 322 5 4 4 NUM iajs-3553 322 6 ] ] PUNCT iajs-3553 322 7 . . PUNCT iajs-3553 323 1 𝑑2 𝑑2 PROPN iajs-3553 324 1 [ [ X iajs-3553 324 2 4 4 NUM iajs-3553 324 3 ] ] PUNCT iajs-3553 324 4 . . PUNCT iajs-3553 325 1 𝑑3 𝑑3 PROPN iajs-3553 325 2 [ [ X iajs-3553 325 3 4 4 NUM iajs-3553 325 4 ] ] PUNCT iajs-3553 325 5 ) ) PUNCT iajs-3553 325 6 𝑇 𝑇 PROPN iajs-3553 325 7 be be VERB iajs-3553 325 8 the the DET iajs-3553 325 9 eigenvector eigenvector NOUN iajs-3553 325 10 associated associate VERB iajs-3553 325 11 with with ADP iajs-3553 325 12 an an DET iajs-3553 325 13 eigenvalue eigenvalue NOUN iajs-3553 325 14 Ξ»4r1 Ξ»4r1 X iajs-3553 325 15 = = NOUN iajs-3553 325 16 0 0 NUM iajs-3553 325 17 . . PUNCT iajs-3553 326 1 thus,(𝐽4ΜΏ thus,(𝐽4ΜΏ NOUN iajs-3553 326 2 βˆ’ βˆ’ PROPN iajs-3553 326 3 Ξ»4r1 Ξ»4r1 X iajs-3553 326 4 𝐼)𝑇[4 𝐼)𝑇[4 X iajs-3553 326 5 ] ] PUNCT iajs-3553 326 6 = = PUNCT iajs-3553 326 7 0 0 X iajs-3553 326 8 . . PUNCT iajs-3553 327 1 this this PRON iajs-3553 327 2 gives give VERB iajs-3553 327 3 : : PUNCT iajs-3553 327 4 𝑑2 𝑑2 PROPN iajs-3553 327 5 [ [ X iajs-3553 327 6 4 4 NUM iajs-3553 327 7 ] ] X iajs-3553 327 8 = = PUNCT iajs-3553 328 1 𝐼1𝑑1 𝐼1𝑑1 ADJ iajs-3553 328 2 [ [ X iajs-3553 328 3 4 4 NUM iajs-3553 328 4 ] ] PUNCT iajs-3553 328 5 . . PUNCT iajs-3553 329 1 𝑑3 𝑑3 PROPN iajs-3553 329 2 [ [ X iajs-3553 329 3 4 4 NUM iajs-3553 329 4 ] ] X iajs-3553 329 5 = = PUNCT iajs-3553 329 6 𝐼2𝑑1 𝐼2𝑑1 PROPN iajs-3553 329 7 [ [ X iajs-3553 329 8 4 4 NUM iajs-3553 329 9 ] ] PUNCT iajs-3553 329 10 . . PUNCT iajs-3553 330 1 where where SCONJ iajs-3553 330 2 : : PUNCT iajs-3553 330 3 𝐼1 𝐼1 NOUN iajs-3553 330 4 = = SYM iajs-3553 330 5 𝑒21𝑒33 𝑒21𝑒33 NOUN iajs-3553 330 6 𝑒23𝑒32βˆ’π‘’22𝑒33 𝑒23𝑒32βˆ’π‘’22𝑒33 NOUN iajs-3553 330 7 , , PUNCT iajs-3553 330 8 𝐼2 𝐼2 NOUN iajs-3553 330 9 = = PUNCT iajs-3553 330 10 𝑒32𝑒21 𝑒32𝑒21 ADJ iajs-3553 330 11 𝑒33𝑒22βˆ’π‘’23𝑒32 𝑒33𝑒22βˆ’π‘’23𝑒32 NOUN iajs-3553 330 12 . . PUNCT iajs-3553 331 1 and and CCONJ iajs-3553 331 2 𝑑1 𝑑1 VERB iajs-3553 332 1 [ [ X iajs-3553 332 2 4 4 X iajs-3553 332 3 ] ] PUNCT iajs-3553 332 4 any any DET iajs-3553 332 5 real real ADJ iajs-3553 332 6 number number NOUN iajs-3553 332 7 that that PRON iajs-3553 332 8 is be AUX iajs-3553 332 9 not not PART iajs-3553 332 10 zero zero NUM iajs-3553 332 11 . . PUNCT iajs-3553 333 1 let let VERB iajs-3553 333 2 𝑀[4 𝑀[4 PROPN iajs-3553 333 3 ] ] X iajs-3553 333 4 = = SYM iajs-3553 333 5 ( ( PUNCT iajs-3553 333 6 π‘š1 π‘š1 NOUN iajs-3553 333 7 [ [ X iajs-3553 333 8 4 4 NUM iajs-3553 333 9 ] ] PUNCT iajs-3553 333 10 . . PUNCT iajs-3553 334 1 π‘š2 π‘š2 NOUN iajs-3553 334 2 [ [ X iajs-3553 334 3 4 4 NUM iajs-3553 334 4 ] ] PUNCT iajs-3553 334 5 . . PUNCT iajs-3553 335 1 π‘š3 π‘š3 NOUN iajs-3553 336 1 [ [ X iajs-3553 336 2 4 4 NUM iajs-3553 336 3 ] ] PUNCT iajs-3553 336 4 ) ) PUNCT iajs-3553 336 5 𝑇 𝑇 PROPN iajs-3553 336 6 be be VERB iajs-3553 336 7 the the DET iajs-3553 336 8 eigenvector eigenvector NOUN iajs-3553 336 9 associated associate VERB iajs-3553 336 10 with with ADP iajs-3553 336 11 an an DET iajs-3553 336 12 eigenvalue eigenvalue NOUN iajs-3553 336 13 Ξ»4r1 Ξ»4r1 X iajs-3553 336 14 = = SYM iajs-3553 336 15 0 0 NUM iajs-3553 336 16 of of ADP iajs-3553 336 17 the the DET iajs-3553 336 18 matrix matrix NOUN iajs-3553 336 19 𝐽4ΜΏ 𝐽4ΜΏ NOUN iajs-3553 336 20 𝑇.then 𝑇.then NOUN iajs-3553 336 21 . . PUNCT iajs-3553 337 1 ( ( PUNCT iajs-3553 337 2 𝐽4ΜΏ 𝐽4ΜΏ PROPN iajs-3553 337 3 𝑇 𝑇 PROPN iajs-3553 337 4 βˆ’ βˆ’ PROPN iajs-3553 337 5 Ξ»4r1 Ξ»4r1 PRON iajs-3553 337 6 𝐼 𝐼 PROPN iajs-3553 337 7 ) ) PUNCT iajs-3553 337 8 𝑀[4 𝑀[4 PROPN iajs-3553 337 9 ] ] X iajs-3553 337 10 = = SYM iajs-3553 337 11 0 0 NUM iajs-3553 337 12 . . PUNCT iajs-3553 338 1 by by ADP iajs-3553 338 2 solving solve VERB iajs-3553 338 3 this this DET iajs-3553 338 4 equation equation NOUN iajs-3553 338 5 for for ADP iajs-3553 338 6 . . PUNCT iajs-3553 339 1 𝑀[4 𝑀[4 PROPN iajs-3553 339 2 ] ] X iajs-3553 340 1 = = SYM iajs-3553 340 2 ( ( PUNCT iajs-3553 340 3 π‘š1 π‘š1 NOUN iajs-3553 340 4 [ [ X iajs-3553 340 5 4 4 NUM iajs-3553 340 6 ] ] PUNCT iajs-3553 340 7 . . PUNCT iajs-3553 341 1 0 0 X iajs-3553 341 2 . . NOUN iajs-3553 341 3 0 0 X iajs-3553 341 4 ) ) PUNCT iajs-3553 341 5 𝑇 𝑇 NOUN iajs-3553 341 6 and and CCONJ iajs-3553 341 7 π‘š1 π‘š1 NOUN iajs-3553 342 1 [ [ X iajs-3553 342 2 4 4 X iajs-3553 342 3 ] ] PUNCT iajs-3553 342 4 any any DET iajs-3553 342 5 real real ADJ iajs-3553 342 6 number number NOUN iajs-3553 342 7 that that PRON iajs-3553 342 8 is be AUX iajs-3553 342 9 not not PART iajs-3553 342 10 zero zero NUM iajs-3553 342 11 . . PUNCT iajs-3553 343 1 now now ADV iajs-3553 343 2 consider consider VERB iajs-3553 343 3 this this PRON iajs-3553 343 4 : : PUNCT iajs-3553 343 5 πœ•π‘“ πœ•π‘“ ADJ iajs-3553 343 6 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3553 343 7 = = NOUN iajs-3553 343 8 π‘“π‘Ÿ1(𝑋. π‘“π‘Ÿ1(𝑋. NUM iajs-3553 343 9 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 343 10 ) ) PUNCT iajs-3553 343 11 = = SYM iajs-3553 343 12 ( ( PUNCT iajs-3553 343 13 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3553 343 14 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3553 343 15 . . PUNCT iajs-3553 344 1 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3553 344 2 πœ•π‘Ÿ1 πœ•π‘Ÿ1 PUNCT iajs-3553 344 3 . . PUNCT iajs-3553 345 1 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3553 345 2 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3553 345 3 ) ) PUNCT iajs-3553 345 4 𝑇 𝑇 PROPN iajs-3553 345 5 = = SYM iajs-3553 345 6 ( ( PUNCT iajs-3553 345 7 𝑅1 𝑅1 PROPN iajs-3553 345 8 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 345 9 . . PUNCT iajs-3553 346 1 0 0 NUM iajs-3553 346 2 . . NOUN iajs-3553 346 3 0 0 NUM iajs-3553 346 4 ) ) PUNCT iajs-3553 346 5 𝑇 𝑇 NOUN iajs-3553 346 6 . . PUNCT iajs-3553 347 1 so so ADV iajs-3553 347 2 . . PUNCT iajs-3553 348 1 π‘“π‘Ÿ1(𝐴4 π‘“π‘Ÿ1(𝐴4 PROPN iajs-3553 348 2 . . PUNCT iajs-3553 348 3 οΏ½ οΏ½ PROPN iajs-3553 348 4 ΜΏ ΜΏ NOUN iajs-3553 348 5 οΏ½ οΏ½ PROPN iajs-3553 348 6 1 1 NUM iajs-3553 348 7 ) ) PUNCT iajs-3553 348 8 = = SYM iajs-3553 348 9 ( ( PUNCT iajs-3553 348 10 0 0 NUM iajs-3553 348 11 . . NOUN iajs-3553 348 12 0 0 NUM iajs-3553 348 13 . . X iajs-3553 348 14 0)𝑇and 0)𝑇and PUNCT iajs-3553 349 1 hence hence ADV iajs-3553 349 2 ( ( PUNCT iajs-3553 349 3 𝑀[4])π‘‡π‘“π‘Ÿ1(𝐴4 𝑀[4])π‘‡π‘“π‘Ÿ1(𝐴4 PROPN iajs-3553 349 4 . . PUNCT iajs-3553 349 5 οΏ½ οΏ½ PROPN iajs-3553 349 6 ΜΏ ΜΏ NOUN iajs-3553 349 7 οΏ½ οΏ½ PROPN iajs-3553 349 8 1 1 NUM iajs-3553 349 9 ) ) PUNCT iajs-3553 349 10 = = SYM iajs-3553 350 1 0 0 X iajs-3553 350 2 . . X iajs-3553 350 3 using use VERB iajs-3553 350 4 sotomayor sotomayor NOUN iajs-3553 350 5 's 's PART iajs-3553 350 6 theorem theorem NOUN iajs-3553 350 7 , , PUNCT iajs-3553 350 8 it it PRON iajs-3553 350 9 is be AUX iajs-3553 350 10 impossible impossible ADJ iajs-3553 350 11 to to PART iajs-3553 350 12 satisfy satisfy VERB iajs-3553 350 13 the the DET iajs-3553 350 14 saddle saddle NOUN iajs-3553 350 15 - - PUNCT iajs-3553 350 16 node node NOUN iajs-3553 350 17 bifurcation bifurcation NOUN iajs-3553 350 18 condition condition NOUN iajs-3553 350 19 . . PUNCT iajs-3553 351 1 the the DET iajs-3553 351 2 first first ADJ iajs-3553 351 3 condition condition NOUN iajs-3553 351 4 for for ADP iajs-3553 351 5 transcritical transcritical ADJ iajs-3553 351 6 bifurcation bifurcation NOUN iajs-3553 351 7 is be AUX iajs-3553 351 8 therefore therefore ADV iajs-3553 351 9 satisfied satisfied ADJ iajs-3553 351 10 . . PUNCT iajs-3553 352 1 now now ADV iajs-3553 352 2 π·π‘“π‘Ÿ1(𝑋. π·π‘“π‘Ÿ1(𝑋. VERB iajs-3553 352 3 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 352 4 ) ) PUNCT iajs-3553 353 1 = = PUNCT iajs-3553 353 2 [ [ PUNCT iajs-3553 353 3 1 1 NUM iajs-3553 353 4 1 1 NUM iajs-3553 353 5 + + CCONJ iajs-3553 353 6 𝐾1 𝐾1 PROPN iajs-3553 353 7 οΏ½ οΏ½ PROPN iajs-3553 353 8 ΜΏ ΜΏ NOUN iajs-3553 353 9 οΏ½ οΏ½ PROPN iajs-3553 353 10 2 2 NUM iajs-3553 353 11 0 0 NUM iajs-3553 353 12 0 0 NUM iajs-3553 353 13 0 0 NUM iajs-3553 353 14 0 0 NUM iajs-3553 353 15 0 0 NUM iajs-3553 353 16 0 0 NUM iajs-3553 353 17 0 0 NUM iajs-3553 353 18 0 0 NUM iajs-3553 353 19 ] ] PUNCT iajs-3553 353 20 . . PUNCT iajs-3553 354 1 where where SCONJ iajs-3553 354 2 , , PUNCT iajs-3553 354 3 π·π‘“π‘Ÿ1(𝑋. π·π‘“π‘Ÿ1(𝑋. NOUN iajs-3553 354 4 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 354 5 ) ) PUNCT iajs-3553 354 6 represents represent VERB iajs-3553 354 7 the the DET iajs-3553 354 8 derivative derivative NOUN iajs-3553 354 9 of of ADP iajs-3553 354 10 π‘“π‘Ÿ1(𝑋. π‘“π‘Ÿ1(𝑋. NOUN iajs-3553 354 11 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 354 12 ) ) PUNCT iajs-3553 354 13 with with ADP iajs-3553 354 14 respect respect NOUN iajs-3553 354 15 to to ADP iajs-3553 354 16 𝑋 𝑋 PROPN iajs-3553 354 17 = = SYM iajs-3553 354 18 ( ( PUNCT iajs-3553 354 19 𝑅1 𝑅1 PROPN iajs-3553 354 20 . . PUNCT iajs-3553 354 21 𝑅2 𝑅2 PROPN iajs-3553 354 22 . . PUNCT iajs-3553 355 1 𝑅3 𝑅3 PROPN iajs-3553 355 2 ) ) PUNCT iajs-3553 355 3 𝑇 𝑇 PROPN iajs-3553 355 4 . . PUNCT iajs-3553 356 1 furthermore furthermore ADV iajs-3553 356 2 , , PUNCT iajs-3553 356 3 it it PRON iajs-3553 356 4 is be AUX iajs-3553 356 5 observed observe VERB iajs-3553 356 6 that that SCONJ iajs-3553 356 7 : : PUNCT iajs-3553 356 8 π·π‘“π‘Ÿ1(𝐴4 π·π‘“π‘Ÿ1(𝐴4 PROPN iajs-3553 356 9 . . PUNCT iajs-3553 357 1 π‘Ÿ1)𝑇 π‘Ÿ1)𝑇 X iajs-3553 358 1 [ [ X iajs-3553 358 2 4 4 X iajs-3553 358 3 ] ] X iajs-3553 358 4 = = PUNCT iajs-3553 358 5 [ [ PUNCT iajs-3553 358 6 1 1 NUM iajs-3553 358 7 1 1 NUM iajs-3553 358 8 + + CCONJ iajs-3553 358 9 𝐾1𝑅2 𝐾1𝑅2 ADP iajs-3553 358 10 0 0 NUM iajs-3553 358 11 0 0 NUM iajs-3553 358 12 0 0 NUM iajs-3553 358 13 0 0 NUM iajs-3553 358 14 0 0 NUM iajs-3553 358 15 0 0 NUM iajs-3553 358 16 0 0 NUM iajs-3553 358 17 0 0 NUM iajs-3553 358 18 ] ] PUNCT iajs-3553 359 1 [ [ PUNCT iajs-3553 359 2 𝑑1 𝑑1 NOUN iajs-3553 359 3 [ [ X iajs-3553 359 4 4 4 NUM iajs-3553 359 5 ] ] X iajs-3553 359 6 𝐼1𝑑1 𝐼1𝑑1 ADV iajs-3553 359 7 [ [ X iajs-3553 359 8 4 4 NUM iajs-3553 359 9 ] ] X iajs-3553 359 10 𝐼2𝑑1 𝐼2𝑑1 PROPN iajs-3553 359 11 [ [ X iajs-3553 359 12 4 4 NUM iajs-3553 359 13 ] ] PUNCT iajs-3553 359 14 ] ] PUNCT iajs-3553 360 1 = = PUNCT iajs-3553 360 2 [ [ PUNCT iajs-3553 360 3 𝑑1 𝑑1 NOUN iajs-3553 360 4 [ [ X iajs-3553 360 5 4 4 NUM iajs-3553 360 6 ] ] SYM iajs-3553 360 7 1 1 NUM iajs-3553 360 8 + + CCONJ iajs-3553 360 9 𝐾1 𝐾1 PROPN iajs-3553 360 10 οΏ½ οΏ½ PROPN iajs-3553 360 11 ΜΏ ΜΏ NOUN iajs-3553 360 12 οΏ½ οΏ½ PROPN iajs-3553 360 13 2 2 NUM iajs-3553 360 14 0 0 NUM iajs-3553 360 15 0 0 NUM iajs-3553 360 16 ] ] PUNCT iajs-3553 360 17 ( ( PUNCT iajs-3553 360 18 𝑀[4 𝑀[4 PROPN iajs-3553 360 19 ] ] X iajs-3553 360 20 ) ) PUNCT iajs-3553 360 21 𝑇 𝑇 PROPN iajs-3553 361 1 [ [ X iajs-3553 361 2 π·π‘“π‘Ÿ1(𝐴4 π·π‘“π‘Ÿ1(𝐴4 PROPN iajs-3553 361 3 . . PUNCT iajs-3553 361 4 οΏ½ οΏ½ PROPN iajs-3553 361 5 ΜΏ ΜΏ NOUN iajs-3553 361 6 οΏ½ οΏ½ PROPN iajs-3553 361 7 1)𝑇 1)𝑇 NOUN iajs-3553 361 8 [ [ X iajs-3553 361 9 4 4 NUM iajs-3553 361 10 ] ] X iajs-3553 361 11 ] ] X iajs-3553 361 12 = = SYM iajs-3553 361 13 ( ( PUNCT iajs-3553 361 14 π‘š1 π‘š1 NOUN iajs-3553 362 1 [ [ X iajs-3553 362 2 4 4 NUM iajs-3553 362 3 ] ] PUNCT iajs-3553 362 4 . . PUNCT iajs-3553 363 1 0 0 X iajs-3553 363 2 . . NOUN iajs-3553 363 3 0 0 NUM iajs-3553 363 4 ) ) PUNCT iajs-3553 363 5 ( ( PUNCT iajs-3553 363 6 𝑑1 𝑑1 NOUN iajs-3553 363 7 [ [ X iajs-3553 363 8 4 4 NUM iajs-3553 363 9 ] ] SYM iajs-3553 363 10 1+𝐾1 1+𝐾1 NUM iajs-3553 363 11 οΏ½ οΏ½ PROPN iajs-3553 363 12 ΜΏ ΜΏ NOUN iajs-3553 363 13 οΏ½ οΏ½ NOUN iajs-3553 363 14 2 2 NUM iajs-3553 363 15 . . NOUN iajs-3553 363 16 0 0 NUM iajs-3553 363 17 . . NOUN iajs-3553 363 18 0 0 X iajs-3553 363 19 ) ) PUNCT iajs-3553 363 20 𝑇 𝑇 NOUN iajs-3553 363 21 = = SYM iajs-3553 363 22 π‘š1 π‘š1 NOUN iajs-3553 364 1 [ [ X iajs-3553 364 2 4 4 NUM iajs-3553 364 3 ] ] X iajs-3553 364 4 𝑑1 𝑑1 NOUN iajs-3553 365 1 [ [ X iajs-3553 365 2 4 4 NUM iajs-3553 365 3 ] ] SYM iajs-3553 365 4 1+𝐾1 1+𝐾1 NUM iajs-3553 365 5 οΏ½ οΏ½ PROPN iajs-3553 365 6 ΜΏ ΜΏ NOUN iajs-3553 365 7 οΏ½ οΏ½ PROPN iajs-3553 365 8 2 2 NUM iajs-3553 365 9 β‰  β‰  PROPN iajs-3553 365 10 0 0 NUM iajs-3553 365 11 , , PUNCT iajs-3553 365 12 by by ADP iajs-3553 365 13 substituting substitute VERB iajs-3553 365 14 𝑇[4 𝑇[4 PROPN iajs-3553 365 15 ] ] PUNCT iajs-3553 365 16 in in ADP iajs-3553 365 17 equation equation NOUN iajs-3553 365 18 ( ( PUNCT iajs-3553 365 19 3 3 X iajs-3553 365 20 ) ) PUNCT iajs-3553 365 21 we we PRON iajs-3553 365 22 get get VERB iajs-3553 365 23 : : PUNCT iajs-3553 365 24 𝐷2πΉπœ‡(𝐴4 𝐷2πΉπœ‡(𝐴4 PROPN iajs-3553 365 25 . . PUNCT iajs-3553 365 26 οΏ½ οΏ½ PROPN iajs-3553 365 27 ΜΏ ΜΏ NOUN iajs-3553 365 28 οΏ½ οΏ½ NOUN iajs-3553 365 29 1)(𝑇 1)(𝑇 NUM iajs-3553 365 30 [ [ X iajs-3553 365 31 4 4 NUM iajs-3553 365 32 ] ] PUNCT iajs-3553 365 33 . . PUNCT iajs-3553 366 1 𝑇[4 𝑇[4 PROPN iajs-3553 366 2 ] ] PUNCT iajs-3553 366 3 ) ) PUNCT iajs-3553 366 4 = = SYM iajs-3553 366 5 ( ( PUNCT iajs-3553 366 6 οΏ½ οΏ½ PROPN iajs-3553 366 7 ΜΏ ΜΏ NOUN iajs-3553 366 8 οΏ½ οΏ½ PROPN iajs-3553 366 9 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 366 10 ) ) PUNCT iajs-3553 366 11 , , PUNCT iajs-3553 366 12 ( ( PUNCT iajs-3553 366 13 19 19 NUM iajs-3553 366 14 ) ) PUNCT iajs-3553 366 15 οΏ½ οΏ½ PROPN iajs-3553 366 16 ΜΏ ΜΏ NOUN iajs-3553 366 17 οΏ½ οΏ½ PROPN iajs-3553 366 18 11 11 NUM iajs-3553 366 19 = = SYM iajs-3553 366 20 2(𝑑1 2(𝑑1 NUM iajs-3553 367 1 [ [ X iajs-3553 367 2 4 4 NUM iajs-3553 367 3 ] ] PUNCT iajs-3553 367 4 ) ) PUNCT iajs-3553 367 5 2 2 NUM iajs-3553 368 1 [ [ X iajs-3553 368 2 ( ( PUNCT iajs-3553 368 3 𝑅2π‘Ž1𝐼1 𝑅2π‘Ž1𝐼1 ADJ iajs-3553 368 4 2 2 NUM iajs-3553 368 5 ( ( PUNCT iajs-3553 368 6 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 368 7 βˆ’ βˆ’ NUM iajs-3553 368 8 𝐼1𝐼2 𝐼1𝐼2 PROPN iajs-3553 368 9 ( ( PUNCT iajs-3553 368 10 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 368 11 ) ) PUNCT iajs-3553 368 12 + + CCONJ iajs-3553 368 13 π‘Ž2𝑅2𝐼2 π‘Ž2𝑅2𝐼2 PROPN iajs-3553 368 14 2 2 NUM iajs-3553 368 15 ( ( PUNCT iajs-3553 368 16 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 368 17 ) ) PUNCT iajs-3553 368 18 𝛽1 𝛽1 NOUN iajs-3553 368 19 ( ( PUNCT iajs-3553 368 20 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 368 21 ) ) PUNCT iajs-3553 368 22 βˆ’ βˆ’ PROPN iajs-3553 368 23 π‘š1𝐼1 π‘š1𝐼1 SYM iajs-3553 368 24 2 2 NUM iajs-3553 368 25 + + CCONJ iajs-3553 368 26 ( ( PUNCT iajs-3553 368 27 𝐾1𝑅1𝐼1 𝐾1𝑅1𝐼1 PROPN iajs-3553 368 28 2 2 NUM iajs-3553 368 29 ( ( PUNCT iajs-3553 368 30 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 368 31 ) ) PUNCT iajs-3553 368 32 βˆ’πΌ1𝐼2 βˆ’πΌ1𝐼2 NUM iajs-3553 368 33 ) ) PUNCT iajs-3553 368 34 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 PROPN iajs-3553 368 35 ( ( PUNCT iajs-3553 368 36 1+𝐾1𝑅2)2 1+𝐾1𝑅2)2 NUM iajs-3553 368 37 ] ] PUNCT iajs-3553 368 38 , , PUNCT iajs-3553 368 39 οΏ½ οΏ½ PROPN iajs-3553 368 40 ΜΏ ΜΏ NOUN iajs-3553 368 41 οΏ½ οΏ½ PROPN iajs-3553 368 42 21 21 NUM iajs-3553 368 43 = = SYM iajs-3553 368 44 2(𝑑1 2(𝑑1 NUM iajs-3553 369 1 [ [ X iajs-3553 369 2 4 4 NUM iajs-3553 369 3 ] ] PUNCT iajs-3553 369 4 ) ) PUNCT iajs-3553 369 5 2 2 NUM iajs-3553 370 1 [ [ X iajs-3553 370 2 ( ( PUNCT iajs-3553 370 3 𝐼1𝐼2 𝐼1𝐼2 X iajs-3553 370 4 ( ( PUNCT iajs-3553 370 5 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 370 6 ) ) PUNCT iajs-3553 370 7 βˆ’ βˆ’ NOUN iajs-3553 370 8 r2π‘Ž1𝐼1 r2π‘Ž1𝐼1 NOUN iajs-3553 370 9 2 2 NUM iajs-3553 370 10 ( ( PUNCT iajs-3553 370 11 1+a1r1)2 1+a1r1)2 NUM iajs-3553 370 12 βˆ’ βˆ’ NOUN iajs-3553 370 13 r1π‘Ž2𝐼2 r1π‘Ž2𝐼2 NOUN iajs-3553 370 14 2 2 NUM iajs-3553 370 15 ( ( PUNCT iajs-3553 370 16 1+a2r2)2 1+a2r2)2 NUM iajs-3553 370 17 ) ) PUNCT iajs-3553 370 18 𝑙1 𝑙1 NOUN iajs-3553 370 19 ( ( PUNCT iajs-3553 370 20 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 370 21 ) ) PUNCT iajs-3553 370 22 + + CCONJ iajs-3553 370 23 ( ( PUNCT iajs-3553 370 24 𝐾2𝑅2 𝐾2𝑅2 NOUN iajs-3553 370 25 ( ( PUNCT iajs-3553 370 26 1+k2r3 1+k2r3 NUM iajs-3553 370 27 ) ) PUNCT iajs-3553 370 28 βˆ’ βˆ’ NOUN iajs-3553 370 29 𝐼2 𝐼2 NOUN iajs-3553 370 30 ) ) PUNCT iajs-3553 370 31 r2k2 r2k2 NOUN iajs-3553 370 32 ( ( PUNCT iajs-3553 370 33 1+k2r3)2 1+k2r3)2 NOUN iajs-3553 370 34 + + CCONJ iajs-3553 370 35 ( ( PUNCT iajs-3553 370 36 a3r3𝐼2 a3r3𝐼2 ADJ iajs-3553 370 37 2 2 NUM iajs-3553 370 38 ( ( PUNCT iajs-3553 370 39 1+a3r2)2 1+a3r2)2 NUM iajs-3553 370 40 + + SYM iajs-3553 370 41 𝐼2 𝐼2 NOUN iajs-3553 370 42 ( ( PUNCT iajs-3553 370 43 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 370 44 ) ) PUNCT iajs-3553 370 45 + + CCONJ iajs-3553 370 46 a4𝑅2 a4𝑅2 PRON iajs-3553 370 47 ( ( PUNCT iajs-3553 370 48 1+a4r3)2 1+a4r3)2 X iajs-3553 370 49 ) ) PUNCT iajs-3553 370 50 𝛽2 𝛽2 PROPN iajs-3553 370 51 ( ( PUNCT iajs-3553 370 52 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 370 53 ) ) PUNCT iajs-3553 370 54 βˆ’π‘š2𝐼2 βˆ’π‘š2𝐼2 ADP iajs-3553 370 55 2 2 NUM iajs-3553 370 56 ] ] PUNCT iajs-3553 370 57 , , PUNCT iajs-3553 370 58 ihjpas ihjpas PROPN iajs-3553 370 59 . . PUNCT iajs-3553 371 1 2025 2025 NUM iajs-3553 371 2 , , PUNCT iajs-3553 371 3 38(2 38(2 NUM iajs-3553 371 4 ) ) PUNCT iajs-3553 371 5 324 324 NUM iajs-3553 371 6 οΏ½ οΏ½ PROPN iajs-3553 371 7 ΜΏ ΜΏ NOUN iajs-3553 371 8 οΏ½ οΏ½ PROPN iajs-3553 371 9 31 31 NUM iajs-3553 371 10 = = SYM iajs-3553 372 1 2(𝑑1 2(𝑑1 NUM iajs-3553 373 1 [ [ X iajs-3553 373 2 4 4 NUM iajs-3553 373 3 ] ] PUNCT iajs-3553 373 4 ) ) PUNCT iajs-3553 373 5 2 2 NUM iajs-3553 373 6 𝑙2 𝑙2 NOUN iajs-3553 373 7 ( ( PUNCT iajs-3553 373 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 373 9 ) ) PUNCT iajs-3553 373 10 ( ( PUNCT iajs-3553 373 11 𝐼2 𝐼2 NOUN iajs-3553 373 12 ( ( PUNCT iajs-3553 373 13 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 373 14 ) ) PUNCT iajs-3553 373 15 βˆ’ βˆ’ NOUN iajs-3553 374 1 𝑅3π‘Ž3𝐼2 𝑅3π‘Ž3𝐼2 ADJ iajs-3553 374 2 2 2 NUM iajs-3553 374 3 ( ( PUNCT iajs-3553 374 4 1+a3r2)2 1+a3r2)2 NUM iajs-3553 374 5 βˆ’ βˆ’ NOUN iajs-3553 374 6 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 374 7 ( ( PUNCT iajs-3553 374 8 1+a4r3)2 1+a4r3)2 NUM iajs-3553 374 9 ) ) PUNCT iajs-3553 374 10 . . PUNCT iajs-3553 375 1 hence hence ADV iajs-3553 375 2 , , PUNCT iajs-3553 375 3 it it PRON iajs-3553 375 4 was be AUX iajs-3553 375 5 obtained obtain VERB iajs-3553 375 6 ( ( PUNCT iajs-3553 375 7 𝑀[4 𝑀[4 PROPN iajs-3553 375 8 ] ] X iajs-3553 375 9 ) ) PUNCT iajs-3553 375 10 𝑇 𝑇 PROPN iajs-3553 376 1 [ [ X iajs-3553 376 2 𝐷2πΉπœ‡(𝐴4 𝐷2πΉπœ‡(𝐴4 PROPN iajs-3553 376 3 . . PUNCT iajs-3553 376 4 οΏ½ οΏ½ PROPN iajs-3553 376 5 ΜΏ ΜΏ NOUN iajs-3553 376 6 οΏ½ οΏ½ NOUN iajs-3553 376 7 1)(𝑇 1)(𝑇 NUM iajs-3553 376 8 [ [ X iajs-3553 376 9 4 4 NUM iajs-3553 376 10 ] ] PUNCT iajs-3553 376 11 . . PUNCT iajs-3553 377 1 𝑇[4 𝑇[4 PROPN iajs-3553 377 2 ] ] PUNCT iajs-3553 377 3 ) ) PUNCT iajs-3553 377 4 ] ] PUNCT iajs-3553 378 1 = = SYM iajs-3553 378 2 2 2 X iajs-3553 378 3 ( ( PUNCT iajs-3553 378 4 𝑑1 𝑑1 NOUN iajs-3553 378 5 [ [ X iajs-3553 378 6 4 4 NUM iajs-3553 378 7 ] ] PUNCT iajs-3553 378 8 ) ) PUNCT iajs-3553 378 9 2 2 NUM iajs-3553 378 10 π‘š1 π‘š1 NOUN iajs-3553 378 11 [ [ X iajs-3553 378 12 4 4 NUM iajs-3553 378 13 ] ] PUNCT iajs-3553 378 14 ( ( PUNCT iajs-3553 378 15 𝑍1 𝑍1 PROPN iajs-3553 378 16 βˆ’ βˆ’ PROPN iajs-3553 378 17 𝑍2 𝑍2 PROPN iajs-3553 378 18 ) ) PUNCT iajs-3553 378 19 β‰  β‰  PROPN iajs-3553 378 20 0 0 NUM iajs-3553 378 21 . . PUNCT iajs-3553 379 1 this this PRON iajs-3553 379 2 indicates indicate VERB iajs-3553 379 3 that that SCONJ iajs-3553 379 4 the the DET iajs-3553 379 5 system system NOUN iajs-3553 379 6 ( ( PUNCT iajs-3553 379 7 1 1 X iajs-3553 379 8 ) ) PUNCT iajs-3553 379 9 exhibits exhibit VERB iajs-3553 379 10 a a DET iajs-3553 379 11 transcritical transcritical ADJ iajs-3553 379 12 bifurcation bifurcation NOUN iajs-3553 379 13 at at ADP iajs-3553 379 14 𝐴4 𝐴4 PROPN iajs-3553 379 15 with with ADP iajs-3553 379 16 a a DET iajs-3553 379 17 parameter parameter NOUN iajs-3553 379 18 οΏ½ οΏ½ PROPN iajs-3553 379 19 ΜΏ ΜΏ NOUN iajs-3553 379 20 οΏ½ οΏ½ PROPN iajs-3553 379 21 1 1 NUM iajs-3553 379 22 = = SYM iajs-3553 379 23 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 379 24 , , PUNCT iajs-3553 379 25 and and CCONJ iajs-3553 379 26 no no DET iajs-3553 379 27 pitch pitch NOUN iajs-3553 379 28 fork fork NOUN iajs-3553 379 29 bifurcation bifurcation NOUN iajs-3553 379 30 at at ADP iajs-3553 379 31 𝐴4 𝐴4 PROPN iajs-3553 379 32 where where SCONJ iajs-3553 379 33 οΏ½ οΏ½ PROPN iajs-3553 379 34 ΜΏ ΜΏ NOUN iajs-3553 379 35 οΏ½ οΏ½ PROPN iajs-3553 379 36 1 1 NUM iajs-3553 379 37 = = SYM iajs-3553 379 38 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 379 39 . . PUNCT iajs-3553 380 1 by by ADP iajs-3553 380 2 substituting substitute VERB iajs-3553 380 3 𝑇[4 𝑇[4 PROPN iajs-3553 380 4 ] ] PUNCT iajs-3553 380 5 in in ADP iajs-3553 380 6 equation equation NOUN iajs-3553 380 7 ( ( PUNCT iajs-3553 380 8 4 4 X iajs-3553 380 9 ) ) PUNCT iajs-3553 380 10 we we PRON iajs-3553 380 11 get get VERB iajs-3553 380 12 : : PUNCT iajs-3553 380 13 𝐷3πΉπœ‡(𝐴4 𝐷3πΉπœ‡(𝐴4 PROPN iajs-3553 380 14 . . PROPN iajs-3553 380 15 οΏ½ οΏ½ PROPN iajs-3553 380 16 ΜΏ ΜΏ NOUN iajs-3553 380 17 οΏ½ οΏ½ NOUN iajs-3553 380 18 1)(𝑇 1)(𝑇 NUM iajs-3553 380 19 [ [ X iajs-3553 380 20 4 4 NUM iajs-3553 380 21 ] ] PUNCT iajs-3553 380 22 . . PUNCT iajs-3553 381 1 𝑇[4 𝑇[4 PROPN iajs-3553 381 2 ] ] PUNCT iajs-3553 381 3 . . PUNCT iajs-3553 382 1 𝑇[4 𝑇[4 PROPN iajs-3553 382 2 ] ] PUNCT iajs-3553 382 3 ) ) PUNCT iajs-3553 382 4 = = SYM iajs-3553 382 5 ( ( PUNCT iajs-3553 382 6 οΏ½ οΏ½ PROPN iajs-3553 382 7 ΜΏ ΜΏ NOUN iajs-3553 382 8 οΏ½ οΏ½ PROPN iajs-3553 382 9 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 382 10 ) ) PUNCT iajs-3553 382 11 , , PUNCT iajs-3553 382 12 ( ( PUNCT iajs-3553 382 13 20 20 NUM iajs-3553 382 14 ) ) PUNCT iajs-3553 382 15 οΏ½ οΏ½ PROPN iajs-3553 382 16 ΜΏ ΜΏ NOUN iajs-3553 382 17 οΏ½ οΏ½ PROPN iajs-3553 382 18 11 11 NUM iajs-3553 382 19 = = SYM iajs-3553 382 20 6(𝑑1 6(𝑑1 NUM iajs-3553 383 1 [ [ X iajs-3553 383 2 4 4 NUM iajs-3553 383 3 ] ] PUNCT iajs-3553 383 4 ) ) PUNCT iajs-3553 383 5 3 3 NUM iajs-3553 384 1 [ [ X iajs-3553 384 2 ( ( PUNCT iajs-3553 384 3 π‘Ž2𝐼1𝐼2 π‘Ž2𝐼1𝐼2 ADP iajs-3553 384 4 2 2 NUM iajs-3553 384 5 ( ( PUNCT iajs-3553 384 6 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)2 NUM iajs-3553 384 7 βˆ’ βˆ’ NOUN iajs-3553 384 8 𝐼2 𝐼2 NOUN iajs-3553 384 9 3π‘Ž2 3π‘Ž2 NUM iajs-3553 384 10 2𝑅1 2𝑅1 NUM iajs-3553 384 11 ( ( PUNCT iajs-3553 384 12 1+π‘Ž2𝑅2)3 1+π‘Ž2𝑅2)3 NUM iajs-3553 384 13 βˆ’ βˆ’ NOUN iajs-3553 384 14 r2π‘Ž1 r2π‘Ž1 NOUN iajs-3553 384 15 2𝐼1 2𝐼1 NUM iajs-3553 384 16 3 3 NUM iajs-3553 384 17 ( ( PUNCT iajs-3553 384 18 1+π‘Ž1𝑅1)3 1+π‘Ž1𝑅1)3 NUM iajs-3553 384 19 + + CCONJ iajs-3553 384 20 π‘Ž1𝐼1 π‘Ž1𝐼1 NUM iajs-3553 384 21 2𝐼2 2𝐼2 NUM iajs-3553 384 22 ( ( PUNCT iajs-3553 384 23 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)2(1+π‘Ž2𝑅2 NUM iajs-3553 384 24 ) ) PUNCT iajs-3553 384 25 ) ) PUNCT iajs-3553 384 26 𝛽1 𝛽1 NOUN iajs-3553 384 27 ( ( PUNCT iajs-3553 384 28 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 384 29 ) ) PUNCT iajs-3553 384 30 + + CCONJ iajs-3553 384 31 ( ( PUNCT iajs-3553 384 32 𝐼1 𝐼1 NOUN iajs-3553 384 33 βˆ’ βˆ’ X iajs-3553 385 1 𝐾1𝑅1𝐼1 𝐾1𝑅1𝐼1 PROPN iajs-3553 386 1 ( ( PUNCT iajs-3553 386 2 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 386 3 ) ) PUNCT iajs-3553 386 4 ) ) PUNCT iajs-3553 387 1 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 387 2 2𝐼2 2𝐼2 NUM iajs-3553 387 3 2 2 NUM iajs-3553 387 4 ( ( PUNCT iajs-3553 387 5 1+𝐾1𝑅2)3 1+𝐾1𝑅2)3 NUM iajs-3553 387 6 ] ] X iajs-3553 387 7 , , PUNCT iajs-3553 387 8 οΏ½ οΏ½ PROPN iajs-3553 387 9 ΜΏ ΜΏ NOUN iajs-3553 387 10 οΏ½ οΏ½ PROPN iajs-3553 387 11 21 21 NUM iajs-3553 387 12 = = SYM iajs-3553 387 13 6(𝑑1 6(𝑑1 NUM iajs-3553 388 1 [ [ X iajs-3553 388 2 4 4 NUM iajs-3553 388 3 ] ] PUNCT iajs-3553 388 4 ) ) PUNCT iajs-3553 388 5 3 3 NUM iajs-3553 389 1 [ [ X iajs-3553 389 2 ( ( PUNCT iajs-3553 389 3 𝐼2 𝐼2 PROPN iajs-3553 389 4 βˆ’ βˆ’ PROPN iajs-3553 389 5 k2𝑅2 k2𝑅2 PROPN iajs-3553 389 6 ( ( PUNCT iajs-3553 389 7 1+k2r3 1+k2r3 NUM iajs-3553 389 8 ) ) PUNCT iajs-3553 389 9 ) ) PUNCT iajs-3553 389 10 r2k2 r2k2 X iajs-3553 389 11 2 2 NUM iajs-3553 389 12 ( ( PUNCT iajs-3553 389 13 1+k2r3)3 1+k2r3)3 VERB iajs-3553 389 14 βˆ’ βˆ’ PROPN iajs-3553 389 15 𝛽1π‘Ž2 𝛽1π‘Ž2 ADP iajs-3553 389 16 2𝑅1 2𝑅1 NUM iajs-3553 389 17 ( ( PUNCT iajs-3553 389 18 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2)4 NUM iajs-3553 389 19 + + CCONJ iajs-3553 389 20 ( ( PUNCT iajs-3553 389 21 r2π‘Ž1 r2π‘Ž1 NUM iajs-3553 389 22 2𝐼1 2𝐼1 NUM iajs-3553 389 23 3 3 NUM iajs-3553 389 24 ( ( PUNCT iajs-3553 389 25 1+a1r1)3 1+a1r1)3 NOUN iajs-3553 390 1 + + CCONJ iajs-3553 390 2 π‘Ž1𝐼1 π‘Ž1𝐼1 NUM iajs-3553 390 3 2𝐼2 2𝐼2 NUM iajs-3553 390 4 ( ( PUNCT iajs-3553 390 5 1+a1r1)2(1+a2r2 1+a1r1)2(1+a2r2 NUM iajs-3553 390 6 ) ) PUNCT iajs-3553 390 7 βˆ’ βˆ’ NOUN iajs-3553 391 1 π‘Ž2𝐼1𝐼2 π‘Ž2𝐼1𝐼2 ADP iajs-3553 391 2 2 2 NUM iajs-3553 391 3 ( ( PUNCT iajs-3553 391 4 1+a1r1)(1+a2r2)2 1+a1r1)(1+a2r2)2 NUM iajs-3553 391 5 + + CCONJ iajs-3553 391 6 r1π‘Ž2 r1π‘Ž2 PROPN iajs-3553 391 7 2𝐼2 2𝐼2 NUM iajs-3553 391 8 3 3 NUM iajs-3553 391 9 ( ( PUNCT iajs-3553 391 10 1+a2r2)3 1+a2r2)3 NUM iajs-3553 391 11 ) ) PUNCT iajs-3553 391 12 𝑙1 𝑙1 NOUN iajs-3553 391 13 ( ( PUNCT iajs-3553 391 14 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 391 15 ) ) PUNCT iajs-3553 391 16 + + CCONJ iajs-3553 391 17 ( ( PUNCT iajs-3553 391 18 π‘Ž4𝐼2 π‘Ž4𝐼2 X iajs-3553 391 19 ( ( PUNCT iajs-3553 391 20 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3)2 1+π‘Ž3𝑅2)(1+π‘Ž4𝑅3)2 NUM iajs-3553 391 21 + + CCONJ iajs-3553 391 22 a4𝐼2 a4𝐼2 PROPN iajs-3553 391 23 2 2 NUM iajs-3553 391 24 ( ( PUNCT iajs-3553 391 25 1+a3r2)(1+a4r3)2 1+a3r2)(1+a4r3)2 NUM iajs-3553 391 26 βˆ’ βˆ’ NOUN iajs-3553 391 27 r3π‘Ž3 r3π‘Ž3 NOUN iajs-3553 391 28 2𝐼2 2𝐼2 NUM iajs-3553 391 29 3 3 NUM iajs-3553 391 30 ( ( PUNCT iajs-3553 391 31 1+a3r2)3 1+a3r2)3 NUM iajs-3553 391 32 ) ) PUNCT iajs-3553 391 33 𝛽2𝐼 𝛽2𝐼 ADV iajs-3553 391 34 ( ( PUNCT iajs-3553 391 35 1+π‘Ž3𝑅2)2(1+π‘Ž4𝑅3 1+π‘Ž3𝑅2)2(1+π‘Ž4𝑅3 NUM iajs-3553 391 36 ) ) PUNCT iajs-3553 391 37 ] ] PUNCT iajs-3553 391 38 , , PUNCT iajs-3553 391 39 οΏ½ οΏ½ PROPN iajs-3553 391 40 ΜΏ ΜΏ NOUN iajs-3553 391 41 οΏ½ οΏ½ PROPN iajs-3553 391 42 31 31 NUM iajs-3553 391 43 = = SYM iajs-3553 392 1 6(𝑑1 6(𝑑1 NUM iajs-3553 393 1 [ [ X iajs-3553 393 2 4 4 NUM iajs-3553 393 3 ] ] PUNCT iajs-3553 393 4 ) ) PUNCT iajs-3553 393 5 3 3 NUM iajs-3553 393 6 𝑙2 𝑙2 NOUN iajs-3553 393 7 ( ( PUNCT iajs-3553 393 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 393 9 ) ) PUNCT iajs-3553 393 10 ( ( PUNCT iajs-3553 393 11 𝑅3π‘Ž3 𝑅3π‘Ž3 NOUN iajs-3553 393 12 2𝐼2 2𝐼2 NUM iajs-3553 393 13 3 3 NUM iajs-3553 393 14 ( ( PUNCT iajs-3553 393 15 1+a3r2)3 1+a3r2)3 NUM iajs-3553 393 16 + + NUM iajs-3553 393 17 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 393 18 2 2 NUM iajs-3553 393 19 ( ( PUNCT iajs-3553 393 20 1+a4r3)3 1+a4r3)3 NUM iajs-3553 393 21 βˆ’ βˆ’ PROPN iajs-3553 393 22 π‘Ž4𝐼2 π‘Ž4𝐼2 PROPN iajs-3553 393 23 ( ( PUNCT iajs-3553 393 24 1+a3r2)(1+a4r3)2 1+a3r2)(1+a4r3)2 NUM iajs-3553 393 25 βˆ’ βˆ’ NOUN iajs-3553 393 26 π‘Ž3𝐼2 π‘Ž3𝐼2 PROPN iajs-3553 393 27 2 2 NUM iajs-3553 393 28 ( ( PUNCT iajs-3553 393 29 1+a3r2)2(1+a4r3 1+a3r2)2(1+a4r3 NUM iajs-3553 393 30 ) ) PUNCT iajs-3553 393 31 ) ) PUNCT iajs-3553 393 32 . . PUNCT iajs-3553 394 1 hence hence ADV iajs-3553 394 2 , , PUNCT iajs-3553 394 3 ( ( PUNCT iajs-3553 394 4 𝑀[4 𝑀[4 X iajs-3553 394 5 ] ] X iajs-3553 394 6 ) ) PUNCT iajs-3553 394 7 𝑇 𝑇 PROPN iajs-3553 395 1 [ [ X iajs-3553 395 2 𝐷3πΉπœ‡(𝐴4 𝐷3πΉπœ‡(𝐴4 PROPN iajs-3553 395 3 . . PROPN iajs-3553 395 4 οΏ½ οΏ½ PROPN iajs-3553 395 5 ΜΏ ΜΏ NOUN iajs-3553 395 6 οΏ½ οΏ½ NOUN iajs-3553 395 7 1)(𝑇 1)(𝑇 NUM iajs-3553 395 8 [ [ X iajs-3553 395 9 4 4 NUM iajs-3553 395 10 ] ] PUNCT iajs-3553 395 11 . . PUNCT iajs-3553 396 1 𝑇[4 𝑇[4 PROPN iajs-3553 396 2 ] ] PUNCT iajs-3553 396 3 . . PUNCT iajs-3553 397 1 𝑇[4 𝑇[4 PROPN iajs-3553 397 2 ] ] PUNCT iajs-3553 397 3 ) ) PUNCT iajs-3553 397 4 ] ] PUNCT iajs-3553 398 1 = = SYM iajs-3553 398 2 6 6 NUM iajs-3553 398 3 ( ( PUNCT iajs-3553 398 4 𝑑1 𝑑1 NOUN iajs-3553 398 5 [ [ X iajs-3553 398 6 4 4 NUM iajs-3553 398 7 ] ] PUNCT iajs-3553 398 8 ) ) PUNCT iajs-3553 398 9 3 3 NUM iajs-3553 398 10 π‘š1 π‘š1 NOUN iajs-3553 399 1 [ [ X iajs-3553 399 2 4 4 NUM iajs-3553 399 3 ] ] PUNCT iajs-3553 399 4 ( ( PUNCT iajs-3553 399 5 οΏ½ οΏ½ PROPN iajs-3553 399 6 ΜΏ ΜΏ NOUN iajs-3553 399 7 οΏ½ οΏ½ PROPN iajs-3553 399 8 3 3 NUM iajs-3553 399 9 βˆ’ βˆ’ PROPN iajs-3553 399 10 οΏ½ οΏ½ PROPN iajs-3553 399 11 ΜΏ ΜΏ NOUN iajs-3553 399 12 οΏ½ οΏ½ NOUN iajs-3553 399 13 4 4 NUM iajs-3553 399 14 ) ) PUNCT iajs-3553 399 15 β‰  β‰  PROPN iajs-3553 399 16 0 0 NUM iajs-3553 400 1 if if SCONJ iajs-3553 400 2 condition condition NOUN iajs-3553 400 3 ( ( PUNCT iajs-3553 400 4 17 17 NUM iajs-3553 400 5 ) ) PUNCT iajs-3553 400 6 . . PUNCT iajs-3553 401 1 which which DET iajs-3553 401 2 grantee grantee NOUN iajs-3553 401 3 that that SCONJ iajs-3553 401 4 there there PRON iajs-3553 401 5 is be VERB iajs-3553 401 6 pitch pitch NOUN iajs-3553 401 7 fork fork NOUN iajs-3553 401 8 bifurcation bifurcation NOUN iajs-3553 401 9 at at ADP iajs-3553 401 10 𝐴4 𝐴4 PROPN iajs-3553 401 11 where where SCONJ iajs-3553 401 12 οΏ½ οΏ½ PROPN iajs-3553 401 13 ΜΏ ΜΏ NOUN iajs-3553 401 14 οΏ½ οΏ½ PROPN iajs-3553 401 15 1 1 NUM iajs-3553 401 16 = = SYM iajs-3553 401 17 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 401 18 . . PUNCT iajs-3553 402 1 theorem theorem NOUN iajs-3553 402 2 ( ( PUNCT iajs-3553 402 3 5 5 NUM iajs-3553 402 4 ): ): PUNCT iajs-3553 402 5 suppose suppose VERB iajs-3553 402 6 that that SCONJ iajs-3553 402 7 conditions condition NOUN iajs-3553 402 8 ( ( PUNCT iajs-3553 402 9 1.11b 1.11b NUM iajs-3553 402 10 ) ) PUNCT iajs-3553 402 11 , , PUNCT iajs-3553 402 12 ( ( PUNCT iajs-3553 402 13 1.11e 1.11e NUM iajs-3553 402 14 ) ) PUNCT iajs-3553 402 15 and and CCONJ iajs-3553 402 16 , , PUNCT iajs-3553 402 17 reverse reverse ADJ iajs-3553 402 18 condition condition NOUN iajs-3553 402 19 ( ( PUNCT iajs-3553 402 20 1.11c 1.11c NOUN iajs-3553 402 21 ) ) PUNCT iajs-3553 402 22 in in ADP iajs-3553 402 23 ( ( PUNCT iajs-3553 402 24 17 17 NUM iajs-3553 402 25 ) ) PUNCT iajs-3553 402 26 with with ADP iajs-3553 402 27 the the DET iajs-3553 402 28 following follow VERB iajs-3553 402 29 conditions condition NOUN iajs-3553 402 30 are be AUX iajs-3553 402 31 satisfied satisfied ADJ iajs-3553 402 32 : : PUNCT iajs-3553 402 33 π‘Ÿ11π‘Ÿ22 π‘Ÿ11π‘Ÿ22 VERB iajs-3553 402 34 < < X iajs-3553 402 35 π‘Ÿ12π‘Ÿ21 π‘Ÿ12π‘Ÿ21 NOUN iajs-3553 402 36 . . PUNCT iajs-3553 403 1 ( ( PUNCT iajs-3553 403 2 21 21 NUM iajs-3553 403 3 ) ) PUNCT iajs-3553 403 4 οΏ½ οΏ½ PROPN iajs-3553 403 5 Μƒ Μƒ PROPN iajs-3553 403 6 οΏ½ οΏ½ PROPN iajs-3553 403 7 1 1 NUM iajs-3553 403 8 β‰  β‰  PROPN iajs-3553 403 9 οΏ½ οΏ½ PROPN iajs-3553 403 10 Μƒ Μƒ PROPN iajs-3553 403 11 οΏ½ οΏ½ NOUN iajs-3553 403 12 2 2 NUM iajs-3553 403 13 . . PUNCT iajs-3553 404 1 ( ( PUNCT iajs-3553 404 2 22 22 NUM iajs-3553 404 3 ) ) PUNCT iajs-3553 405 1 where where SCONJ iajs-3553 405 2 , , PUNCT iajs-3553 405 3 οΏ½ οΏ½ PROPN iajs-3553 405 4 Μƒ Μƒ NOUN iajs-3553 405 5 οΏ½ οΏ½ NOUN iajs-3553 405 6 1 1 NUM iajs-3553 405 7 = = SYM iajs-3553 405 8 𝑙2𝛾2 𝑙2𝛾2 X iajs-3553 405 9 ( ( PUNCT iajs-3553 405 10 1+a3r2)2(1+a4r3)2 1+a3r2)2(1+a4r3)2 NUM iajs-3553 405 11 ، ، X iajs-3553 405 12 οΏ½ οΏ½ PROPN iajs-3553 405 13 Μƒ Μƒ NOUN iajs-3553 405 14 οΏ½ οΏ½ NOUN iajs-3553 405 15 2 2 NUM iajs-3553 405 16 = = SYM iajs-3553 405 17 ( ( PUNCT iajs-3553 405 18 𝑙2𝑅3π‘Ž3𝛾2 𝑙2𝑅3π‘Ž3𝛾2 NOUN iajs-3553 405 19 2 2 NUM iajs-3553 405 20 ( ( PUNCT iajs-3553 405 21 1+a3r2)3(1+a4r3 1+a3r2)3(1+a4r3 NUM iajs-3553 405 22 ) ) PUNCT iajs-3553 405 23 + + NUM iajs-3553 405 24 𝑙2𝑅2π‘Ž4 𝑙2𝑅2π‘Ž4 NOUN iajs-3553 405 25 ( ( PUNCT iajs-3553 405 26 1+a3r2)(1+a4r3)3 1+a3r2)(1+a4r3)3 NUM iajs-3553 405 27 ) ) PUNCT iajs-3553 405 28 ، ، NOUN iajs-3553 405 29 𝛾2 𝛾2 NOUN iajs-3553 405 30 = = PUNCT iajs-3553 405 31 βˆ’ βˆ’ NUM iajs-3553 405 32 οΏ½ οΏ½ PROPN iajs-3553 405 33 Μƒ Μƒ NOUN iajs-3553 405 34 οΏ½ οΏ½ PROPN iajs-3553 405 35 33 33 NUM iajs-3553 405 36 οΏ½ οΏ½ PROPN iajs-3553 405 37 Μƒ Μƒ PROPN iajs-3553 405 38 οΏ½ οΏ½ NOUN iajs-3553 405 39 32 32 NUM iajs-3553 405 40 < < X iajs-3553 405 41 0 0 NUM iajs-3553 405 42 . . PUNCT iajs-3553 406 1 then then ADV iajs-3553 406 2 the the DET iajs-3553 406 3 system system NOUN iajs-3553 406 4 near near ADP iajs-3553 406 5 𝐴5 𝐴5 NOUN iajs-3553 406 6 = = SYM iajs-3553 406 7 ( ( PUNCT iajs-3553 406 8 οΏ½ οΏ½ PROPN iajs-3553 406 9 Μƒ Μƒ NOUN iajs-3553 406 10 οΏ½ οΏ½ NOUN iajs-3553 406 11 1 1 NUM iajs-3553 406 12 . . PUNCT iajs-3553 406 13 οΏ½ οΏ½ PROPN iajs-3553 406 14 Μƒ Μƒ PROPN iajs-3553 406 15 οΏ½ οΏ½ PROPN iajs-3553 406 16 2 2 NUM iajs-3553 406 17 . . PUNCT iajs-3553 406 18 οΏ½ οΏ½ PROPN iajs-3553 406 19 Μƒ Μƒ PROPN iajs-3553 406 20 οΏ½ οΏ½ NOUN iajs-3553 406 21 3 3 NUM iajs-3553 406 22 ) ) PUNCT iajs-3553 406 23 . . PUNCT iajs-3553 406 24 has have VERB iajs-3553 406 25 a a DET iajs-3553 406 26 saddle saddle NOUN iajs-3553 406 27 – – PUNCT iajs-3553 406 28 node node ADJ iajs-3553 406 29 bifurcation bifurcation NOUN iajs-3553 406 30 at at ADP iajs-3553 406 31 οΏ½ οΏ½ PROPN iajs-3553 406 32 Μƒ Μƒ NOUN iajs-3553 406 33 οΏ½ οΏ½ NOUN iajs-3553 406 34 2 2 NUM iajs-3553 406 35 = = SYM iajs-3553 406 36 𝑑2 𝑑2 NOUN iajs-3553 406 37 = = SYM iajs-3553 406 38 οΏ½ οΏ½ PROPN iajs-3553 406 39 Μƒ Μƒ PROPN iajs-3553 406 40 οΏ½ οΏ½ PROPN iajs-3553 406 41 11 11 NUM iajs-3553 406 42 οΏ½ οΏ½ PROPN iajs-3553 406 43 Μƒ Μƒ PROPN iajs-3553 406 44 οΏ½ οΏ½ PROPN iajs-3553 406 45 32 32 NUM iajs-3553 406 46 οΏ½ οΏ½ PROPN iajs-3553 406 47 Μƒ Μƒ PROPN iajs-3553 406 48 οΏ½ οΏ½ PROPN iajs-3553 406 49 23 23 NUM iajs-3553 406 50 οΏ½ οΏ½ PROPN iajs-3553 406 51 Μƒ Μƒ PROPN iajs-3553 406 52 οΏ½ οΏ½ PROPN iajs-3553 406 53 11 11 NUM iajs-3553 406 54 οΏ½ οΏ½ PROPN iajs-3553 406 55 Μƒ Μƒ PROPN iajs-3553 406 56 οΏ½ οΏ½ PROPN iajs-3553 406 57 22βˆ’ 22βˆ’ NUM iajs-3553 406 58 οΏ½ οΏ½ PROPN iajs-3553 406 59 Μƒ Μƒ PROPN iajs-3553 406 60 οΏ½ οΏ½ PROPN iajs-3553 406 61 12 12 NUM iajs-3553 406 62 οΏ½ οΏ½ PROPN iajs-3553 406 63 Μƒ Μƒ NOUN iajs-3553 406 64 οΏ½ οΏ½ NOUN iajs-3553 406 65 21 21 NUM iajs-3553 406 66 . . PUNCT iajs-3553 407 1 proof proof NOUN iajs-3553 407 2 : : PUNCT iajs-3553 407 3 by by ADP iajs-3553 407 4 using use VERB iajs-3553 407 5 the the DET iajs-3553 407 6 jacobian jacobian ADJ iajs-3553 407 7 matrix matrix NOUN iajs-3553 407 8 in in ADP iajs-3553 407 9 equation equation NOUN iajs-3553 407 10 ( ( PUNCT iajs-3553 407 11 1.11 1.11 NUM iajs-3553 407 12 ) ) PUNCT iajs-3553 407 13 in in ADP iajs-3553 407 14 ( ( PUNCT iajs-3553 407 15 20 20 NUM iajs-3553 407 16 ) ) PUNCT iajs-3553 407 17 𝐽5 𝐽5 NOUN iajs-3553 407 18 = = SYM iajs-3553 408 1 𝐽5(𝐴5 𝐽5(𝐴5 NUM iajs-3553 408 2 . . PUNCT iajs-3553 409 1 οΏ½ οΏ½ PROPN iajs-3553 409 2 Μƒ Μƒ PROPN iajs-3553 409 3 οΏ½ οΏ½ NOUN iajs-3553 409 4 2 2 NUM iajs-3553 409 5 ) ) PUNCT iajs-3553 410 1 = = PUNCT iajs-3553 411 1 [ [ X iajs-3553 411 2 οΏ½ οΏ½ PROPN iajs-3553 411 3 Μƒ Μƒ PROPN iajs-3553 411 4 οΏ½ οΏ½ PROPN iajs-3553 411 5 𝑖𝑗]3Γ—3 𝑖𝑗]3Γ—3 PROPN iajs-3553 411 6 , , PUNCT iajs-3553 411 7 where where SCONJ iajs-3553 411 8 οΏ½ οΏ½ PROPN iajs-3553 411 9 Μƒ Μƒ NOUN iajs-3553 411 10 οΏ½ οΏ½ NOUN iajs-3553 411 11 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 411 12 = = NOUN iajs-3553 411 13 π‘Ÿπ‘–π‘— π‘Ÿπ‘–π‘— VERB iajs-3553 411 14 . . PUNCT iajs-3553 412 1 except except SCONJ iajs-3553 412 2 οΏ½ οΏ½ PROPN iajs-3553 412 3 Μƒ Μƒ PROPN iajs-3553 412 4 οΏ½ οΏ½ PROPN iajs-3553 412 5 33 33 NUM iajs-3553 412 6 = = SYM iajs-3553 412 7 𝑙2 𝑙2 PROPN iajs-3553 412 8 οΏ½ οΏ½ PROPN iajs-3553 412 9 Μƒ Μƒ PROPN iajs-3553 412 10 οΏ½ οΏ½ NOUN iajs-3553 412 11 2 2 NUM iajs-3553 412 12 ( ( PUNCT iajs-3553 412 13 1+π‘Ž3 1+π‘Ž3 NUM iajs-3553 412 14 οΏ½ οΏ½ SYM iajs-3553 412 15 Μƒ Μƒ PROPN iajs-3553 412 16 οΏ½ οΏ½ PROPN iajs-3553 412 17 2)(1+π‘Ž4 2)(1+π‘Ž4 NUM iajs-3553 412 18 οΏ½ οΏ½ PROPN iajs-3553 412 19 Μƒ Μƒ PROPN iajs-3553 412 20 οΏ½ οΏ½ PROPN iajs-3553 412 21 3)2 3)2 NUM iajs-3553 412 22 βˆ’ βˆ’ PROPN iajs-3553 412 23 οΏ½ οΏ½ PROPN iajs-3553 412 24 Μƒ Μƒ PROPN iajs-3553 412 25 οΏ½ οΏ½ NOUN iajs-3553 412 26 2 2 NUM iajs-3553 412 27 .then .then NOUN iajs-3553 412 28 , , PUNCT iajs-3553 412 29 the the DET iajs-3553 412 30 characterising characterise VERB iajs-3553 412 31 equation equation NOUN iajs-3553 412 32 for for ADP iajs-3553 412 33 𝐽5 𝐽5 PROPN iajs-3553 412 34 has have VERB iajs-3553 412 35 an an DET iajs-3553 412 36 eigenvalue eigenvalue NOUN iajs-3553 412 37 of of ADP iajs-3553 412 38 zero zero NUM iajs-3553 412 39 , , PUNCT iajs-3553 412 40 which which PRON iajs-3553 412 41 is be AUX iajs-3553 412 42 ( ( PUNCT iajs-3553 412 43 say say VERB iajs-3553 412 44 Ξ»5r3 Ξ»5r3 X iajs-3553 412 45 = = SYM iajs-3553 412 46 0 0 NUM iajs-3553 412 47 ) ) PUNCT iajs-3553 413 1 if if SCONJ iajs-3553 413 2 and and CCONJ iajs-3553 413 3 if if SCONJ iajs-3553 413 4 𝜌3 𝜌3 VERB iajs-3553 413 5 = = SYM iajs-3553 413 6 0 0 X iajs-3553 413 7 . . PUNCT iajs-3553 414 1 now now ADV iajs-3553 414 2 , , PUNCT iajs-3553 414 3 let let VERB iajs-3553 414 4 𝑇[5 𝑇[5 PROPN iajs-3553 414 5 ] ] X iajs-3553 415 1 = = PUNCT iajs-3553 415 2 ( ( PUNCT iajs-3553 415 3 𝑑1 𝑑1 NOUN iajs-3553 415 4 [ [ X iajs-3553 415 5 5 5 NUM iajs-3553 415 6 ] ] PUNCT iajs-3553 415 7 . . PUNCT iajs-3553 416 1 𝑑2 𝑑2 PROPN iajs-3553 416 2 [ [ X iajs-3553 416 3 5 5 NUM iajs-3553 416 4 ] ] PUNCT iajs-3553 416 5 . . PUNCT iajs-3553 417 1 𝑑3 𝑑3 PROPN iajs-3553 417 2 [ [ X iajs-3553 417 3 5 5 NUM iajs-3553 417 4 ] ] PUNCT iajs-3553 417 5 ) ) PUNCT iajs-3553 417 6 𝑇 𝑇 PROPN iajs-3553 417 7 be be VERB iajs-3553 417 8 the the DET iajs-3553 417 9 eigenvector eigenvector NOUN iajs-3553 417 10 associated associate VERB iajs-3553 417 11 with with ADP iajs-3553 417 12 an an DET iajs-3553 417 13 eigenvalue eigenvalue ADJ iajs-3553 417 14 Ξ»5r3 Ξ»5r3 SYM iajs-3553 417 15 = = SYM iajs-3553 417 16 0 0 PROPN iajs-3553 417 17 . . NUM iajs-3553 417 18 thus,(𝐽5 thus,(𝐽5 VERB iajs-3553 417 19 βˆ’ βˆ’ PROPN iajs-3553 417 20 Ξ»5r3 Ξ»5r3 SYM iajs-3553 417 21 𝐼)𝑇[5 𝐼)𝑇[5 X iajs-3553 417 22 ] ] X iajs-3553 417 23 = = SYM iajs-3553 417 24 0 0 X iajs-3553 417 25 . . PUNCT iajs-3553 418 1 this this PRON iajs-3553 418 2 gives give VERB iajs-3553 418 3 : : PUNCT iajs-3553 418 4 𝑑1 𝑑1 NOUN iajs-3553 418 5 [ [ X iajs-3553 418 6 5 5 NUM iajs-3553 418 7 ] ] PUNCT iajs-3553 418 8 = = PUNCT iajs-3553 418 9 𝛾1𝑑3 𝛾1𝑑3 PUNCT iajs-3553 419 1 [ [ X iajs-3553 419 2 5 5 NUM iajs-3553 419 3 ] ] PUNCT iajs-3553 419 4 . . PUNCT iajs-3553 420 1 𝑑2 𝑑2 PROPN iajs-3553 421 1 [ [ X iajs-3553 421 2 5 5 NUM iajs-3553 421 3 ] ] PUNCT iajs-3553 421 4 = = SYM iajs-3553 421 5 𝛾2𝑑3 𝛾2𝑑3 X iajs-3553 422 1 [ [ X iajs-3553 422 2 5 5 NUM iajs-3553 422 3 ] ] PUNCT iajs-3553 422 4 . . PUNCT iajs-3553 423 1 and and CCONJ iajs-3553 423 2 𝑑3 𝑑3 PROPN iajs-3553 423 3 [ [ X iajs-3553 423 4 5 5 NUM iajs-3553 423 5 ] ] PUNCT iajs-3553 423 6 any any DET iajs-3553 423 7 real real ADJ iajs-3553 423 8 number number NOUN iajs-3553 423 9 that that PRON iajs-3553 423 10 is be AUX iajs-3553 423 11 not not PART iajs-3553 423 12 zero zero NUM iajs-3553 423 13 . . PUNCT iajs-3553 424 1 𝐴𝑛𝑑 𝐴𝑛𝑑 PROPN iajs-3553 424 2 𝛾1 𝛾1 PROPN iajs-3553 424 3 = = SYM iajs-3553 424 4 οΏ½ οΏ½ PROPN iajs-3553 424 5 Μƒ Μƒ PROPN iajs-3553 424 6 οΏ½ οΏ½ PROPN iajs-3553 424 7 12 12 NUM iajs-3553 424 8 οΏ½ οΏ½ PROPN iajs-3553 424 9 Μƒ Μƒ PROPN iajs-3553 424 10 οΏ½ οΏ½ PROPN iajs-3553 424 11 33 33 NUM iajs-3553 424 12 οΏ½ οΏ½ PROPN iajs-3553 424 13 Μƒ Μƒ PROPN iajs-3553 424 14 οΏ½ οΏ½ PROPN iajs-3553 424 15 11 11 NUM iajs-3553 424 16 οΏ½ οΏ½ PROPN iajs-3553 424 17 Μƒ Μƒ NOUN iajs-3553 424 18 οΏ½ οΏ½ NOUN iajs-3553 424 19 32 32 NUM iajs-3553 424 20 . . PUNCT iajs-3553 425 1 let let VERB iajs-3553 425 2 𝑀[5 𝑀[5 PROPN iajs-3553 425 3 ] ] X iajs-3553 426 1 = = SYM iajs-3553 426 2 ( ( PUNCT iajs-3553 426 3 π‘š1 π‘š1 X iajs-3553 426 4 [ [ X iajs-3553 426 5 5 5 NUM iajs-3553 426 6 ] ] PUNCT iajs-3553 426 7 . . PUNCT iajs-3553 427 1 π‘š2 π‘š2 NOUN iajs-3553 428 1 [ [ X iajs-3553 428 2 5 5 NUM iajs-3553 428 3 ] ] PUNCT iajs-3553 428 4 . . PUNCT iajs-3553 429 1 π‘š3 π‘š3 NOUN iajs-3553 430 1 [ [ X iajs-3553 430 2 5 5 NUM iajs-3553 430 3 ] ] PUNCT iajs-3553 430 4 ) ) PUNCT iajs-3553 430 5 𝑇 𝑇 PROPN iajs-3553 430 6 be be VERB iajs-3553 430 7 the the DET iajs-3553 430 8 eigenvector eigenvector NOUN iajs-3553 430 9 associated associate VERB iajs-3553 430 10 with with ADP iajs-3553 430 11 an an DET iajs-3553 430 12 eigenvalue eigenvalue ADJ iajs-3553 430 13 Ξ»5r3 Ξ»5r3 SYM iajs-3553 430 14 = = SYM iajs-3553 430 15 0 0 NUM iajs-3553 430 16 of of ADP iajs-3553 430 17 the the DET iajs-3553 430 18 matrix matrix NOUN iajs-3553 430 19 𝐽5 𝐽5 VERB iajs-3553 430 20 𝑇.then 𝑇.then ADV iajs-3553 430 21 . . PUNCT iajs-3553 431 1 ( ( PUNCT iajs-3553 431 2 𝐽5 𝐽5 PROPN iajs-3553 431 3 𝑇 𝑇 PROPN iajs-3553 431 4 βˆ’ βˆ’ PROPN iajs-3553 431 5 Ξ»5r3 Ξ»5r3 PROPN iajs-3553 431 6 𝐼 𝐼 PROPN iajs-3553 431 7 ) ) PUNCT iajs-3553 431 8 𝑀[5 𝑀[5 PROPN iajs-3553 431 9 ] ] X iajs-3553 431 10 = = SYM iajs-3553 431 11 0 0 PUNCT iajs-3553 431 12 . . PUNCT iajs-3553 432 1 by by ADP iajs-3553 432 2 solving solve VERB iajs-3553 432 3 this this DET iajs-3553 432 4 equation equation NOUN iajs-3553 432 5 for for ADP iajs-3553 432 6 . . PUNCT iajs-3553 433 1 𝑀[5 𝑀[5 PROPN iajs-3553 433 2 ] ] X iajs-3553 433 3 = = SYM iajs-3553 433 4 ( ( PUNCT iajs-3553 433 5 𝛾3π‘š3 𝛾3π‘š3 X iajs-3553 434 1 [ [ X iajs-3553 434 2 5 5 NUM iajs-3553 434 3 ] ] PUNCT iajs-3553 434 4 . . PUNCT iajs-3553 435 1 𝛾4π‘š3 𝛾4π‘š3 PUNCT iajs-3553 436 1 [ [ X iajs-3553 436 2 5 5 NUM iajs-3553 436 3 ] ] PUNCT iajs-3553 436 4 . . PUNCT iajs-3553 437 1 π‘š3 π‘š3 NOUN iajs-3553 438 1 [ [ X iajs-3553 438 2 5 5 NUM iajs-3553 438 3 ] ] PUNCT iajs-3553 438 4 ) ) PUNCT iajs-3553 438 5 𝑇 𝑇 PROPN iajs-3553 438 6 . . PUNCT iajs-3553 439 1 where where SCONJ iajs-3553 439 2 π‘š3 π‘š3 NOUN iajs-3553 439 3 [ [ X iajs-3553 439 4 5 5 NUM iajs-3553 439 5 ] ] PUNCT iajs-3553 439 6 any any DET iajs-3553 439 7 real real ADJ iajs-3553 439 8 number number NOUN iajs-3553 439 9 that that PRON iajs-3553 439 10 is be AUX iajs-3553 439 11 not not PART iajs-3553 439 12 zero zero NUM iajs-3553 439 13 , , PUNCT iajs-3553 439 14 and and CCONJ iajs-3553 439 15 𝛾3 𝛾3 PROPN iajs-3553 439 16 = = SYM iajs-3553 439 17 οΏ½ οΏ½ PROPN iajs-3553 439 18 Μƒ Μƒ PROPN iajs-3553 439 19 οΏ½ οΏ½ PROPN iajs-3553 439 20 21 21 NUM iajs-3553 439 21 οΏ½ οΏ½ PROPN iajs-3553 439 22 Μƒ Μƒ PROPN iajs-3553 439 23 οΏ½ οΏ½ PROPN iajs-3553 439 24 33 33 NUM iajs-3553 439 25 οΏ½ οΏ½ PROPN iajs-3553 439 26 Μƒ Μƒ PROPN iajs-3553 439 27 οΏ½ οΏ½ PROPN iajs-3553 439 28 11 11 NUM iajs-3553 439 29 οΏ½ οΏ½ PROPN iajs-3553 439 30 Μƒ Μƒ NOUN iajs-3553 439 31 οΏ½ οΏ½ NOUN iajs-3553 439 32 23 23 NUM iajs-3553 439 33 > > SYM iajs-3553 439 34 0 0 X iajs-3553 439 35 . . PUNCT iajs-3553 439 36 𝛾4 𝛾4 PROPN iajs-3553 439 37 = = SYM iajs-3553 439 38 βˆ’ βˆ’ NUM iajs-3553 439 39 οΏ½ οΏ½ PROPN iajs-3553 439 40 Μƒ Μƒ NOUN iajs-3553 439 41 οΏ½ οΏ½ PROPN iajs-3553 439 42 33 33 NUM iajs-3553 439 43 οΏ½ οΏ½ PROPN iajs-3553 439 44 Μƒ Μƒ PROPN iajs-3553 439 45 οΏ½ οΏ½ NOUN iajs-3553 439 46 23 23 NUM iajs-3553 439 47 < < X iajs-3553 439 48 0 0 X iajs-3553 439 49 . . PUNCT iajs-3553 439 50 ihjpas ihjpas PROPN iajs-3553 439 51 . . PUNCT iajs-3553 440 1 2025 2025 NUM iajs-3553 440 2 , , PUNCT iajs-3553 440 3 38(2 38(2 NUM iajs-3553 440 4 ) ) PUNCT iajs-3553 440 5 325 325 NUM iajs-3553 440 6 now now ADV iajs-3553 440 7 consider consider VERB iajs-3553 440 8 this this PRON iajs-3553 440 9 : : PUNCT iajs-3553 440 10 πœ•π‘“ πœ•π‘“ ADJ iajs-3553 440 11 πœ•π‘‘2 πœ•π‘‘2 NOUN iajs-3553 440 12 = = SYM iajs-3553 440 13 𝑓𝑑2 𝑓𝑑2 X iajs-3553 440 14 ( ( PUNCT iajs-3553 440 15 𝑋. 𝑋. PROPN iajs-3553 440 16 𝑑2 𝑑2 PROPN iajs-3553 440 17 ) ) PUNCT iajs-3553 440 18 = = SYM iajs-3553 440 19 ( ( PUNCT iajs-3553 440 20 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3553 440 21 πœ•π‘‘2 πœ•π‘‘2 PROPN iajs-3553 440 22 . . PUNCT iajs-3553 441 1 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3553 441 2 πœ•π‘‘2 πœ•π‘‘2 VERB iajs-3553 441 3 . . PUNCT iajs-3553 442 1 πœ•π‘“3 πœ•π‘“3 PROPN iajs-3553 442 2 πœ•π‘‘2 πœ•π‘‘2 PROPN iajs-3553 442 3 ) ) PUNCT iajs-3553 443 1 𝑇 𝑇 PROPN iajs-3553 443 2 = = SYM iajs-3553 443 3 ( ( PUNCT iajs-3553 443 4 0 0 NUM iajs-3553 443 5 . . NOUN iajs-3553 443 6 0 0 NUM iajs-3553 443 7 . . PUNCT iajs-3553 443 8 βˆ’π‘…3 βˆ’π‘…3 NOUN iajs-3553 443 9 ) ) PUNCT iajs-3553 443 10 𝑇. 𝑇. PROPN iajs-3553 443 11 so so ADV iajs-3553 443 12 . . PUNCT iajs-3553 444 1 𝑓𝑑2 𝑓𝑑2 PROPN iajs-3553 444 2 ( ( PUNCT iajs-3553 444 3 𝐴5 𝐴5 PROPN iajs-3553 444 4 . . PUNCT iajs-3553 444 5 οΏ½ οΏ½ PROPN iajs-3553 444 6 Μƒ Μƒ PROPN iajs-3553 444 7 οΏ½ οΏ½ NOUN iajs-3553 444 8 2 2 NUM iajs-3553 444 9 ) ) PUNCT iajs-3553 444 10 = = SYM iajs-3553 444 11 ( ( PUNCT iajs-3553 444 12 0 0 NUM iajs-3553 444 13 . . NOUN iajs-3553 444 14 0 0 NUM iajs-3553 444 15 . . PUNCT iajs-3553 445 1 βˆ’ βˆ’ NUM iajs-3553 445 2 οΏ½ οΏ½ PROPN iajs-3553 445 3 Μƒ Μƒ NOUN iajs-3553 445 4 οΏ½ οΏ½ NOUN iajs-3553 445 5 3 3 NUM iajs-3553 445 6 ) ) PUNCT iajs-3553 445 7 𝑇 𝑇 NOUN iajs-3553 445 8 and and CCONJ iajs-3553 445 9 hence hence ADV iajs-3553 445 10 ( ( PUNCT iajs-3553 445 11 𝑀[5 𝑀[5 PROPN iajs-3553 445 12 ] ] PUNCT iajs-3553 445 13 ) ) PUNCT iajs-3553 445 14 𝑇 𝑇 PROPN iajs-3553 445 15 𝑓𝑑2 𝑓𝑑2 NOUN iajs-3553 445 16 ( ( PUNCT iajs-3553 445 17 𝐴5 𝐴5 PROPN iajs-3553 445 18 . . PUNCT iajs-3553 445 19 οΏ½ οΏ½ PROPN iajs-3553 445 20 Μƒ Μƒ PROPN iajs-3553 445 21 οΏ½ οΏ½ NOUN iajs-3553 445 22 2 2 NUM iajs-3553 445 23 ) ) PUNCT iajs-3553 445 24 = = SYM iajs-3553 446 1 βˆ’ βˆ’ NUM iajs-3553 446 2 οΏ½ οΏ½ PROPN iajs-3553 446 3 Μƒ Μƒ PROPN iajs-3553 446 4 οΏ½ οΏ½ NOUN iajs-3553 446 5 3π‘š3 3π‘š3 NUM iajs-3553 446 6 [ [ X iajs-3553 446 7 5 5 NUM iajs-3553 446 8 ] ] PUNCT iajs-3553 446 9 β‰  β‰  PROPN iajs-3553 446 10 0 0 NUM iajs-3553 446 11 . . PUNCT iajs-3553 447 1 by by ADP iajs-3553 447 2 substituting substitute VERB iajs-3553 447 3 𝑇[5 𝑇[5 PROPN iajs-3553 447 4 ] ] PUNCT iajs-3553 447 5 in in ADP iajs-3553 447 6 equation equation NOUN iajs-3553 447 7 ( ( PUNCT iajs-3553 447 8 3 3 X iajs-3553 447 9 ) ) PUNCT iajs-3553 447 10 we we PRON iajs-3553 447 11 get get VERB iajs-3553 447 12 : : PUNCT iajs-3553 447 13 𝐷2πΉπœ‡(𝐴5 𝐷2πΉπœ‡(𝐴5 PROPN iajs-3553 447 14 . . PUNCT iajs-3553 448 1 οΏ½ οΏ½ PROPN iajs-3553 448 2 Μƒ Μƒ PROPN iajs-3553 448 3 οΏ½ οΏ½ NOUN iajs-3553 448 4 2)(𝑇 2)(𝑇 PRON iajs-3553 449 1 [ [ X iajs-3553 449 2 5 5 NUM iajs-3553 449 3 ] ] PUNCT iajs-3553 449 4 . . PUNCT iajs-3553 450 1 𝑇[5 𝑇[5 PROPN iajs-3553 450 2 ] ] PUNCT iajs-3553 450 3 ) ) PUNCT iajs-3553 450 4 = = SYM iajs-3553 450 5 ( ( PUNCT iajs-3553 450 6 οΏ½ οΏ½ PROPN iajs-3553 450 7 Μƒ Μƒ PROPN iajs-3553 450 8 οΏ½ οΏ½ NOUN iajs-3553 450 9 𝑖𝑗 𝑖𝑗 NOUN iajs-3553 450 10 ) ) PUNCT iajs-3553 450 11 . . PUNCT iajs-3553 451 1 ( ( PUNCT iajs-3553 451 2 23 23 NUM iajs-3553 451 3 ) ) PUNCT iajs-3553 451 4 οΏ½ οΏ½ PROPN iajs-3553 451 5 Μƒ Μƒ PROPN iajs-3553 451 6 οΏ½ οΏ½ NOUN iajs-3553 451 7 11 11 NUM iajs-3553 451 8 = = SYM iajs-3553 451 9 2(𝑑3 2(𝑑3 NUM iajs-3553 452 1 [ [ X iajs-3553 452 2 5 5 NUM iajs-3553 452 3 ] ] PUNCT iajs-3553 452 4 ) ) PUNCT iajs-3553 452 5 2 2 NUM iajs-3553 453 1 [ [ X iajs-3553 453 2 ( ( PUNCT iajs-3553 453 3 𝐾1𝑅1𝛾1 𝐾1𝑅1𝛾1 ADJ iajs-3553 453 4 2 2 NUM iajs-3553 453 5 ( ( PUNCT iajs-3553 453 6 1+𝐾1𝑅2 1+𝐾1𝑅2 NUM iajs-3553 453 7 ) ) PUNCT iajs-3553 453 8 βˆ’π›Ύ1𝛾2 βˆ’π›Ύ1𝛾2 NOUN iajs-3553 453 9 ) ) PUNCT iajs-3553 453 10 π‘Ÿ1𝐾1 π‘Ÿ1𝐾1 NUM iajs-3553 453 11 ( ( PUNCT iajs-3553 453 12 1+𝐾1𝑅2)2 1+𝐾1𝑅2)2 NUM iajs-3553 453 13 βˆ’ βˆ’ NOUN iajs-3553 453 14 π‘š1𝛾1 π‘š1𝛾1 PROPN iajs-3553 453 15 2 2 NUM iajs-3553 453 16 + + CCONJ iajs-3553 453 17 ( ( PUNCT iajs-3553 453 18 𝑅2π‘Ž1𝛾1 𝑅2π‘Ž1𝛾1 NUM iajs-3553 453 19 2 2 NUM iajs-3553 453 20 ( ( PUNCT iajs-3553 453 21 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 453 22 βˆ’ βˆ’ NOUN iajs-3553 453 23 𝛾1𝛾2 𝛾1𝛾2 X iajs-3553 453 24 ( ( PUNCT iajs-3553 453 25 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 453 26 ) ) PUNCT iajs-3553 453 27 + + NUM iajs-3553 453 28 π‘Ž2𝑅2𝛾2 π‘Ž2𝑅2𝛾2 NOUN iajs-3553 453 29 2 2 NUM iajs-3553 453 30 ( ( PUNCT iajs-3553 453 31 1+π‘Ž2𝑅2)2 1+π‘Ž2𝑅2)2 NUM iajs-3553 453 32 ) ) PUNCT iajs-3553 453 33 𝛽1 𝛽1 NOUN iajs-3553 453 34 ( ( PUNCT iajs-3553 453 35 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 453 36 ) ) PUNCT iajs-3553 453 37 ] ] PUNCT iajs-3553 453 38 , , PUNCT iajs-3553 453 39 οΏ½ οΏ½ PROPN iajs-3553 453 40 Μƒ Μƒ PROPN iajs-3553 453 41 οΏ½ οΏ½ NOUN iajs-3553 453 42 21 21 NUM iajs-3553 453 43 = = SYM iajs-3553 453 44 2(𝑑3 2(𝑑3 NUM iajs-3553 454 1 [ [ X iajs-3553 454 2 5 5 NUM iajs-3553 454 3 ] ] PUNCT iajs-3553 454 4 ) ) PUNCT iajs-3553 454 5 2 2 NUM iajs-3553 455 1 [ [ X iajs-3553 455 2 ( ( PUNCT iajs-3553 455 3 𝛾1𝛾2 𝛾1𝛾2 X iajs-3553 455 4 ( ( PUNCT iajs-3553 455 5 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 455 6 ) ) PUNCT iajs-3553 455 7 βˆ’ βˆ’ NOUN iajs-3553 455 8 r2π‘Ž1𝛾1 r2π‘Ž1𝛾1 NOUN iajs-3553 455 9 2 2 NUM iajs-3553 455 10 ( ( PUNCT iajs-3553 455 11 1+a1r1)2 1+a1r1)2 NUM iajs-3553 455 12 βˆ’ βˆ’ ADP iajs-3553 455 13 r1π‘Ž2𝛾2 r1π‘Ž2𝛾2 VERB iajs-3553 455 14 2 2 NUM iajs-3553 455 15 ( ( PUNCT iajs-3553 455 16 1+a2r2)2 1+a2r2)2 NUM iajs-3553 455 17 ) ) PUNCT iajs-3553 455 18 𝑙1 𝑙1 NOUN iajs-3553 455 19 ( ( PUNCT iajs-3553 455 20 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 1+π‘Ž1𝑅1)(1+π‘Ž2𝑅2 NUM iajs-3553 455 21 ) ) PUNCT iajs-3553 455 22 + + CCONJ iajs-3553 455 23 ( ( PUNCT iajs-3553 455 24 𝐾2𝑅2 𝐾2𝑅2 NOUN iajs-3553 455 25 ( ( PUNCT iajs-3553 455 26 1+k2r3 1+k2r3 NUM iajs-3553 455 27 ) ) PUNCT iajs-3553 455 28 βˆ’ βˆ’ NOUN iajs-3553 455 29 𝛾2 𝛾2 NOUN iajs-3553 455 30 ) ) PUNCT iajs-3553 455 31 r2k2 r2k2 NOUN iajs-3553 455 32 ( ( PUNCT iajs-3553 455 33 1+k2r3)2 1+k2r3)2 NOUN iajs-3553 455 34 + + CCONJ iajs-3553 455 35 ( ( PUNCT iajs-3553 455 36 a3r3𝛾2 a3r3𝛾2 PROPN iajs-3553 455 37 2 2 NUM iajs-3553 455 38 ( ( PUNCT iajs-3553 455 39 1+a3r2)2 1+a3r2)2 NUM iajs-3553 455 40 + + CCONJ iajs-3553 455 41 𝛾2 𝛾2 PROPN iajs-3553 455 42 ( ( PUNCT iajs-3553 455 43 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 455 44 ) ) PUNCT iajs-3553 455 45 + + CCONJ iajs-3553 455 46 a4𝑅2 a4𝑅2 PRON iajs-3553 455 47 ( ( PUNCT iajs-3553 455 48 1+a4r3)2 1+a4r3)2 X iajs-3553 455 49 ) ) PUNCT iajs-3553 455 50 𝛽2 𝛽2 PROPN iajs-3553 455 51 ( ( PUNCT iajs-3553 455 52 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 455 53 ) ) PUNCT iajs-3553 455 54 βˆ’π‘š2𝛾2 βˆ’π‘š2𝛾2 PROPN iajs-3553 455 55 2 2 NUM iajs-3553 455 56 ] ] PUNCT iajs-3553 455 57 , , PUNCT iajs-3553 455 58 οΏ½ οΏ½ PROPN iajs-3553 455 59 Μƒ Μƒ PROPN iajs-3553 455 60 οΏ½ οΏ½ NOUN iajs-3553 455 61 31 31 NUM iajs-3553 455 62 = = SYM iajs-3553 455 63 2(𝑑3 2(𝑑3 NUM iajs-3553 456 1 [ [ X iajs-3553 456 2 5 5 NUM iajs-3553 456 3 ] ] PUNCT iajs-3553 456 4 ) ) PUNCT iajs-3553 456 5 2 2 NUM iajs-3553 456 6 𝑙2 𝑙2 NOUN iajs-3553 456 7 ( ( PUNCT iajs-3553 456 8 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 456 9 ) ) PUNCT iajs-3553 456 10 ( ( PUNCT iajs-3553 456 11 𝛾2 𝛾2 VERB iajs-3553 456 12 ( ( PUNCT iajs-3553 456 13 1+a3r2)(1+a4r3 1+a3r2)(1+a4r3 NUM iajs-3553 456 14 ) ) PUNCT iajs-3553 456 15 βˆ’ βˆ’ NOUN iajs-3553 456 16 𝑅3π‘Ž3𝛾2 𝑅3π‘Ž3𝛾2 NOUN iajs-3553 456 17 2 2 NUM iajs-3553 456 18 ( ( PUNCT iajs-3553 456 19 1+a3r2)2 1+a3r2)2 NUM iajs-3553 456 20 βˆ’ βˆ’ NOUN iajs-3553 456 21 𝑅2π‘Ž4 𝑅2π‘Ž4 NOUN iajs-3553 456 22 ( ( PUNCT iajs-3553 456 23 1+a4r3)2 1+a4r3)2 NUM iajs-3553 456 24 ) ) PUNCT iajs-3553 456 25 . . PUNCT iajs-3553 457 1 hence hence ADV iajs-3553 457 2 , , PUNCT iajs-3553 457 3 it it PRON iajs-3553 457 4 was be AUX iajs-3553 457 5 obtained obtain VERB iajs-3553 457 6 . . PUNCT iajs-3553 458 1 ( ( PUNCT iajs-3553 458 2 𝑀[5 𝑀[5 PROPN iajs-3553 458 3 ] ] NOUN iajs-3553 458 4 ) ) PUNCT iajs-3553 458 5 𝑇 𝑇 PROPN iajs-3553 459 1 [ [ X iajs-3553 459 2 𝐷2πΉπœ‡(𝐴5 𝐷2πΉπœ‡(𝐴5 NUM iajs-3553 459 3 . . PUNCT iajs-3553 459 4 οΏ½ οΏ½ PROPN iajs-3553 459 5 Μƒ Μƒ PROPN iajs-3553 459 6 οΏ½ οΏ½ NOUN iajs-3553 459 7 2)(𝑇 2)(𝑇 PRON iajs-3553 459 8 [ [ X iajs-3553 459 9 5 5 NUM iajs-3553 459 10 ] ] PUNCT iajs-3553 459 11 . . PUNCT iajs-3553 460 1 𝑇[5 𝑇[5 PROPN iajs-3553 460 2 ] ] PUNCT iajs-3553 460 3 . . PUNCT iajs-3553 461 1 𝑇[5 𝑇[5 PROPN iajs-3553 461 2 ] ] PUNCT iajs-3553 461 3 ) ) PUNCT iajs-3553 461 4 ] ] PUNCT iajs-3553 462 1 = = SYM iajs-3553 462 2 2 2 X iajs-3553 462 3 ( ( PUNCT iajs-3553 462 4 𝑑3 𝑑3 PROPN iajs-3553 462 5 [ [ X iajs-3553 462 6 5 5 NUM iajs-3553 462 7 ] ] PUNCT iajs-3553 462 8 ) ) PUNCT iajs-3553 462 9 2 2 NUM iajs-3553 462 10 π‘š3 π‘š3 NOUN iajs-3553 462 11 [ [ X iajs-3553 462 12 5 5 NUM iajs-3553 462 13 ] ] X iajs-3553 462 14 ( ( PUNCT iajs-3553 462 15 οΏ½ οΏ½ PROPN iajs-3553 462 16 Μƒ Μƒ NOUN iajs-3553 462 17 οΏ½ οΏ½ PROPN iajs-3553 462 18 1 1 NUM iajs-3553 462 19 βˆ’ βˆ’ PROPN iajs-3553 462 20 οΏ½ οΏ½ PROPN iajs-3553 462 21 Μƒ Μƒ NOUN iajs-3553 462 22 οΏ½ οΏ½ NOUN iajs-3553 462 23 2 2 NUM iajs-3553 462 24 ) ) PUNCT iajs-3553 462 25 β‰  β‰  PROPN iajs-3553 462 26 0 0 NUM iajs-3553 462 27 . . PUNCT iajs-3553 463 1 so so ADV iajs-3553 463 2 , , PUNCT iajs-3553 463 3 by by ADP iajs-3553 463 4 condition condition NOUN iajs-3553 463 5 ( ( PUNCT iajs-3553 463 6 21 21 NUM iajs-3553 463 7 ) ) PUNCT iajs-3553 463 8 system system NOUN iajs-3553 463 9 ( ( PUNCT iajs-3553 463 10 1 1 X iajs-3553 463 11 ) ) PUNCT iajs-3553 463 12 exhibits exhibit VERB iajs-3553 463 13 a a DET iajs-3553 463 14 saddle saddle NOUN iajs-3553 463 15 - - PUNCT iajs-3553 463 16 node node NOUN iajs-3553 463 17 bifurcation bifurcation NOUN iajs-3553 463 18 at at ADP iajs-3553 463 19 𝐴5 𝐴5 NOUN iajs-3553 463 20 for for ADP iajs-3553 463 21 any any DET iajs-3553 463 22 value value NOUN iajs-3553 463 23 of of ADP iajs-3553 463 24 the the DET iajs-3553 463 25 parameter parameter PROPN iajs-3553 463 26 οΏ½ οΏ½ PROPN iajs-3553 463 27 Μƒ Μƒ PROPN iajs-3553 463 28 οΏ½ οΏ½ NOUN iajs-3553 463 29 2 2 NUM iajs-3553 463 30 = = NOUN iajs-3553 464 1 𝑑2 𝑑2 NOUN iajs-3553 465 1 but but CCONJ iajs-3553 465 2 no no DET iajs-3553 465 3 pitch pitch NOUN iajs-3553 465 4 fork fork NOUN iajs-3553 465 5 bifurcation bifurcation NOUN iajs-3553 465 6 at at ADP iajs-3553 465 7 𝐴5 𝐴5 PROPN iajs-3553 465 8 where where SCONJ iajs-3553 465 9 οΏ½ οΏ½ PROPN iajs-3553 465 10 Μƒ Μƒ PROPN iajs-3553 465 11 οΏ½ οΏ½ NOUN iajs-3553 465 12 2 2 NUM iajs-3553 465 13 = = SYM iajs-3553 465 14 𝑑2 𝑑2 NOUN iajs-3553 465 15 . . PUNCT iajs-3553 466 1 3 3 X iajs-3553 466 2 . . X iajs-3553 466 3 numerical numerical PROPN iajs-3553 466 4 simulation simulation PROPN iajs-3553 466 5 in in ADP iajs-3553 466 6 this this DET iajs-3553 466 7 section section NOUN iajs-3553 466 8 , , PUNCT iajs-3553 466 9 the the DET iajs-3553 466 10 dynamical dynamical ADJ iajs-3553 466 11 behavior behavior NOUN iajs-3553 466 12 of of ADP iajs-3553 466 13 the the DET iajs-3553 466 14 system system NOUN iajs-3553 466 15 ( ( PUNCT iajs-3553 466 16 1 1 X iajs-3553 466 17 ) ) PUNCT iajs-3553 466 18 was be AUX iajs-3553 466 19 investigated investigate VERB iajs-3553 466 20 . . PUNCT iajs-3553 467 1 calculations calculation NOUN iajs-3553 467 2 may may AUX iajs-3553 467 3 be be AUX iajs-3553 467 4 performed perform VERB iajs-3553 467 5 for for ADP iajs-3553 467 6 a a DET iajs-3553 467 7 one one NUM iajs-3553 467 8 set set NOUN iajs-3553 467 9 of of ADP iajs-3553 467 10 parameters parameter NOUN iajs-3553 467 11 with with ADP iajs-3553 467 12 a a DET iajs-3553 467 13 different different ADJ iajs-3553 467 14 starting starting NOUN iajs-3553 467 15 point point NOUN iajs-3553 467 16 to to PART iajs-3553 467 17 examine examine VERB iajs-3553 467 18 the the DET iajs-3553 467 19 analytical analytical ADJ iajs-3553 467 20 results result NOUN iajs-3553 467 21 and and CCONJ iajs-3553 467 22 understand understand VERB iajs-3553 467 23 how how SCONJ iajs-3553 467 24 the the DET iajs-3553 467 25 parameters parameter NOUN iajs-3553 467 26 impact impact VERB iajs-3553 467 27 the the DET iajs-3553 467 28 dynamic dynamic ADJ iajs-3553 467 29 model model NOUN iajs-3553 467 30 . . PUNCT iajs-3553 468 1 figure figure NOUN iajs-3553 468 2 1 1 NUM iajs-3553 468 3 . . PUNCT iajs-3553 469 1 ( ( PUNCT iajs-3553 469 2 a a DET iajs-3553 469 3 - - PUNCT iajs-3553 469 4 d d NOUN iajs-3553 469 5 ) ) PUNCT iajs-3553 469 6 , , PUNCT iajs-3553 469 7 shows show VERB iajs-3553 469 8 that that DET iajs-3553 469 9 system system NOUN iajs-3553 469 10 ( ( PUNCT iajs-3553 469 11 1 1 X iajs-3553 469 12 ) ) PUNCT iajs-3553 469 13 has have VERB iajs-3553 469 14 positive positive ADJ iajs-3553 469 15 solution solution NOUN iajs-3553 469 16 . . PUNCT iajs-3553 470 1 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 470 2 = = PUNCT iajs-3553 470 3 0.5.π‘š1 0.5.π‘š1 PROPN iajs-3553 471 1 = = SYM iajs-3553 471 2 0.5 0.5 NUM iajs-3553 471 3 . . PUNCT iajs-3553 472 1 𝐾1 𝐾1 PROPN iajs-3553 472 2 = = SYM iajs-3553 472 3 0.003 0.003 NUM iajs-3553 472 4 . . PUNCT iajs-3553 472 5 𝛽1 𝛽1 NOUN iajs-3553 472 6 = = SYM iajs-3553 472 7 0.3 0.3 NUM iajs-3553 472 8 . . PUNCT iajs-3553 473 1 𝑙1 𝑙1 PROPN iajs-3553 473 2 = = PROPN iajs-3553 473 3 0.3 0.3 NUM iajs-3553 473 4 . . PUNCT iajs-3553 474 1 π‘Ž1 π‘Ž1 VERB iajs-3553 474 2 = = NOUN iajs-3553 474 3 0.1 0.1 NUM iajs-3553 474 4 . . PUNCT iajs-3553 475 1 π‘Ž2 π‘Ž2 NOUN iajs-3553 475 2 = = NOUN iajs-3553 475 3 0.02 0.02 NUM iajs-3553 475 4 . . PUNCT iajs-3553 475 5 𝑑1 𝑑1 NOUN iajs-3553 475 6 = = NOUN iajs-3553 475 7 0.1 0.1 NUM iajs-3553 475 8 . . PUNCT iajs-3553 476 1 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3553 476 2 = = SYM iajs-3553 476 3 0.5 0.5 NUM iajs-3553 476 4 . . PUNCT iajs-3553 477 1 π‘š2 π‘š2 NOUN iajs-3553 477 2 = = PUNCT iajs-3553 477 3 0.4 0.4 NUM iajs-3553 477 4 . . PUNCT iajs-3553 478 1 𝐾2 𝐾2 PROPN iajs-3553 478 2 = = NOUN iajs-3553 478 3 0.07 0.07 NUM iajs-3553 478 4 . . PUNCT iajs-3553 479 1 𝛽2 𝛽2 PROPN iajs-3553 479 2 = = SYM iajs-3553 479 3 0.7 0.7 NUM iajs-3553 479 4 . . PUNCT iajs-3553 480 1 𝑙2 𝑙2 NOUN iajs-3553 480 2 = = NOUN iajs-3553 480 3 0.7 0.7 NUM iajs-3553 480 4 . . PUNCT iajs-3553 481 1 π‘Ž3 π‘Ž3 PROPN iajs-3553 481 2 = = PROPN iajs-3553 481 3 0.01 0.01 NUM iajs-3553 481 4 . . PUNCT iajs-3553 482 1 π‘Ž4 π‘Ž4 PROPN iajs-3553 482 2 = = NOUN iajs-3553 482 3 0.02 0.02 NUM iajs-3553 482 4 . . PUNCT iajs-3553 483 1 𝑑2 𝑑2 NOUN iajs-3553 483 2 = = SYM iajs-3553 483 3 0.4 0.4 NUM iajs-3553 483 4 . . PUNCT iajs-3553 484 1 ( ( PUNCT iajs-3553 484 2 24 24 NUM iajs-3553 484 3 ) ) PUNCT iajs-3553 484 4 figure figure NOUN iajs-3553 484 5 𝟏. 𝟏. X iajs-3553 485 1 ( ( PUNCT iajs-3553 485 2 𝐚 𝐚 ADP iajs-3553 485 3 βˆ’ βˆ’ PROPN iajs-3553 485 4 𝐝 𝐝 NOUN iajs-3553 485 5 ) ) PUNCT iajs-3553 485 6 . . PUNCT iajs-3553 486 1 time time NOUN iajs-3553 486 2 series series PROPN iajs-3553 486 3 of of ADP iajs-3553 486 4 system system NOUN iajs-3553 486 5 solution solution NOUN iajs-3553 486 6 ( ( PUNCT iajs-3553 486 7 1 1 X iajs-3553 486 8 ) ) PUNCT iajs-3553 486 9 begin begin VERB iajs-3553 486 10 with with ADP iajs-3553 486 11 different different ADJ iajs-3553 486 12 starting starting NOUN iajs-3553 486 13 points point NOUN iajs-3553 486 14 ( ( PUNCT iajs-3553 486 15 0.5 0.5 NUM iajs-3553 486 16 . . NOUN iajs-3553 486 17 0.6 0.6 NUM iajs-3553 486 18 . . PUNCT iajs-3553 486 19 0.7 0.7 NUM iajs-3553 486 20 ) ) PUNCT iajs-3553 486 21 . . PUNCT iajs-3553 487 1 ( ( PUNCT iajs-3553 487 2 0.4 0.4 NUM iajs-3553 487 3 . . PUNCT iajs-3553 487 4 0.5 0.5 NUM iajs-3553 487 5 . . NOUN iajs-3553 487 6 0.8 0.8 NUM iajs-3553 487 7 ) ) PUNCT iajs-3553 487 8 . . PUNCT iajs-3553 488 1 and and CCONJ iajs-3553 488 2 ( ( PUNCT iajs-3553 488 3 0.2 0.2 NUM iajs-3553 488 4 . . PUNCT iajs-3553 489 1 0.9 0.9 NUM iajs-3553 489 2 . . NOUN iajs-3553 490 1 0.1 0.1 NUM iajs-3553 490 2 ) ) PUNCT iajs-3553 491 1 . . PUNCT iajs-3553 492 1 ( ( PUNCT iajs-3553 492 2 𝐛 𝐛 X iajs-3553 492 3 ) ) PUNCT iajs-3553 492 4 path path NOUN iajs-3553 492 5 of of ADP iajs-3553 492 6 r1 r1 NOUN iajs-3553 492 7 depending depend VERB iajs-3553 492 8 on on ADP iajs-3553 492 9 time time NOUN iajs-3553 492 10 , , PUNCT iajs-3553 492 11 ( ( PUNCT iajs-3553 492 12 𝐜 𝐜 NOUN iajs-3553 492 13 ) ) PUNCT iajs-3553 492 14 path path NOUN iajs-3553 492 15 of of ADP iajs-3553 492 16 r2 r2 PROPN iajs-3553 492 17 depending depend VERB iajs-3553 492 18 on on ADP iajs-3553 492 19 time,(𝐝 time,(𝐝 PROPN iajs-3553 492 20 ) ) PUNCT iajs-3553 492 21 path path NOUN iajs-3553 492 22 of of ADP iajs-3553 492 23 r3 r3 PROPN iajs-3553 492 24 depending depend VERB iajs-3553 492 25 on on ADP iajs-3553 492 26 time time NOUN iajs-3553 492 27 . . PUNCT iajs-3553 493 1 0.2 0.2 NUM iajs-3553 493 2 0.3 0.3 NUM iajs-3553 493 3 0.4 0.4 NUM iajs-3553 493 4 0.5 0.5 NUM iajs-3553 493 5 0.6 0.6 NUM iajs-3553 493 6 0.7 0.7 NUM iajs-3553 493 7 0.8 0.8 NUM iajs-3553 493 8 0.9 0.9 NUM iajs-3553 493 9 0.4 0.4 NUM iajs-3553 493 10 0.5 0.5 NUM iajs-3553 493 11 0.6 0.6 NUM iajs-3553 493 12 0.7 0.7 NUM iajs-3553 493 13 0.8 0.8 NUM iajs-3553 493 14 0.9 0.9 NUM iajs-3553 493 15 1 1 NUM iajs-3553 493 16 0.1 0.1 NUM iajs-3553 493 17 0.2 0.2 NUM iajs-3553 493 18 0.3 0.3 NUM iajs-3553 493 19 0.4 0.4 NUM iajs-3553 493 20 0.5 0.5 NUM iajs-3553 493 21 0.6 0.6 NUM iajs-3553 493 22 0.7 0.7 NUM iajs-3553 493 23 0.8 0.8 NUM iajs-3553 493 24 prey prey NOUN iajs-3553 493 25 a a DET iajs-3553 493 26 intermediate intermediate ADJ iajs-3553 493 27 predator predator NOUN iajs-3553 493 28 t t NOUN iajs-3553 493 29 o o NOUN iajs-3553 494 1 p p X iajs-3553 494 2 p p NOUN iajs-3553 494 3 r r NOUN iajs-3553 494 4 e e NOUN iajs-3553 494 5 d d NOUN iajs-3553 494 6 a a DET iajs-3553 494 7 t t NOUN iajs-3553 494 8 o o NOUN iajs-3553 494 9 r r NOUN iajs-3553 494 10 prey prey NOUN iajs-3553 494 11 intermediate intermediate ADJ iajs-3553 494 12 predator predator NOUN iajs-3553 494 13 top top ADJ iajs-3553 494 14 predator predator NOUN iajs-3553 494 15 ( ( PUNCT iajs-3553 494 16 0.6748,0.5691,0.4943 0.6748,0.5691,0.4943 NUM iajs-3553 494 17 ) ) PUNCT iajs-3553 494 18 ( ( PUNCT iajs-3553 494 19 0.2,0.9,0.1 0.2,0.9,0.1 NOUN iajs-3553 494 20 ) ) PUNCT iajs-3553 494 21 ( ( PUNCT iajs-3553 494 22 0.5,0.6,0.7 0.5,0.6,0.7 NUM iajs-3553 494 23 ) ) PUNCT iajs-3553 494 24 ( ( PUNCT iajs-3553 494 25 0.4,0.5,0.8 0.4,0.5,0.8 NUM iajs-3553 494 26 ) ) PUNCT iajs-3553 494 27 0 0 NUM iajs-3553 494 28 100 100 NUM iajs-3553 494 29 200 200 NUM iajs-3553 494 30 300 300 NUM iajs-3553 494 31 400 400 NUM iajs-3553 494 32 500 500 NUM iajs-3553 494 33 600 600 NUM iajs-3553 494 34 700 700 NUM iajs-3553 494 35 800 800 NUM iajs-3553 494 36 900 900 NUM iajs-3553 494 37 1000 1000 NUM iajs-3553 494 38 0.2 0.2 NUM iajs-3553 494 39 0.3 0.3 NUM iajs-3553 494 40 0.4 0.4 NUM iajs-3553 494 41 0.5 0.5 NUM iajs-3553 494 42 0.6 0.6 NUM iajs-3553 494 43 0.7 0.7 NUM iajs-3553 494 44 0.8 0.8 NUM iajs-3553 494 45 0.9 0.9 NUM iajs-3553 494 46 b b NOUN iajs-3553 494 47 time time NOUN iajs-3553 494 48 p p NOUN iajs-3553 494 49 r r NOUN iajs-3553 494 50 e e X iajs-3553 494 51 y y NOUN iajs-3553 494 52 r r NOUN iajs-3553 494 53 1 1 NUM iajs-3553 494 54 r r NOUN iajs-3553 494 55 2 2 NUM iajs-3553 494 56 r r NOUN iajs-3553 494 57 3 3 NUM iajs-3553 494 58 0 0 NUM iajs-3553 494 59 100 100 NUM iajs-3553 494 60 200 200 NUM iajs-3553 494 61 300 300 NUM iajs-3553 494 62 400 400 NUM iajs-3553 494 63 500 500 NUM iajs-3553 494 64 600 600 NUM iajs-3553 494 65 700 700 NUM iajs-3553 494 66 800 800 NUM iajs-3553 494 67 900 900 NUM iajs-3553 494 68 1000 1000 NUM iajs-3553 494 69 0.4 0.4 NUM iajs-3553 494 70 0.5 0.5 NUM iajs-3553 494 71 0.6 0.6 NUM iajs-3553 494 72 0.7 0.7 NUM iajs-3553 494 73 0.8 0.8 NUM iajs-3553 494 74 0.9 0.9 NUM iajs-3553 494 75 1 1 NUM iajs-3553 494 76 time time NOUN iajs-3553 494 77 in in ADP iajs-3553 494 78 te te ADP iajs-3553 494 79 r r NOUN iajs-3553 494 80 m m NOUN iajs-3553 494 81 e e NOUN iajs-3553 494 82 d d X iajs-3553 494 83 ia ia PROPN iajs-3553 494 84 te te INTJ iajs-3553 494 85 p p NOUN iajs-3553 494 86 r r NOUN iajs-3553 494 87 e e PROPN iajs-3553 494 88 d d X iajs-3553 494 89 a a PRON iajs-3553 494 90 to to ADP iajs-3553 494 91 r r NOUN iajs-3553 494 92 c c NOUN iajs-3553 494 93 r r NOUN iajs-3553 494 94 1 1 NUM iajs-3553 494 95 r r NOUN iajs-3553 494 96 2 2 NUM iajs-3553 494 97 r r NOUN iajs-3553 494 98 3 3 NUM iajs-3553 494 99 0 0 NUM iajs-3553 494 100 100 100 NUM iajs-3553 494 101 200 200 NUM iajs-3553 494 102 300 300 NUM iajs-3553 494 103 400 400 NUM iajs-3553 494 104 500 500 NUM iajs-3553 494 105 600 600 NUM iajs-3553 494 106 700 700 NUM iajs-3553 494 107 800 800 NUM iajs-3553 494 108 900 900 NUM iajs-3553 494 109 1000 1000 NUM iajs-3553 494 110 0.1 0.1 NUM iajs-3553 494 111 0.2 0.2 NUM iajs-3553 494 112 0.3 0.3 NUM iajs-3553 494 113 0.4 0.4 NUM iajs-3553 494 114 0.5 0.5 NUM iajs-3553 494 115 0.6 0.6 NUM iajs-3553 494 116 0.7 0.7 NUM iajs-3553 494 117 0.8 0.8 NUM iajs-3553 494 118 d d NOUN iajs-3553 494 119 time time NOUN iajs-3553 494 120 to to ADP iajs-3553 494 121 p p NOUN iajs-3553 494 122 p p NOUN iajs-3553 494 123 r r NOUN iajs-3553 494 124 e e PROPN iajs-3553 494 125 d d X iajs-3553 494 126 a a DET iajs-3553 494 127 to to ADP iajs-3553 494 128 r r NOUN iajs-3553 494 129 r r NOUN iajs-3553 494 130 1 1 NUM iajs-3553 494 131 r r NOUN iajs-3553 494 132 2 2 NUM iajs-3553 494 133 r r NOUN iajs-3553 494 134 3 3 NUM iajs-3553 494 135 ihjpas ihjpa NOUN iajs-3553 494 136 . . PUNCT iajs-3553 495 1 2025 2025 NUM iajs-3553 495 2 , , PUNCT iajs-3553 495 3 38(2 38(2 NUM iajs-3553 495 4 ) ) PUNCT iajs-3553 495 5 326 326 NUM iajs-3553 495 6 now now ADV iajs-3553 495 7 , , PUNCT iajs-3553 495 8 to to PART iajs-3553 495 9 study study VERB iajs-3553 495 10 the the DET iajs-3553 495 11 effect effect NOUN iajs-3553 495 12 of of ADP iajs-3553 495 13 parameters parameter NOUN iajs-3553 495 14 on on ADP iajs-3553 495 15 the the DET iajs-3553 495 16 dynamic dynamic ADJ iajs-3553 495 17 behavior behavior NOUN iajs-3553 495 18 of of ADP iajs-3553 495 19 the the DET iajs-3553 495 20 system system NOUN iajs-3553 495 21 ( ( PUNCT iajs-3553 495 22 1 1 NUM iajs-3553 495 23 ) ) PUNCT iajs-3553 495 24 , , PUNCT iajs-3553 495 25 the the DET iajs-3553 495 26 system system NOUN iajs-3553 495 27 ( ( PUNCT iajs-3553 495 28 1 1 X iajs-3553 495 29 ) ) PUNCT iajs-3553 495 30 was be AUX iajs-3553 495 31 numerically numerically ADV iajs-3553 495 32 solved solve VERB iajs-3553 495 33 to to ADP iajs-3553 495 34 the the DET iajs-3553 495 35 data datum NOUN iajs-3553 495 36 in in ADP iajs-3553 495 37 ( ( PUNCT iajs-3553 495 38 24 24 NUM iajs-3553 495 39 ) ) PUNCT iajs-3553 495 40 by by ADP iajs-3553 495 41 changing change VERB iajs-3553 495 42 a a DET iajs-3553 495 43 single single ADJ iajs-3553 495 44 parameter parameter NOUN iajs-3553 495 45 each each DET iajs-3553 495 46 time time NOUN iajs-3553 495 47 the the DET iajs-3553 495 48 results result NOUN iajs-3553 495 49 are be AUX iajs-3553 495 50 given give VERB iajs-3553 495 51 in in ADP iajs-3553 495 52 table table NOUN iajs-3553 495 53 ( ( PUNCT iajs-3553 495 54 2 2 NUM iajs-3553 495 55 ) ) PUNCT iajs-3553 495 56 . . PUNCT iajs-3553 496 1 the the DET iajs-3553 496 2 result result NOUN iajs-3553 496 3 of of ADP iajs-3553 496 4 changing change VERB iajs-3553 496 5 the the DET iajs-3553 496 6 value value NOUN iajs-3553 496 7 of of ADP iajs-3553 496 8 the the DET iajs-3553 496 9 parameter parameter NOUN iajs-3553 496 10 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3553 496 11 in in ADP iajs-3553 496 12 the the DET iajs-3553 496 13 range range NOUN iajs-3553 496 14 0.15 0.15 NUM iajs-3553 496 15 ≀ ≀ NUM iajs-3553 496 16 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 496 17 < < X iajs-3553 496 18 3.7 3.7 NUM iajs-3553 496 19 solution solution NOUN iajs-3553 496 20 approaches approach VERB iajs-3553 496 21 a4 a4 NOUN iajs-3553 496 22 , , PUNCT iajs-3553 496 23 as as SCONJ iajs-3553 496 24 described describe VERB iajs-3553 496 25 in in ADP iajs-3553 496 26 figure figure NOUN iajs-3553 496 27 2 2 NUM iajs-3553 496 28 , , PUNCT iajs-3553 496 29 ( ( PUNCT iajs-3553 496 30 a)for a)for ADP iajs-3553 496 31 typical typical ADJ iajs-3553 496 32 value value NOUN iajs-3553 496 33 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 496 34 = = SYM iajs-3553 496 35 2 2 X iajs-3553 496 36 . . PUNCT iajs-3553 497 1 in in ADP iajs-3553 497 2 the the DET iajs-3553 497 3 range range NOUN iajs-3553 497 4 0.1 0.1 NUM iajs-3553 497 5 ≀ ≀ NUM iajs-3553 497 6 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 497 7 < < X iajs-3553 497 8 0.15 0.15 NUM iajs-3553 497 9 the the DET iajs-3553 497 10 solution solution NOUN iajs-3553 497 11 approaches approach NOUN iajs-3553 497 12 to to ADP iajs-3553 497 13 a5 a5 NOUN iajs-3553 497 14 , , PUNCT iajs-3553 497 15 as as SCONJ iajs-3553 497 16 described describe VERB iajs-3553 497 17 in in ADP iajs-3553 497 18 figure figure NOUN iajs-3553 497 19 2,(b 2,(b NUM iajs-3553 497 20 ) ) PUNCT iajs-3553 497 21 for for ADP iajs-3553 497 22 typical typical ADJ iajs-3553 497 23 value value NOUN iajs-3553 497 24 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 497 25 = = SYM iajs-3553 497 26 0.14 0.14 NUM iajs-3553 497 27 . . PUNCT iajs-3553 498 1 figure figure NOUN iajs-3553 498 2 2 2 NUM iajs-3553 498 3 . . PUNCT iajs-3553 498 4 ( ( PUNCT iajs-3553 498 5 a).(t.s a).(t.s NOUN iajs-3553 498 6 . . PUNCT iajs-3553 498 7 ) ) PUNCT iajs-3553 498 8 of of ADP iajs-3553 498 9 system system NOUN iajs-3553 498 10 solution solution NOUN iajs-3553 498 11 ( ( PUNCT iajs-3553 498 12 1 1 NUM iajs-3553 498 13 ) ) PUNCT iajs-3553 498 14 for for ADP iajs-3553 498 15 the the DET iajs-3553 498 16 given give VERB iajs-3553 498 17 values value NOUN iajs-3553 498 18 in in ADP iajs-3553 498 19 the the DET iajs-3553 498 20 equation equation NOUN iajs-3553 498 21 ( ( PUNCT iajs-3553 498 22 24 24 NUM iajs-3553 498 23 ) ) PUNCT iajs-3553 498 24 with with ADP iajs-3553 498 25 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 498 26 = = SYM iajs-3553 498 27 2 2 NUM iajs-3553 498 28 , , PUNCT iajs-3553 498 29 which which PRON iajs-3553 498 30 approaches approach VERB iajs-3553 498 31 to to ADP iajs-3553 498 32 𝐴4 𝐴4 PROPN iajs-3553 498 33 = = PUNCT iajs-3553 498 34 ( ( PUNCT iajs-3553 498 35 0 0 NUM iajs-3553 498 36 . . NOUN iajs-3553 499 1 0.5716 0.5716 NUM iajs-3553 499 2 . . PUNCT iajs-3553 499 3 0.2356 0.2356 NUM iajs-3553 499 4 ) ) PUNCT iajs-3553 499 5 . . PUNCT iajs-3553 500 1 ( ( PUNCT iajs-3553 500 2 b b NOUN iajs-3553 500 3 ) ) PUNCT iajs-3553 500 4 . . PUNCT iajs-3553 501 1 ( ( PUNCT iajs-3553 501 2 t.s t.s PROPN iajs-3553 501 3 . . PROPN iajs-3553 501 4 ) ) PUNCT iajs-3553 501 5 of of ADP iajs-3553 501 6 system system NOUN iajs-3553 501 7 solution solution NOUN iajs-3553 501 8 ( ( PUNCT iajs-3553 501 9 1 1 NUM iajs-3553 501 10 ) ) PUNCT iajs-3553 501 11 for for ADP iajs-3553 501 12 the the DET iajs-3553 501 13 given give VERB iajs-3553 501 14 values value NOUN iajs-3553 501 15 in in ADP iajs-3553 501 16 the the DET iajs-3553 501 17 equation equation NOUN iajs-3553 501 18 ( ( PUNCT iajs-3553 501 19 24 24 NUM iajs-3553 501 20 ) ) PUNCT iajs-3553 501 21 with with ADP iajs-3553 501 22 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 501 23 = = PUNCT iajs-3553 501 24 0.14 0.14 NUM iajs-3553 501 25 , , PUNCT iajs-3553 501 26 which which PRON iajs-3553 501 27 approaches approach VERB iajs-3553 501 28 to to ADP iajs-3553 501 29 a5 a5 PROPN iajs-3553 501 30 = = PUNCT iajs-3553 501 31 ( ( PUNCT iajs-3553 501 32 2.25226 2.25226 NUM iajs-3553 501 33 . . PUNCT iajs-3553 502 1 0.5627 0.5627 NUM iajs-3553 502 2 . . PUNCT iajs-3553 503 1 1.0574 1.0574 NUM iajs-3553 503 2 ) ) PUNCT iajs-3553 503 3 . . PUNCT iajs-3553 504 1 the the DET iajs-3553 504 2 impact impact NOUN iajs-3553 504 3 of of ADP iajs-3553 504 4 varying vary VERB iajs-3553 504 5 the the DET iajs-3553 504 6 parameter parameter NOUN iajs-3553 504 7 𝑑1 𝑑1 NOUN iajs-3553 504 8 in in ADP iajs-3553 504 9 the the DET iajs-3553 504 10 range range NOUN iajs-3553 504 11 0.001 0.001 NUM iajs-3553 504 12 ≀ ≀ NOUN iajs-3553 504 13 𝑑1 𝑑1 NOUN iajs-3553 504 14 < < X iajs-3553 504 15 0.475 0.475 NUM iajs-3553 504 16 the the DET iajs-3553 504 17 solution solution NOUN iajs-3553 504 18 approaches approach VERB iajs-3553 504 19 to to ADP iajs-3553 504 20 a5 a5 NOUN iajs-3553 504 21 , , PUNCT iajs-3553 504 22 as as SCONJ iajs-3553 504 23 shown show VERB iajs-3553 504 24 in in ADP iajs-3553 504 25 figure figure NOUN iajs-3553 504 26 3 3 NUM iajs-3553 504 27 , , PUNCT iajs-3553 504 28 ( ( PUNCT iajs-3553 504 29 a a X iajs-3553 504 30 ) ) PUNCT iajs-3553 504 31 for for ADP iajs-3553 504 32 typical typical ADJ iajs-3553 504 33 value value NOUN iajs-3553 504 34 𝑑1 𝑑1 NOUN iajs-3553 504 35 = = NOUN iajs-3553 504 36 0.2 0.2 NUM iajs-3553 504 37 . . PUNCT iajs-3553 505 1 in in ADP iajs-3553 505 2 the the DET iajs-3553 505 3 range range NOUN iajs-3553 505 4 0.475 0.475 NUM iajs-3553 505 5 ≀ ≀ NOUN iajs-3553 505 6 𝑑1 𝑑1 NOUN iajs-3553 505 7 < < X iajs-3553 505 8 0.799 0.799 NUM iajs-3553 505 9 solution solution NOUN iajs-3553 505 10 approaches approach NOUN iajs-3553 505 11 a3 a3 NOUN iajs-3553 505 12 , , PUNCT iajs-3553 505 13 as as SCONJ iajs-3553 505 14 described describe VERB iajs-3553 505 15 in in ADP iajs-3553 505 16 figure figure NOUN iajs-3553 505 17 3 3 NUM iajs-3553 505 18 , , PUNCT iajs-3553 505 19 ( ( PUNCT iajs-3553 505 20 b b NOUN iajs-3553 505 21 ) ) PUNCT iajs-3553 505 22 for for ADP iajs-3553 505 23 typical typical ADJ iajs-3553 505 24 value value NOUN iajs-3553 505 25 𝑑1 𝑑1 NOUN iajs-3553 505 26 = = PROPN iajs-3553 505 27 0.4799 0.4799 NUM iajs-3553 505 28 . . PUNCT iajs-3553 506 1 in in ADP iajs-3553 506 2 the the DET iajs-3553 506 3 range range NOUN iajs-3553 506 4 0.799 0.799 NUM iajs-3553 506 5 ≀ ≀ NUM iajs-3553 506 6 𝑑1 𝑑1 NOUN iajs-3553 506 7 < < X iajs-3553 506 8 1 1 NUM iajs-3553 506 9 solution solution NOUN iajs-3553 506 10 approaches approach VERB iajs-3553 506 11 a1 a1 PROPN iajs-3553 506 12 , , PUNCT iajs-3553 506 13 as as SCONJ iajs-3553 506 14 shown show VERB iajs-3553 506 15 in in ADP iajs-3553 506 16 figure figure NOUN iajs-3553 506 17 3 3 NUM iajs-3553 506 18 , , PUNCT iajs-3553 506 19 ( ( PUNCT iajs-3553 506 20 c c NOUN iajs-3553 506 21 ) ) PUNCT iajs-3553 506 22 for for ADP iajs-3553 506 23 typical typical ADJ iajs-3553 506 24 value value NOUN iajs-3553 506 25 𝑑1 𝑑1 NOUN iajs-3553 506 26 = = SYM iajs-3553 506 27 0.18 0.18 NUM iajs-3553 506 28 . . PUNCT iajs-3553 506 29 figure figure NOUN iajs-3553 506 30 3 3 NUM iajs-3553 506 31 . . PUNCT iajs-3553 506 32 ( ( PUNCT iajs-3553 506 33 a),(t.s a),(t.s NOUN iajs-3553 506 34 . . PUNCT iajs-3553 506 35 ) ) PUNCT iajs-3553 506 36 of of ADP iajs-3553 506 37 system system NOUN iajs-3553 506 38 solution solution NOUN iajs-3553 506 39 ( ( PUNCT iajs-3553 506 40 1 1 NUM iajs-3553 506 41 ) ) PUNCT iajs-3553 506 42 for for ADP iajs-3553 506 43 the the DET iajs-3553 506 44 given give VERB iajs-3553 506 45 values value NOUN iajs-3553 506 46 in in ADP iajs-3553 506 47 the the DET iajs-3553 506 48 equation equation NOUN iajs-3553 506 49 ( ( PUNCT iajs-3553 506 50 24 24 NUM iajs-3553 506 51 ) ) PUNCT iajs-3553 506 52 with with ADP iajs-3553 506 53 𝑑1 𝑑1 NOUN iajs-3553 506 54 = = NUM iajs-3553 506 55 0.2 0.2 NUM iajs-3553 506 56 , , PUNCT iajs-3553 506 57 which which PRON iajs-3553 506 58 approaches approach VERB iajs-3553 506 59 to to ADP iajs-3553 506 60 a5 a5 PROPN iajs-3553 506 61 = = SYM iajs-3553 506 62 ( ( PUNCT iajs-3553 506 63 0.6981 0.6981 NUM iajs-3553 506 64 . . PUNCT iajs-3553 506 65 0.53 0.53 NUM iajs-3553 506 66 . . PUNCT iajs-3553 506 67 0.19 0.19 NUM iajs-3553 506 68 ) ) PUNCT iajs-3553 506 69 . . PUNCT iajs-3553 507 1 ( ( PUNCT iajs-3553 507 2 b b X iajs-3553 507 3 ) ) PUNCT iajs-3553 507 4 ( ( PUNCT iajs-3553 507 5 t.s t.s PROPN iajs-3553 507 6 . . PROPN iajs-3553 507 7 ) ) PUNCT iajs-3553 507 8 . . PUNCT iajs-3553 508 1 of of ADP iajs-3553 508 2 system system NOUN iajs-3553 508 3 solution solution NOUN iajs-3553 508 4 ( ( PUNCT iajs-3553 508 5 1 1 NUM iajs-3553 508 6 ) ) PUNCT iajs-3553 508 7 for for ADP iajs-3553 508 8 the the DET iajs-3553 508 9 given give VERB iajs-3553 508 10 values value NOUN iajs-3553 508 11 in in ADP iajs-3553 508 12 the the DET iajs-3553 508 13 equation equation NOUN iajs-3553 508 14 ( ( PUNCT iajs-3553 508 15 24 24 NUM iajs-3553 508 16 ) ) PUNCT iajs-3553 508 17 with with ADP iajs-3553 508 18 𝑑1 𝑑1 NOUN iajs-3553 508 19 = = PROPN iajs-3553 508 20 0.4799 0.4799 NUM iajs-3553 508 21 , , PUNCT iajs-3553 508 22 which which PRON iajs-3553 508 23 approaches approach VERB iajs-3553 508 24 to to PART iajs-3553 508 25 a3 a3 VERB iajs-3553 508 26 = = SYM iajs-3553 508 27 ( ( PUNCT iajs-3553 508 28 0.6932 0.6932 NUM iajs-3553 508 29 . . PUNCT iajs-3553 508 30 0.5383 0.5383 NUM iajs-3553 508 31 . . NOUN iajs-3553 508 32 0 0 NUM iajs-3553 508 33 ) ) PUNCT iajs-3553 508 34 . . PUNCT iajs-3553 509 1 ( ( PUNCT iajs-3553 509 2 c c X iajs-3553 509 3 ) ) PUNCT iajs-3553 509 4 ( ( PUNCT iajs-3553 509 5 t.s t.s PROPN iajs-3553 509 6 . . PROPN iajs-3553 509 7 ) ) PUNCT iajs-3553 509 8 of of ADP iajs-3553 509 9 system system NOUN iajs-3553 509 10 solution solution NOUN iajs-3553 509 11 ( ( PUNCT iajs-3553 509 12 1 1 NUM iajs-3553 509 13 ) ) PUNCT iajs-3553 509 14 for for ADP iajs-3553 509 15 the the DET iajs-3553 509 16 given give VERB iajs-3553 509 17 values value NOUN iajs-3553 509 18 in in ADP iajs-3553 509 19 the the DET iajs-3553 509 20 equation equation NOUN iajs-3553 509 21 ( ( PUNCT iajs-3553 509 22 24 24 NUM iajs-3553 509 23 ) ) PUNCT iajs-3553 509 24 with with ADP iajs-3553 509 25 𝑑1 𝑑1 NOUN iajs-3553 509 26 = = NOUN iajs-3553 509 27 0.81 0.81 NUM iajs-3553 509 28 , , PUNCT iajs-3553 509 29 which which PRON iajs-3553 509 30 approaches approach VERB iajs-3553 509 31 to to ADP iajs-3553 509 32 a1 a1 NOUN iajs-3553 509 33 = = PUNCT iajs-3553 509 34 ( ( PUNCT iajs-3553 509 35 0.999 0.999 NUM iajs-3553 509 36 . . NOUN iajs-3553 510 1 0 0 NUM iajs-3553 510 2 . . NOUN iajs-3553 510 3 0 0 NUM iajs-3553 510 4 ) ) PUNCT iajs-3553 510 5 . . PUNCT iajs-3553 511 1 0 0 NUM iajs-3553 512 1 100 100 NUM iajs-3553 512 2 200 200 NUM iajs-3553 512 3 300 300 NUM iajs-3553 512 4 400 400 NUM iajs-3553 512 5 500 500 NUM iajs-3553 512 6 600 600 NUM iajs-3553 512 7 700 700 NUM iajs-3553 512 8 800 800 NUM iajs-3553 512 9 900 900 NUM iajs-3553 512 10 1000 1000 NUM iajs-3553 512 11 0 0 NUM iajs-3553 512 12 0.5 0.5 NUM iajs-3553 512 13 1 1 NUM iajs-3553 512 14 1.5 1.5 NUM iajs-3553 512 15 2 2 NUM iajs-3553 512 16 2.5 2.5 NUM iajs-3553 512 17 3 3 NUM iajs-3553 512 18 time time NOUN iajs-3553 512 19 p p X iajs-3553 512 20 o o X iajs-3553 512 21 p p X iajs-3553 512 22 u u X iajs-3553 512 23 la la X iajs-3553 512 24 ti ti X iajs-3553 512 25 o o NOUN iajs-3553 512 26 n n ADP iajs-3553 512 27 prey prey VERB iajs-3553 512 28 intermediate intermediate ADJ iajs-3553 512 29 predator predator NOUN iajs-3553 512 30 top top ADJ iajs-3553 512 31 predator predator NOUN iajs-3553 512 32 0 0 NUM iajs-3553 512 33 100 100 NUM iajs-3553 512 34 200 200 NUM iajs-3553 512 35 300 300 NUM iajs-3553 512 36 400 400 NUM iajs-3553 512 37 500 500 NUM iajs-3553 512 38 600 600 NUM iajs-3553 512 39 700 700 NUM iajs-3553 512 40 800 800 NUM iajs-3553 512 41 900 900 NUM iajs-3553 512 42 1000 1000 NUM iajs-3553 512 43 0 0 NUM iajs-3553 512 44 0.5 0.5 NUM iajs-3553 512 45 1 1 NUM iajs-3553 512 46 1.5 1.5 NUM iajs-3553 512 47 2 2 NUM iajs-3553 512 48 2.5 2.5 NUM iajs-3553 512 49 3 3 NUM iajs-3553 512 50 time time NOUN iajs-3553 512 51 p p X iajs-3553 512 52 o o X iajs-3553 512 53 p p X iajs-3553 512 54 u u X iajs-3553 512 55 la la X iajs-3553 512 56 ti ti X iajs-3553 512 57 o o NOUN iajs-3553 512 58 n n ADP iajs-3553 512 59 prey prey VERB iajs-3553 512 60 intermediate intermediate ADJ iajs-3553 512 61 predator predator NOUN iajs-3553 512 62 top top ADJ iajs-3553 512 63 predator predator NOUN iajs-3553 512 64 0 0 NUM iajs-3553 512 65 100 100 NUM iajs-3553 512 66 200 200 NUM iajs-3553 512 67 300 300 NUM iajs-3553 512 68 400 400 NUM iajs-3553 512 69 500 500 NUM iajs-3553 512 70 600 600 NUM iajs-3553 512 71 700 700 NUM iajs-3553 512 72 800 800 NUM iajs-3553 512 73 900 900 NUM iajs-3553 512 74 1000 1000 NUM iajs-3553 512 75 0 0 NUM iajs-3553 512 76 0.1 0.1 NUM iajs-3553 512 77 0.2 0.2 NUM iajs-3553 512 78 0.3 0.3 NUM iajs-3553 512 79 0.4 0.4 NUM iajs-3553 512 80 0.5 0.5 NUM iajs-3553 512 81 0.6 0.6 NUM iajs-3553 512 82 0.7 0.7 NUM iajs-3553 512 83 0.8 0.8 NUM iajs-3553 512 84 0.9 0.9 NUM iajs-3553 512 85 1 1 NUM iajs-3553 512 86 time time NOUN iajs-3553 513 1 p p X iajs-3553 513 2 o o X iajs-3553 513 3 p p X iajs-3553 513 4 u u X iajs-3553 513 5 la la X iajs-3553 513 6 ti ti X iajs-3553 513 7 o o NOUN iajs-3553 513 8 n n ADP iajs-3553 513 9 prey prey VERB iajs-3553 513 10 intermediate intermediate ADJ iajs-3553 513 11 predator predator NOUN iajs-3553 513 12 top top ADJ iajs-3553 513 13 predator predator NOUN iajs-3553 513 14 0 0 NUM iajs-3553 513 15 100 100 NUM iajs-3553 513 16 200 200 NUM iajs-3553 513 17 300 300 NUM iajs-3553 513 18 400 400 NUM iajs-3553 513 19 500 500 NUM iajs-3553 513 20 600 600 NUM iajs-3553 513 21 700 700 NUM iajs-3553 513 22 800 800 NUM iajs-3553 513 23 900 900 NUM iajs-3553 513 24 1000 1000 NUM iajs-3553 513 25 0 0 NUM iajs-3553 513 26 0.1 0.1 NUM iajs-3553 513 27 0.2 0.2 NUM iajs-3553 513 28 0.3 0.3 NUM iajs-3553 513 29 0.4 0.4 NUM iajs-3553 513 30 0.5 0.5 NUM iajs-3553 513 31 0.6 0.6 NUM iajs-3553 513 32 0.7 0.7 NUM iajs-3553 513 33 0.8 0.8 NUM iajs-3553 513 34 0.9 0.9 NUM iajs-3553 513 35 time time NOUN iajs-3553 513 36 p p X iajs-3553 513 37 o o X iajs-3553 513 38 p p X iajs-3553 513 39 u u X iajs-3553 513 40 la la X iajs-3553 513 41 ti ti X iajs-3553 513 42 o o NOUN iajs-3553 513 43 n n ADP iajs-3553 513 44 prey prey VERB iajs-3553 513 45 intermediate intermediate ADJ iajs-3553 513 46 predator predator NOUN iajs-3553 513 47 top top ADJ iajs-3553 513 48 predator predator NOUN iajs-3553 513 49 0 0 NUM iajs-3553 513 50 100 100 NUM iajs-3553 513 51 200 200 NUM iajs-3553 514 1 300 300 NUM iajs-3553 514 2 400 400 NUM iajs-3553 514 3 500 500 NUM iajs-3553 514 4 600 600 NUM iajs-3553 514 5 700 700 NUM iajs-3553 514 6 800 800 NUM iajs-3553 514 7 900 900 NUM iajs-3553 514 8 1000 1000 NUM iajs-3553 514 9 0 0 NUM iajs-3553 514 10 0.1 0.1 NUM iajs-3553 514 11 0.2 0.2 NUM iajs-3553 514 12 0.3 0.3 NUM iajs-3553 514 13 0.4 0.4 NUM iajs-3553 514 14 0.5 0.5 NUM iajs-3553 514 15 0.6 0.6 NUM iajs-3553 514 16 0.7 0.7 NUM iajs-3553 514 17 0.8 0.8 NUM iajs-3553 514 18 0.9 0.9 NUM iajs-3553 514 19 time time NOUN iajs-3553 514 20 p p X iajs-3553 514 21 o o X iajs-3553 514 22 p p X iajs-3553 514 23 u u X iajs-3553 514 24 la la X iajs-3553 514 25 ti ti X iajs-3553 514 26 o o NOUN iajs-3553 514 27 n n ADP iajs-3553 514 28 prey prey VERB iajs-3553 514 29 intermediate intermediate ADJ iajs-3553 514 30 predator predator NOUN iajs-3553 514 31 top top ADJ iajs-3553 514 32 predator predator NOUN iajs-3553 514 33 ihjpas ihjpa NOUN iajs-3553 514 34 . . PUNCT iajs-3553 515 1 2025 2025 NUM iajs-3553 515 2 , , PUNCT iajs-3553 515 3 38(2 38(2 NUM iajs-3553 515 4 ) ) PUNCT iajs-3553 515 5 327 327 NUM iajs-3553 515 6 for for ADP iajs-3553 515 7 the the DET iajs-3553 515 8 parameter parameter NOUN iajs-3553 515 9 𝑑2 𝑑2 NOUN iajs-3553 515 10 in in ADP iajs-3553 515 11 the the DET iajs-3553 515 12 range range NOUN iajs-3553 515 13 0.1 0.1 NUM iajs-3553 515 14 ≀ ≀ NOUN iajs-3553 516 1 𝑑2 𝑑2 VERB iajs-3553 516 2 < < X iajs-3553 516 3 0.869 0.869 NUM iajs-3553 516 4 solution solution NOUN iajs-3553 516 5 approaches approach VERB iajs-3553 516 6 a5 a5 PROPN iajs-3553 516 7 , , PUNCT iajs-3553 516 8 as as SCONJ iajs-3553 516 9 shown show VERB iajs-3553 516 10 in in ADP iajs-3553 516 11 figure figure NOUN iajs-3553 516 12 4,(a 4,(a NUM iajs-3553 516 13 ) ) PUNCT iajs-3553 516 14 for for ADP iajs-3553 516 15 typical typical ADJ iajs-3553 516 16 value value NOUN iajs-3553 516 17 𝑑2 𝑑2 NOUN iajs-3553 516 18 = = SYM iajs-3553 516 19 0.5 0.5 NUM iajs-3553 516 20 , , PUNCT iajs-3553 516 21 the the DET iajs-3553 516 22 range range NOUN iajs-3553 516 23 growing grow VERB iajs-3553 516 24 further far ADV iajs-3553 516 25 0.869 0.869 NUM iajs-3553 516 26 ≀ ≀ NOUN iajs-3553 516 27 𝑑2 𝑑2 NOUN iajs-3553 516 28 < < X iajs-3553 516 29 1 1 NUM iajs-3553 516 30 solution solution NOUN iajs-3553 516 31 approaches approach NOUN iajs-3553 516 32 a3 a3 VERB iajs-3553 516 33 as as SCONJ iajs-3553 516 34 described describe VERB iajs-3553 516 35 in in ADP iajs-3553 516 36 figure figure NOUN iajs-3553 516 37 4,(b 4,(b PROPN iajs-3553 516 38 ) ) PUNCT iajs-3553 516 39 for for ADP iajs-3553 516 40 typical typical ADJ iajs-3553 516 41 value value NOUN iajs-3553 516 42 𝑑2 𝑑2 NOUN iajs-3553 516 43 = = NOUN iajs-3553 516 44 0.921 0.921 NUM iajs-3553 516 45 , , PUNCT iajs-3553 516 46 in in ADP iajs-3553 516 47 the the DET iajs-3553 516 48 range range NOUN iajs-3553 516 49 figure figure NOUN iajs-3553 516 50 4 4 NUM iajs-3553 516 51 . . PUNCT iajs-3553 517 1 ( ( PUNCT iajs-3553 517 2 a a X iajs-3553 517 3 ) ) PUNCT iajs-3553 517 4 ( ( PUNCT iajs-3553 517 5 t.s t.s PROPN iajs-3553 517 6 . . PROPN iajs-3553 517 7 ) ) PUNCT iajs-3553 517 8 of of ADP iajs-3553 517 9 system system NOUN iajs-3553 517 10 solution solution NOUN iajs-3553 517 11 ( ( PUNCT iajs-3553 517 12 1 1 NUM iajs-3553 517 13 ) ) PUNCT iajs-3553 517 14 for for ADP iajs-3553 517 15 the the DET iajs-3553 517 16 given give VERB iajs-3553 517 17 values value NOUN iajs-3553 517 18 in in ADP iajs-3553 517 19 the the DET iajs-3553 517 20 equation equation NOUN iajs-3553 517 21 ( ( PUNCT iajs-3553 517 22 24 24 NUM iajs-3553 517 23 ) ) PUNCT iajs-3553 517 24 with with ADP iajs-3553 517 25 d2 d2 PROPN iajs-3553 517 26 = = PROPN iajs-3553 517 27 0.5,which 0.5,which NUM iajs-3553 517 28 approaches approach VERB iajs-3553 517 29 to to ADP iajs-3553 517 30 a5 a5 PROPN iajs-3553 517 31 = = SYM iajs-3553 517 32 ( ( PUNCT iajs-3553 517 33 0.5875 0.5875 NUM iajs-3553 517 34 . . PUNCT iajs-3553 518 1 0.7139 0.7139 NUM iajs-3553 518 2 . . PUNCT iajs-3553 519 1 0.3857 0.3857 NUM iajs-3553 519 2 ) ) PUNCT iajs-3553 519 3 . . PUNCT iajs-3553 520 1 ( ( PUNCT iajs-3553 520 2 b b X iajs-3553 520 3 ) ) PUNCT iajs-3553 520 4 ( ( PUNCT iajs-3553 520 5 t.s t.s PROPN iajs-3553 520 6 . . PROPN iajs-3553 520 7 ) ) PUNCT iajs-3553 520 8 of of ADP iajs-3553 520 9 system system NOUN iajs-3553 520 10 solution solution NOUN iajs-3553 520 11 ( ( PUNCT iajs-3553 520 12 1 1 NUM iajs-3553 520 13 ) ) PUNCT iajs-3553 520 14 for for ADP iajs-3553 520 15 the the DET iajs-3553 520 16 given give VERB iajs-3553 520 17 values value NOUN iajs-3553 520 18 in in ADP iajs-3553 520 19 the the DET iajs-3553 520 20 equation equation NOUN iajs-3553 520 21 ( ( PUNCT iajs-3553 520 22 24 24 NUM iajs-3553 520 23 ) ) PUNCT iajs-3553 520 24 with with ADP iajs-3553 520 25 d2 d2 PROPN iajs-3553 520 26 = = PROPN iajs-3553 520 27 0.921 0.921 NUM iajs-3553 520 28 , , PUNCT iajs-3553 520 29 which which PRON iajs-3553 520 30 approaches approach VERB iajs-3553 520 31 to to PART iajs-3553 520 32 a3 a3 VERB iajs-3553 520 33 = = SYM iajs-3553 520 34 ( ( PUNCT iajs-3553 520 35 0.2753 0.2753 NUM iajs-3553 520 36 . . PUNCT iajs-3553 521 1 1.2058 1.2058 NUM iajs-3553 521 2 . . NOUN iajs-3553 521 3 0 0 NUM iajs-3553 521 4 ) ) PUNCT iajs-3553 521 5 . . PUNCT iajs-3553 522 1 table table NOUN iajs-3553 522 2 2 2 NUM iajs-3553 522 3 . . PUNCT iajs-3553 523 1 the the DET iajs-3553 523 2 numerical numerical ADJ iajs-3553 523 3 result result PROPN iajs-3553 523 4 . . PUNCT iajs-3553 524 1 range range NOUN iajs-3553 524 2 of of ADP iajs-3553 524 3 parameter parameter NOUN iajs-3553 524 4 stable stable ADJ iajs-3553 524 5 the the DET iajs-3553 524 6 bifurcation bifurcation NOUN iajs-3553 524 7 point point NOUN iajs-3553 524 8 0.1 0.1 NUM iajs-3553 524 9 ≀ ≀ NOUN iajs-3553 524 10 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 524 11 < < X iajs-3553 524 12 0.15 0.15 NUM iajs-3553 524 13 0.15 0.15 NUM iajs-3553 524 14 ≀ ≀ VERB iajs-3553 524 15 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3553 524 16 < < X iajs-3553 524 17 3.7 3.7 NUM iajs-3553 524 18 𝐴4 𝐴4 NOUN iajs-3553 524 19 𝐴5 𝐴5 PROPN iajs-3553 524 20 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3553 524 21 = = PUNCT iajs-3553 524 22 0.15 0.15 NUM iajs-3553 524 23 0.0001 0.0001 NUM iajs-3553 524 24 ≀ ≀ NUM iajs-3553 524 25 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3553 524 26 < < X iajs-3553 524 27 0.1 0.1 NUM iajs-3553 524 28 0.1 0.1 NUM iajs-3553 524 29 ≀ ≀ NOUN iajs-3553 524 30 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3553 524 31 < < X iajs-3553 524 32 0.5 0.5 NUM iajs-3553 524 33 𝐴2 𝐴2 PROPN iajs-3553 524 34 𝐴5 𝐴5 NOUN iajs-3553 524 35 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3553 524 36 = = NUM iajs-3553 524 37 0.1 0.1 NUM iajs-3553 524 38 0.003 0.003 NUM iajs-3553 524 39 ≀ ≀ NOUN iajs-3553 524 40 𝐾1 𝐾1 PROPN iajs-3553 524 41 ≀ ≀ NUM iajs-3553 524 42 0.999 0.999 NUM iajs-3553 524 43 0.999 0.999 NUM iajs-3553 524 44 ≀ ≀ NUM iajs-3553 524 45 𝐾1 𝐾1 PROPN iajs-3553 524 46 < < X iajs-3553 524 47 1 1 NUM iajs-3553 524 48 𝐴5 𝐴5 PROPN iajs-3553 524 49 𝐴4 𝐴4 PROPN iajs-3553 524 50 𝐾1 𝐾1 PROPN iajs-3553 525 1 = = PUNCT iajs-3553 525 2 0.999 0.999 NUM iajs-3553 525 3 0.00007 0.00007 NUM iajs-3553 525 4 ≀ ≀ NUM iajs-3553 525 5 𝐾2 𝐾2 NOUN iajs-3553 525 6 < < X iajs-3553 525 7 0.07 0.07 NUM iajs-3553 525 8 𝐴5 𝐴5 NOUN iajs-3553 525 9 0.0009 0.0009 NUM iajs-3553 525 10 ≀ ≀ NUM iajs-3553 525 11 π‘š1 π‘š1 NOUN iajs-3553 525 12 < < X iajs-3553 525 13 0.001 0.001 NUM iajs-3553 525 14 0.001 0.001 NUM iajs-3553 525 15 ≀ ≀ NUM iajs-3553 525 16 π‘š1 π‘š1 NOUN iajs-3553 525 17 < < X iajs-3553 525 18 0.5 0.5 NUM iajs-3553 525 19 𝐴1 𝐴1 PROPN iajs-3553 525 20 𝐴5 𝐴5 PROPN iajs-3553 525 21 π‘š1 π‘š1 NOUN iajs-3553 525 22 = = SYM iajs-3553 525 23 0.001 0.001 NUM iajs-3553 525 24 0.4 0.4 NUM iajs-3553 525 25 ≀ ≀ NUM iajs-3553 525 26 π‘š2 π‘š2 NOUN iajs-3553 525 27 < < X iajs-3553 525 28 1.2 1.2 NUM iajs-3553 525 29 𝐴5 𝐴5 NOUN iajs-3553 525 30 0.3 0.3 NUM iajs-3553 525 31 ≀ ≀ NUM iajs-3553 525 32 𝛽1 𝛽1 NOUN iajs-3553 525 33 < < X iajs-3553 525 34 0.99 0.99 NUM iajs-3553 525 35 0.99 0.99 NUM iajs-3553 525 36 ≀ ≀ NOUN iajs-3553 525 37 𝛽1 𝛽1 NOUN iajs-3553 525 38 < < X iajs-3553 525 39 2 2 NUM iajs-3553 525 40 𝐴5 𝐴5 NOUN iajs-3553 525 41 𝐴1 𝐴1 PROPN iajs-3553 525 42 𝛽 𝛽 NOUN iajs-3553 525 43 1 1 NUM iajs-3553 525 44 = = SYM iajs-3553 525 45 0.99 0.99 NUM iajs-3553 525 46 0.7 0.7 NUM iajs-3553 525 47 ≀ ≀ NOUN iajs-3553 525 48 𝛽2 𝛽2 NOUN iajs-3553 525 49 < < X iajs-3553 525 50 2.5 2.5 NUM iajs-3553 525 51 𝐴5 𝐴5 NOUN iajs-3553 525 52 0.0001 0.0001 NUM iajs-3553 525 53 ≀ ≀ PUNCT iajs-3553 525 54 𝑙1 𝑙1 NOUN iajs-3553 525 55 < < X iajs-3553 525 56 0.3 0.3 NUM iajs-3553 525 57 𝐴5 𝐴5 NOUN iajs-3553 525 58 0.1 0.1 NUM iajs-3553 525 59 ≀ ≀ NUM iajs-3553 525 60 𝑙2 𝑙2 NOUN iajs-3553 525 61 < < X iajs-3553 525 62 0.7 0.7 NUM iajs-3553 525 63 𝐴5 𝐴5 NOUN iajs-3553 525 64 0.0001 0.0001 NUM iajs-3553 525 65 ≀ ≀ PUNCT iajs-3553 526 1 π‘Ž1 π‘Ž1 ADP iajs-3553 526 2 < < X iajs-3553 526 3 1.6 1.6 NUM iajs-3553 526 4 𝐴5 𝐴5 NOUN iajs-3553 526 5 0.008 0.008 NUM iajs-3553 526 6 ≀ ≀ NOUN iajs-3553 527 1 π‘Ž2 π‘Ž2 NOUN iajs-3553 527 2 < < X iajs-3553 527 3 2 2 NUM iajs-3553 527 4 𝐴5 𝐴5 NOUN iajs-3553 527 5 0.0006 0.0006 NUM iajs-3553 527 6 ≀ ≀ NUM iajs-3553 527 7 π‘Ž3 π‘Ž3 NOUN iajs-3553 527 8 < < X iajs-3553 527 9 1.2 1.2 NUM iajs-3553 527 10 𝐴5 𝐴5 NOUN iajs-3553 527 11 0.0002 0.0002 NUM iajs-3553 527 12 ≀ ≀ NUM iajs-3553 527 13 π‘Ž4 π‘Ž4 PROPN iajs-3553 527 14 < < X iajs-3553 527 15 3 3 NUM iajs-3553 527 16 𝐴5 𝐴5 NOUN iajs-3553 527 17 0.001 0.001 NUM iajs-3553 527 18 ≀ ≀ NUM iajs-3553 527 19 𝑑1 𝑑1 NOUN iajs-3553 527 20 < < X iajs-3553 527 21 0.475 0.475 NUM iajs-3553 527 22 0.475 0.475 NUM iajs-3553 527 23 ≀ ≀ NUM iajs-3553 527 24 𝑑1 𝑑1 NOUN iajs-3553 527 25 < < X iajs-3553 527 26 0.799 0.799 NUM iajs-3553 527 27 0.799 0.799 NUM iajs-3553 527 28 ≀ ≀ NOUN iajs-3553 527 29 𝑑1 𝑑1 NOUN iajs-3553 527 30 < < X iajs-3553 527 31 1 1 NUM iajs-3553 527 32 𝐴5 𝐴5 PROPN iajs-3553 527 33 𝐴3 𝐴3 PROPN iajs-3553 527 34 𝐴1 𝐴1 PROPN iajs-3553 527 35 𝑑1 𝑑1 NOUN iajs-3553 527 36 = = NOUN iajs-3553 527 37 0.475 0.475 NUM iajs-3553 527 38 𝑑1 𝑑1 NOUN iajs-3553 527 39 = = SYM iajs-3553 527 40 0.799 0.799 NUM iajs-3553 527 41 0.01 0.01 NUM iajs-3553 527 42 ≀ ≀ NOUN iajs-3553 528 1 𝑑2 𝑑2 VERB iajs-3553 528 2 < < X iajs-3553 528 3 0.869 0.869 NUM iajs-3553 528 4 0.869 0.869 NUM iajs-3553 528 5 ≀ ≀ NOUN iajs-3553 529 1 𝑑2 𝑑2 NOUN iajs-3553 529 2 < < X iajs-3553 529 3 1 1 NUM iajs-3553 529 4 𝐴5 𝐴5 PROPN iajs-3553 529 5 𝐴3 𝐴3 PROPN iajs-3553 529 6 𝑑2 𝑑2 NOUN iajs-3553 529 7 = = NOUN iajs-3553 529 8 0.869 0.869 NUM iajs-3553 529 9 5 5 NUM iajs-3553 529 10 . . PUNCT iajs-3553 529 11 conclusion conclusion NOUN iajs-3553 529 12 in in ADP iajs-3553 529 13 this this DET iajs-3553 529 14 work work NOUN iajs-3553 529 15 , , PUNCT iajs-3553 529 16 the the DET iajs-3553 529 17 occurrence occurrence NOUN iajs-3553 529 18 of of ADP iajs-3553 529 19 local local ADJ iajs-3553 529 20 bifurcation bifurcation NOUN iajs-3553 529 21 have have AUX iajs-3553 529 22 been be AUX iajs-3553 529 23 discussed discuss VERB iajs-3553 529 24 with with ADP iajs-3553 529 25 an an DET iajs-3553 529 26 appropriate appropriate ADJ iajs-3553 529 27 conditions condition NOUN iajs-3553 529 28 of of ADP iajs-3553 529 29 food food NOUN iajs-3553 529 30 chain chain NOUN iajs-3553 529 31 which which PRON iajs-3553 529 32 contains contain VERB iajs-3553 529 33 a a DET iajs-3553 529 34 prey prey NOUN iajs-3553 529 35 – – PUNCT iajs-3553 529 36 intermediate intermediate ADJ iajs-3553 529 37 predator predator NOUN iajs-3553 529 38 - - PUNCT iajs-3553 529 39 top top ADJ iajs-3553 529 40 predator predator NOUN iajs-3553 529 41 model model NOUN iajs-3553 529 42 with with ADP iajs-3553 529 43 fear fear NOUN iajs-3553 529 44 and and CCONJ iajs-3553 529 45 crowley crowley PROPN iajs-3553 529 46 - - PUNCT iajs-3553 529 47 martin martin PROPN iajs-3553 529 48 - - PUNCT iajs-3553 529 49 type type NOUN iajs-3553 529 50 of of ADP iajs-3553 529 51 functional functional ADJ iajs-3553 529 52 response response NOUN iajs-3553 529 53 have have AUX iajs-3553 529 54 been be AUX iajs-3553 529 55 studied study VERB iajs-3553 529 56 , , PUNCT iajs-3553 529 57 transcritical transcritical ADJ iajs-3553 529 58 and and CCONJ iajs-3553 529 59 pitch pitch NOUN iajs-3553 529 60 fork fork NOUN iajs-3553 529 61 bifurcation bifurcation NOUN iajs-3553 529 62 occurrence occurrence NOUN iajs-3553 529 63 near near ADP iajs-3553 529 64 𝐴1 𝐴1 PROPN iajs-3553 529 65 . . PUNCT iajs-3553 530 1 𝐴2 𝐴2 PROPN iajs-3553 530 2 . . PUNCT iajs-3553 531 1 𝐴3 𝐴3 PROPN iajs-3553 531 2 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-3553 531 3 𝐴4 𝐴4 PROPN iajs-3553 531 4 , , PUNCT iajs-3553 531 5 while while SCONJ iajs-3553 531 6 a a DET iajs-3553 531 7 saddle saddle NOUN iajs-3553 531 8 - - PUNCT iajs-3553 531 9 node node NOUN iajs-3553 531 10 bifurcation bifurcation NOUN iajs-3553 531 11 occurs occur VERB iajs-3553 531 12 0 0 NUM iajs-3553 531 13 100 100 NUM iajs-3553 531 14 200 200 NUM iajs-3553 531 15 300 300 NUM iajs-3553 531 16 400 400 NUM iajs-3553 531 17 500 500 NUM iajs-3553 531 18 600 600 NUM iajs-3553 531 19 700 700 NUM iajs-3553 531 20 800 800 NUM iajs-3553 531 21 900 900 NUM iajs-3553 531 22 1000 1000 NUM iajs-3553 531 23 0.35 0.35 NUM iajs-3553 531 24 0.4 0.4 NUM iajs-3553 531 25 0.45 0.45 NUM iajs-3553 531 26 0.5 0.5 NUM iajs-3553 531 27 0.55 0.55 NUM iajs-3553 531 28 0.6 0.6 NUM iajs-3553 531 29 0.65 0.65 NUM iajs-3553 531 30 0.7 0.7 NUM iajs-3553 531 31 0.75 0.75 NUM iajs-3553 531 32 0.8 0.8 NUM iajs-3553 531 33 time time NOUN iajs-3553 531 34 p p X iajs-3553 531 35 o o X iajs-3553 531 36 p p X iajs-3553 531 37 u u X iajs-3553 531 38 la la X iajs-3553 531 39 ti ti X iajs-3553 531 40 o o NOUN iajs-3553 531 41 n n CCONJ iajs-3553 531 42 a a DET iajs-3553 531 43 prey prey NOUN iajs-3553 531 44 intermediate intermediate ADJ iajs-3553 531 45 predator predator NOUN iajs-3553 531 46 top top ADJ iajs-3553 531 47 predator predator NOUN iajs-3553 531 48 0 0 NUM iajs-3553 531 49 100 100 NUM iajs-3553 531 50 200 200 NUM iajs-3553 531 51 300 300 NUM iajs-3553 531 52 400 400 NUM iajs-3553 531 53 500 500 NUM iajs-3553 531 54 600 600 NUM iajs-3553 531 55 700 700 NUM iajs-3553 531 56 800 800 NUM iajs-3553 531 57 900 900 NUM iajs-3553 531 58 1000 1000 NUM iajs-3553 531 59 0 0 NUM iajs-3553 531 60 0.2 0.2 NUM iajs-3553 531 61 0.4 0.4 NUM iajs-3553 531 62 0.6 0.6 NUM iajs-3553 531 63 0.8 0.8 NUM iajs-3553 531 64 1 1 NUM iajs-3553 531 65 1.2 1.2 NUM iajs-3553 531 66 1.4 1.4 NUM iajs-3553 531 67 b b NOUN iajs-3553 531 68 time time NOUN iajs-3553 532 1 p p X iajs-3553 532 2 o o X iajs-3553 532 3 p p X iajs-3553 532 4 u u X iajs-3553 532 5 la la X iajs-3553 532 6 ti ti X iajs-3553 532 7 o o NOUN iajs-3553 532 8 n n ADP iajs-3553 532 9 prey prey VERB iajs-3553 532 10 intermediate intermediate ADJ iajs-3553 532 11 predator predator NOUN iajs-3553 532 12 top top ADJ iajs-3553 532 13 predator predator NOUN iajs-3553 532 14 ihjpas ihjpa NOUN iajs-3553 532 15 . . PUNCT iajs-3553 533 1 2025 2025 NUM iajs-3553 533 2 , , PUNCT iajs-3553 533 3 38(2 38(2 NUM iajs-3553 533 4 ) ) PUNCT iajs-3553 533 5 328 328 NUM iajs-3553 533 6 near near ADP iajs-3553 533 7 𝐴5.finally 𝐴5.finally PROPN iajs-3553 533 8 , , PUNCT iajs-3553 533 9 in in ADP iajs-3553 533 10 order order NOUN iajs-3553 533 11 to to PART iajs-3553 533 12 demonstrate demonstrate VERB iajs-3553 533 13 how how SCONJ iajs-3553 533 14 local local ADJ iajs-3553 533 15 bifurcation bifurcation NOUN iajs-3553 533 16 occurs occur VERB iajs-3553 533 17 in in ADP iajs-3553 533 18 this this DET iajs-3553 533 19 system system NOUN iajs-3553 533 20 , , PUNCT iajs-3553 533 21 numerical numerical ADJ iajs-3553 533 22 simulations simulation NOUN iajs-3553 533 23 are be AUX iajs-3553 533 24 employed employ VERB iajs-3553 533 25 . . PUNCT iajs-3553 534 1 the the DET iajs-3553 534 2 results result NOUN iajs-3553 534 3 of of ADP iajs-3553 534 4 these these DET iajs-3553 534 5 simulations simulation NOUN iajs-3553 534 6 are be AUX iajs-3553 534 7 as as SCONJ iajs-3553 534 8 follows follow VERB iajs-3553 534 9 : : PUNCT iajs-3553 534 10 * * PUNCT iajs-3553 534 11 system system NOUN iajs-3553 534 12 ( ( PUNCT iajs-3553 534 13 1 1 X iajs-3553 534 14 ) ) PUNCT iajs-3553 534 15 has have VERB iajs-3553 534 16 no no DET iajs-3553 534 17 periodic periodic ADJ iajs-3553 534 18 dynamics dynamic NOUN iajs-3553 534 19 * * PUNCT iajs-3553 534 20 the the DET iajs-3553 534 21 parameters parameter NOUN iajs-3553 534 22 π‘Ÿπ‘–. π‘Ÿπ‘–. VERB iajs-3553 534 23 𝑑𝑖 𝑑𝑖 VERB iajs-3553 534 24 . . PUNCT iajs-3553 535 1 𝑖 𝑖 X iajs-3553 535 2 = = NOUN iajs-3553 536 1 1 1 NUM iajs-3553 536 2 . . NUM iajs-3553 536 3 2 2 NUM iajs-3553 536 4 , , PUNCT iajs-3553 536 5 𝐾1 𝐾1 PROPN iajs-3553 536 6 . . PROPN iajs-3553 536 7 π‘š1 π‘š1 PROPN iajs-3553 536 8 . . PUNCT iajs-3553 537 1 𝛽1 𝛽1 PROPN iajs-3553 537 2 have have VERB iajs-3553 537 3 a a DET iajs-3553 537 4 significant significant ADJ iajs-3553 537 5 impact impact NOUN iajs-3553 537 6 in in ADP iajs-3553 537 7 the the DET iajs-3553 537 8 system system NOUN iajs-3553 537 9 dynamics dynamic NOUN iajs-3553 537 10 ( ( PUNCT iajs-3553 537 11 1 1 NUM iajs-3553 537 12 ) ) PUNCT iajs-3553 537 13 , , PUNCT iajs-3553 537 14 as as SCONJ iajs-3553 537 15 opposed oppose VERB iajs-3553 537 16 to to ADP iajs-3553 537 17 other other ADJ iajs-3553 537 18 parameters parameter NOUN iajs-3553 537 19 𝐾2 𝐾2 PROPN iajs-3553 537 20 . . PUNCT iajs-3553 538 1 π‘š2 π‘š2 NOUN iajs-3553 538 2 . . PUNCT iajs-3553 539 1 𝛽2 𝛽2 PROPN iajs-3553 539 2 . . PUNCT iajs-3553 540 1 𝑙𝑖. 𝑙𝑖. PROPN iajs-3553 540 2 𝑖 𝑖 SYM iajs-3553 541 1 = = PUNCT iajs-3553 541 2 1 1 NUM iajs-3553 541 3 . . NOUN iajs-3553 541 4 2 2 NUM iajs-3553 541 5 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-3553 541 6 π‘Žπ‘– π‘Žπ‘– INTJ iajs-3553 541 7 . . PUNCT iajs-3553 542 1 𝑖 𝑖 PUNCT iajs-3553 542 2 = = NOUN iajs-3553 543 1 1 1 NUM iajs-3553 543 2 . . NOUN iajs-3553 544 1 2 2 NUM iajs-3553 544 2 . . NOUN iajs-3553 544 3 3 3 NUM iajs-3553 544 4 . . NOUN iajs-3553 544 5 4 4 NUM iajs-3553 544 6 the the DET iajs-3553 544 7 solution solution NOUN iajs-3553 544 8 continues continue VERB iajs-3553 544 9 to to PART iajs-3553 544 10 approach approach VERB iajs-3553 544 11 the the DET iajs-3553 544 12 equilibrium equilibrium NOUN iajs-3553 544 13 point point NOUN iajs-3553 544 14 . . PUNCT iajs-3553 545 1 acknowledgment acknowledgment NOUN iajs-3553 545 2 our our PRON iajs-3553 545 3 researcher researcher NOUN iajs-3553 545 4 extends extend VERB iajs-3553 545 5 his his PRON iajs-3553 545 6 sincere sincere ADJ iajs-3553 545 7 thanks thank NOUN iajs-3553 545 8 to to ADP iajs-3553 545 9 the the DET iajs-3553 545 10 editor editor NOUN iajs-3553 545 11 and and CCONJ iajs-3553 545 12 members member NOUN iajs-3553 545 13 of of ADP iajs-3553 545 14 the the DET iajs-3553 545 15 preparatory preparatory PROPN iajs-3553 545 16 committee committee NOUN iajs-3553 545 17 of of ADP iajs-3553 545 18 the the DET iajs-3553 545 19 ibn ibn PROPN iajs-3553 545 20 al al PROPN iajs-3553 545 21 - - PUNCT iajs-3553 545 22 haitham haitham PROPN iajs-3553 545 23 journal journal PROPN iajs-3553 545 24 of of ADP iajs-3553 545 25 pure pure ADJ iajs-3553 545 26 and and CCONJ iajs-3553 545 27 applied applied ADJ iajs-3553 545 28 science science NOUN iajs-3553 545 29 . . PUNCT iajs-3553 546 1 conflict conflict NOUN iajs-3553 546 2 of of ADP iajs-3553 546 3 interest interest NOUN iajs-3553 546 4 there there PRON iajs-3553 546 5 are be VERB iajs-3553 546 6 no no DET iajs-3553 546 7 conflicts conflict NOUN iajs-3553 546 8 of of ADP iajs-3553 546 9 interest interest NOUN iajs-3553 546 10 . . PUNCT iajs-3553 547 1 funding funding NOUN iajs-3553 547 2 there there PRON iajs-3553 547 3 is be VERB iajs-3553 547 4 no no DET iajs-3553 547 5 funding funding NOUN iajs-3553 547 6 for for ADP iajs-3553 547 7 the the DET iajs-3553 547 8 article article NOUN iajs-3553 547 9 . . PUNCT iajs-3553 548 1 references reference NOUN iajs-3553 548 2 1 1 NUM iajs-3553 548 3 . . PUNCT iajs-3553 549 1 meng meng PROPN iajs-3553 549 2 xy xy PROPN iajs-3553 549 3 , , PUNCT iajs-3553 549 4 huo huo PROPN iajs-3553 549 5 hf hf PROPN iajs-3553 549 6 , , PUNCT iajs-3553 549 7 zhang zhang PROPN iajs-3553 549 8 xb xb PROPN iajs-3553 549 9 . . PUNCT iajs-3553 550 1 stability stability NOUN iajs-3553 550 2 and and CCONJ iajs-3553 550 3 global global ADJ iajs-3553 550 4 hopf hopf ADJ iajs-3553 550 5 bifurcation bifurcation NOUN iajs-3553 550 6 in in ADP iajs-3553 550 7 a a DET iajs-3553 550 8 delayed delay VERB iajs-3553 550 9 food food NOUN iajs-3553 550 10 web web NOUN iajs-3553 550 11 consisting consisting NOUN iajs-3553 550 12 of of ADP iajs-3553 550 13 a a DET iajs-3553 550 14 prey prey NOUN iajs-3553 550 15 and and CCONJ iajs-3553 550 16 two two NUM iajs-3553 550 17 predators predator NOUN iajs-3553 550 18 . . PUNCT iajs-3553 551 1 commun commun PROPN iajs-3553 551 2 nonlinear nonlinear PROPN iajs-3553 551 3 sci sci PROPN iajs-3553 551 4 numer numer PROPN iajs-3553 551 5 simul simul PROPN iajs-3553 551 6 . . PUNCT iajs-3553 552 1 2011;16:4335–4348 2011;16:4335–4348 NUM iajs-3553 552 2 . . PUNCT iajs-3553 553 1 https://doi.org/10.1016/j.cnsns.2011.03.009 https://doi.org/10.1016/j.cnsns.2011.03.009 X iajs-3553 553 2 2 2 NUM iajs-3553 553 3 . . PUNCT iajs-3553 553 4 arancibia arancibia NOUN iajs-3553 553 5 - - PUNCT iajs-3553 553 6 ibarra ibarra PROPN iajs-3553 553 7 c c PROPN iajs-3553 553 8 , , PUNCT iajs-3553 553 9 aguirre aguirre PROPN iajs-3553 553 10 p p X iajs-3553 553 11 , , PUNCT iajs-3553 553 12 flores flores PROPN iajs-3553 553 13 j j PROPN iajs-3553 553 14 , , PUNCT iajs-3553 553 15 van van PROPN iajs-3553 553 16 heijster heijster PROPN iajs-3553 553 17 p. p. PROPN iajs-3553 553 18 bifurcation bifurcation NOUN iajs-3553 553 19 analysis analysis NOUN iajs-3553 553 20 of of ADP iajs-3553 553 21 a a DET iajs-3553 553 22 predator predator NOUN iajs-3553 553 23 - - PUNCT iajs-3553 553 24 prey prey NOUN iajs-3553 553 25 model model NOUN iajs-3553 553 26 with with ADP iajs-3553 553 27 predator predator NOUN iajs-3553 553 28 intraspecific intraspecific NOUN iajs-3553 553 29 interactions interaction NOUN iajs-3553 553 30 and and CCONJ iajs-3553 553 31 ratio ratio NOUN iajs-3553 553 32 - - PUNCT iajs-3553 553 33 dependent dependent ADJ iajs-3553 553 34 functional functional ADJ iajs-3553 553 35 response response NOUN iajs-3553 553 36 . . PUNCT iajs-3553 554 1 appl appl PROPN iajs-3553 554 2 math math PROPN iajs-3553 554 3 comput comput NOUN iajs-3553 554 4 . . PUNCT iajs-3553 555 1 2021;402:126152 2021;402:126152 NUM iajs-3553 555 2 . . PUNCT iajs-3553 556 1 https://doi.org/10.1016/j.amc.2021.126152 https://doi.org/10.1016/j.amc.2021.126152 PROPN iajs-3553 556 2 3 3 X iajs-3553 556 3 . . PUNCT iajs-3553 556 4 magudeeswaran magudeeswaran PROPN iajs-3553 556 5 s s PROPN iajs-3553 556 6 , , PUNCT iajs-3553 556 7 vinoth vinoth NOUN iajs-3553 556 8 s s NOUN iajs-3553 556 9 , , PUNCT iajs-3553 556 10 sathiyanathan sathiyanathan PROPN iajs-3553 556 11 k k PROPN iajs-3553 556 12 , , PUNCT iajs-3553 556 13 sivabalan sivabalan ADJ iajs-3553 556 14 m. m. NOUN iajs-3553 556 15 impact impact NOUN iajs-3553 556 16 of of ADP iajs-3553 556 17 fear fear NOUN iajs-3553 556 18 on on ADP iajs-3553 556 19 delayed delay VERB iajs-3553 556 20 three three NUM iajs-3553 556 21 species specie NOUN iajs-3553 556 22 food food NOUN iajs-3553 556 23 - - PUNCT iajs-3553 556 24 web web NOUN iajs-3553 556 25 model model NOUN iajs-3553 556 26 with with ADP iajs-3553 556 27 holling holle VERB iajs-3553 556 28 type type NOUN iajs-3553 556 29 - - PUNCT iajs-3553 556 30 ii ii NOUN iajs-3553 556 31 functional functional ADJ iajs-3553 556 32 response response NOUN iajs-3553 556 33 . . PUNCT iajs-3553 557 1 int int NOUN iajs-3553 557 2 j j PROPN iajs-3553 557 3 biomath biomath NOUN iajs-3553 557 4 . . PUNCT iajs-3553 558 1 2022;15:2250014 2022;15:2250014 X iajs-3553 558 2 . . PUNCT iajs-3553 559 1 https://doi.org/10.1142/s1793524522500140 https://doi.org/10.1142/s1793524522500140 NUM iajs-3553 559 2 4 4 NUM iajs-3553 559 3 . . PUNCT iajs-3553 560 1 majeed majeed PROPN iajs-3553 560 2 aa aa PROPN iajs-3553 560 3 , , PUNCT iajs-3553 560 4 ismaeeb ismaeeb PROPN iajs-3553 560 5 mh mh PROPN iajs-3553 560 6 . . PUNCT iajs-3553 561 1 the the DET iajs-3553 561 2 bifurcation bifurcation NOUN iajs-3553 561 3 analysis analysis NOUN iajs-3553 561 4 of of ADP iajs-3553 561 5 prey prey NOUN iajs-3553 561 6 - - PUNCT iajs-3553 561 7 predator predator NOUN iajs-3553 561 8 model model NOUN iajs-3553 561 9 in in ADP iajs-3553 561 10 the the DET iajs-3553 561 11 presence presence NOUN iajs-3553 561 12 of of ADP iajs-3553 561 13 stage stage NOUN iajs-3553 561 14 structured structure VERB iajs-3553 561 15 with with ADP iajs-3553 561 16 harvesting harvesting NOUN iajs-3553 561 17 and and CCONJ iajs-3553 561 18 toxicity toxicity NOUN iajs-3553 561 19 . . PUNCT iajs-3553 562 1 in in ADP iajs-3553 562 2 : : PUNCT iajs-3553 562 3 journal journal PROPN iajs-3553 562 4 of of ADP iajs-3553 562 5 physics physics PROPN iajs-3553 562 6 : : PUNCT iajs-3553 562 7 conference conference NOUN iajs-3553 562 8 series series NOUN iajs-3553 562 9 ; ; PUNCT iajs-3553 562 10 iop iop NOUN iajs-3553 562 11 publishing publishing NOUN iajs-3553 562 12 ; ; PUNCT iajs-3553 562 13 2019 2019 NUM iajs-3553 562 14 ; ; PUNCT iajs-3553 562 15 vol vol NOUN iajs-3553 562 16 . . PROPN iajs-3553 562 17 1362 1362 NUM iajs-3553 562 18 , , PUNCT iajs-3553 562 19 p. p. NOUN iajs-3553 562 20 12155 12155 NUM iajs-3553 562 21 . . PUNCT iajs-3553 563 1 https://doi.org/10.1088/1742-6596/1362/1/012155 https://doi.org/10.1088/1742-6596/1362/1/012155 NOUN iajs-3553 563 2 5 5 X iajs-3553 563 3 . . PUNCT iajs-3553 563 4 din din VERB iajs-3553 563 5 q. q. PROPN iajs-3553 563 6 stability stability PROPN iajs-3553 563 7 , , PUNCT iajs-3553 563 8 bifurcation bifurcation NOUN iajs-3553 563 9 analysis analysis NOUN iajs-3553 563 10 and and CCONJ iajs-3553 563 11 chaos chaos NOUN iajs-3553 563 12 control control NOUN iajs-3553 563 13 for for ADP iajs-3553 563 14 a a DET iajs-3553 563 15 predator predator NOUN iajs-3553 563 16 - - PUNCT iajs-3553 563 17 prey prey NOUN iajs-3553 563 18 system system NOUN iajs-3553 563 19 . . PUNCT iajs-3553 564 1 j j PROPN iajs-3553 564 2 vib vib PROPN iajs-3553 564 3 control control PROPN iajs-3553 564 4 . . PUNCT iajs-3553 565 1 2019;25:612–626 2019;25:612–626 X iajs-3553 565 2 . . PUNCT iajs-3553 566 1 https://doi.org/10.1177/1077546318790871 https://doi.org/10.1177/1077546318790871 PROPN iajs-3553 566 2 6 6 NUM iajs-3553 566 3 . . PUNCT iajs-3553 567 1 prasad prasad PROPN iajs-3553 567 2 kd kd PROPN iajs-3553 567 3 , , PUNCT iajs-3553 567 4 sasmal sasmal NOUN iajs-3553 567 5 sk sk VERB iajs-3553 567 6 . . 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