id sid tid token lemma pos cana-1090 1 1 communications communication NOUN cana-1090 1 2 on on ADP cana-1090 1 3 applied apply VERB cana-1090 1 4 nonlinear nonlinear ADJ cana-1090 1 5 analysis analysis NOUN cana-1090 1 6 issn issn NOUN cana-1090 1 7 : : PUNCT cana-1090 1 8 1074 1074 NUM cana-1090 1 9 - - PUNCT cana-1090 1 10 133x 133x NUM cana-1090 1 11 vol vol NOUN cana-1090 1 12 31 31 NUM cana-1090 1 13 no no NOUN cana-1090 1 14 . . PUNCT cana-1090 2 1 5s 5s NUM cana-1090 2 2 ( ( PUNCT cana-1090 2 3 2024 2024 NUM cana-1090 2 4 ) ) PUNCT cana-1090 2 5 552 552 NUM cana-1090 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 2 7 stability stability NOUN cana-1090 2 8 and and CCONJ cana-1090 2 9 bifurcation bifurcation NOUN cana-1090 2 10 in in ADP cana-1090 2 11 a a DET cana-1090 2 12 second second ADJ cana-1090 2 13 - - PUNCT cana-1090 2 14 order order NOUN cana-1090 2 15 difference difference NOUN cana-1090 2 16 system system NOUN cana-1090 2 17 smitha smitha PROPN cana-1090 2 18 mary mary PROPN cana-1090 2 19 mathew mathew PROPN cana-1090 2 20 𝟏 𝟏 PROPN cana-1090 2 21 , , PUNCT cana-1090 2 22 d.s.dilip d.s.dilip ADJ cana-1090 2 23 𝟐 𝟐 NUM cana-1090 2 24 , , PUNCT cana-1090 2 25 sibi sibi X cana-1090 2 26 c. c. PROPN cana-1090 2 27 babu babu PROPN cana-1090 2 28 πŸ‘ πŸ‘ PROPN cana-1090 2 29 1,2,3department 1,2,3department NUM cana-1090 2 30 of of ADP cana-1090 2 31 mathematics mathematic NOUN cana-1090 2 32 , , PUNCT cana-1090 2 33 st.john st.john NOUN cana-1090 2 34 ’s ’s PART cana-1090 2 35 college college NOUN cana-1090 2 36 , , PUNCT cana-1090 2 37 anchal anchal PROPN cana-1090 2 38 , , PUNCT cana-1090 2 39 kollam kollam PROPN cana-1090 2 40 district district PROPN cana-1090 2 41 , , PUNCT cana-1090 2 42 kerala kerala PROPN cana-1090 2 43 , , PUNCT cana-1090 2 44 india india PROPN cana-1090 2 45 . . PUNCT cana-1090 3 1 1email 1email NUM cana-1090 3 2 : : PUNCT cana-1090 3 3 smathew11@gmail.com smathew11@gmail.com NOUN cana-1090 3 4 , , PUNCT cana-1090 3 5 2email 2email NUM cana-1090 3 6 : : PUNCT cana-1090 3 7 dilip@stjohns.ac.in dilip@stjohns.ac.in NOUN cana-1090 3 8 , , PUNCT cana-1090 3 9 3email 3email NUM cana-1090 3 10 : : PUNCT cana-1090 3 11 sibicbabu@stjohns.ac.in sibicbabu@stjohns.ac.in PROPN cana-1090 3 12 article article NOUN cana-1090 3 13 history history NOUN cana-1090 3 14 : : PUNCT cana-1090 3 15 received receive VERB cana-1090 3 16 : : PUNCT cana-1090 3 17 15 15 NUM cana-1090 3 18 - - SYM cana-1090 3 19 05 05 NUM cana-1090 3 20 - - PUNCT cana-1090 3 21 2024 2024 NUM cana-1090 3 22 revised revise VERB cana-1090 3 23 : : PUNCT cana-1090 3 24 18 18 NUM cana-1090 3 25 - - SYM cana-1090 3 26 06 06 NUM cana-1090 3 27 - - PUNCT cana-1090 3 28 2024 2024 NUM cana-1090 3 29 accepted accept VERB cana-1090 3 30 : : PUNCT cana-1090 3 31 11 11 NUM cana-1090 3 32 - - SYM cana-1090 3 33 07 07 NUM cana-1090 3 34 - - PUNCT cana-1090 3 35 2024 2024 NUM cana-1090 3 36 abstract abstract NOUN cana-1090 3 37 : : PUNCT cana-1090 3 38 in in ADP cana-1090 3 39 the the DET cana-1090 3 40 paper paper NOUN cana-1090 3 41 , , PUNCT cana-1090 3 42 we we PRON cana-1090 3 43 examine examine VERB cana-1090 3 44 the the DET cana-1090 3 45 system system NOUN cana-1090 3 46 π‘₯𝑛+1 π‘₯𝑛+1 X cana-1090 3 47 = = PUNCT cana-1090 3 48 𝛼1 𝛼1 PROPN cana-1090 3 49 + + CCONJ cana-1090 3 50 π‘Ž1π‘’βˆ’π‘₯π‘›βˆ’1 π‘Ž1π‘’βˆ’π‘₯π‘›βˆ’1 NOUN cana-1090 3 51 + + CCONJ cana-1090 3 52 𝑏1π‘¦π‘›π‘’βˆ’π‘¦π‘›βˆ’1 𝑏1π‘¦π‘›π‘’βˆ’π‘¦π‘›βˆ’1 NOUN cana-1090 4 1 + + CCONJ cana-1090 4 2 𝑐1π‘’βˆ’π‘§π‘›βˆ’1 𝑐1π‘’βˆ’π‘§π‘›βˆ’1 NOUN cana-1090 4 3 , , PUNCT cana-1090 4 4 𝑦𝑛+1 𝑦𝑛+1 NUM cana-1090 4 5 = = SYM cana-1090 4 6 𝛼2 𝛼2 PROPN cana-1090 4 7 + + CCONJ cana-1090 4 8 π‘Ž2π‘’βˆ’π‘¦π‘›βˆ’1 π‘Ž2π‘’βˆ’π‘¦π‘›βˆ’1 NOUN cana-1090 4 9 + + CCONJ cana-1090 4 10 𝑏2π‘§π‘›π‘’βˆ’π‘§π‘›βˆ’1 𝑏2π‘§π‘›π‘’βˆ’π‘§π‘›βˆ’1 PROPN cana-1090 5 1 + + CCONJ cana-1090 5 2 𝑐2π‘’βˆ’π‘₯π‘›βˆ’1 𝑐2π‘’βˆ’π‘₯π‘›βˆ’1 NOUN cana-1090 5 3 , , PUNCT cana-1090 5 4 ( ( PUNCT cana-1090 5 5 1 1 X cana-1090 5 6 ) ) PUNCT cana-1090 5 7 𝑧𝑛+1 𝑧𝑛+1 NUM cana-1090 6 1 = = PUNCT cana-1090 6 2 𝛼3 𝛼3 NOUN cana-1090 6 3 + + CCONJ cana-1090 6 4 π‘Ž3π‘’βˆ’π‘§π‘›βˆ’1 π‘Ž3π‘’βˆ’π‘§π‘›βˆ’1 VERB cana-1090 6 5 + + NOUN cana-1090 6 6 𝑏3π‘₯π‘›π‘’βˆ’π‘₯π‘›βˆ’1 𝑏3π‘₯π‘›π‘’βˆ’π‘₯π‘›βˆ’1 NOUN cana-1090 7 1 + + CCONJ cana-1090 7 2 𝑐3π‘’βˆ’π‘¦π‘›βˆ’1 𝑐3π‘’βˆ’π‘¦π‘›βˆ’1 NOUN cana-1090 7 3 , , PUNCT cana-1090 7 4 𝑛 𝑛 PROPN cana-1090 7 5 = = SYM cana-1090 7 6 0,1,2 0,1,2 NUM cana-1090 7 7 , , PUNCT cana-1090 7 8 … … PUNCT cana-1090 7 9 , , PUNCT cana-1090 7 10 where where SCONJ cana-1090 7 11 𝛼1 𝛼1 NOUN cana-1090 7 12 , , PUNCT cana-1090 7 13 𝛼2 𝛼2 PROPN cana-1090 7 14 , , PUNCT cana-1090 7 15 𝛼3 𝛼3 ADV cana-1090 7 16 , , PUNCT cana-1090 7 17 π‘Ž1 π‘Ž1 PRON cana-1090 7 18 , , PUNCT cana-1090 7 19 π‘Ž2 π‘Ž2 PROPN cana-1090 7 20 , , PUNCT cana-1090 7 21 π‘Ž3 π‘Ž3 PROPN cana-1090 7 22 , , PUNCT cana-1090 7 23 𝑏1 𝑏1 NOUN cana-1090 7 24 , , PUNCT cana-1090 7 25 𝑏2 𝑏2 NOUN cana-1090 7 26 , , PUNCT cana-1090 7 27 𝑏3 𝑏3 NOUN cana-1090 7 28 , , PUNCT cana-1090 7 29 𝑐1 𝑐1 NOUN cana-1090 7 30 , , PUNCT cana-1090 7 31 𝑐2 𝑐2 NOUN cana-1090 7 32 , , PUNCT cana-1090 7 33 𝑐3 𝑐3 NOUN cana-1090 7 34 are be AUX cana-1090 7 35 positive positive ADJ cana-1090 7 36 real real ADJ cana-1090 7 37 numbers number NOUN cana-1090 7 38 and and CCONJ cana-1090 7 39 the the DET cana-1090 7 40 initial initial ADJ cana-1090 7 41 conditions condition NOUN cana-1090 7 42 π‘₯βˆ’1 π‘₯βˆ’1 NOUN cana-1090 7 43 , , PUNCT cana-1090 7 44 π‘₯0 π‘₯0 NOUN cana-1090 7 45 , , PUNCT cana-1090 7 46 π‘¦βˆ’1 π‘¦βˆ’1 PROPN cana-1090 7 47 , , PUNCT cana-1090 7 48 𝑦0 𝑦0 NOUN cana-1090 7 49 , , PUNCT cana-1090 7 50 , , PUNCT cana-1090 7 51 π‘§βˆ’1 π‘§βˆ’1 PROPN cana-1090 7 52 , , PUNCT cana-1090 7 53 𝑧0 𝑧0 NOUN cana-1090 7 54 are be AUX cana-1090 7 55 arbitrary arbitrary ADJ cana-1090 7 56 nonnegative nonnegative ADJ cana-1090 7 57 numbers number NOUN cana-1090 7 58 . . PUNCT cana-1090 8 1 we we PRON cana-1090 8 2 investigate investigate VERB cana-1090 8 3 the the DET cana-1090 8 4 persistence persistence NOUN cana-1090 8 5 , , PUNCT cana-1090 8 6 boundedness boundedness NOUN cana-1090 8 7 , , PUNCT cana-1090 8 8 convergence convergence NOUN cana-1090 8 9 , , PUNCT cana-1090 8 10 invariance invariance NOUN cana-1090 8 11 , , PUNCT cana-1090 8 12 and and CCONJ cana-1090 8 13 global global ADJ cana-1090 8 14 asymptotic asymptotic ADJ cana-1090 8 15 character character NOUN cana-1090 8 16 of of ADP cana-1090 8 17 the the DET cana-1090 8 18 positive positive ADJ cana-1090 8 19 solutions solution NOUN cana-1090 8 20 of of ADP cana-1090 8 21 ( ( PUNCT cana-1090 8 22 1 1 NUM cana-1090 8 23 ) ) PUNCT cana-1090 8 24 . . PUNCT cana-1090 9 1 bifurcation bifurcation NOUN cana-1090 9 2 diagrams diagram NOUN cana-1090 9 3 are be AUX cana-1090 9 4 then then ADV cana-1090 9 5 plotted plot VERB cana-1090 9 6 to to PART cana-1090 9 7 visualize visualize VERB cana-1090 9 8 the the DET cana-1090 9 9 periodic periodic ADJ cana-1090 9 10 character character NOUN cana-1090 9 11 . . PUNCT cana-1090 10 1 keywords keyword NOUN cana-1090 10 2 : : PUNCT cana-1090 10 3 persistence persistence NOUN cana-1090 10 4 , , PUNCT cana-1090 10 5 boundedness boundedness NOUN cana-1090 10 6 , , PUNCT cana-1090 10 7 invariance invariance NOUN cana-1090 10 8 , , PUNCT cana-1090 10 9 local local ADJ cana-1090 10 10 property property NOUN cana-1090 10 11 , , PUNCT cana-1090 10 12 global global ADJ cana-1090 10 13 property property NOUN cana-1090 10 14 , , PUNCT cana-1090 10 15 bifurcation bifurcation NOUN cana-1090 10 16 . . PUNCT cana-1090 11 1 ams am NOUN cana-1090 11 2 subject subject VERB cana-1090 11 3 classification classification NOUN cana-1090 11 4 2000 2000 NUM cana-1090 11 5 : : PUNCT cana-1090 11 6 39a22 39a22 NUM cana-1090 11 7 . . PUNCT cana-1090 12 1 1 1 NUM cana-1090 12 2 . . X cana-1090 12 3 introduction introduction NOUN cana-1090 12 4 in in ADP cana-1090 12 5 the the DET cana-1090 12 6 study study NOUN cana-1090 12 7 of of ADP cana-1090 12 8 dynamical dynamical ADJ cana-1090 12 9 systems system NOUN cana-1090 12 10 , , PUNCT cana-1090 12 11 difference difference NOUN cana-1090 12 12 equations equation NOUN cana-1090 12 13 play play VERB cana-1090 12 14 a a DET cana-1090 12 15 crucial crucial ADJ cana-1090 12 16 role role NOUN cana-1090 12 17 in in ADP cana-1090 12 18 modeling model VERB cana-1090 12 19 various various ADJ cana-1090 12 20 phenomena phenomenon NOUN cana-1090 12 21 across across ADP cana-1090 12 22 diverse diverse ADJ cana-1090 12 23 scientific scientific ADJ cana-1090 12 24 disciplines discipline NOUN cana-1090 12 25 , , PUNCT cana-1090 12 26 including include VERB cana-1090 12 27 biology biology NOUN cana-1090 12 28 , , PUNCT cana-1090 12 29 economics economic NOUN cana-1090 12 30 , , PUNCT cana-1090 12 31 engineering engineering NOUN cana-1090 12 32 , , PUNCT cana-1090 12 33 and and CCONJ cana-1090 12 34 physics.(see physics.(see VERB cana-1090 13 1 [ [ X cana-1090 13 2 16],[21],[22],[27 16],[21],[22],[27 PROPN cana-1090 13 3 ] ] PUNCT cana-1090 13 4 , , PUNCT cana-1090 13 5 [ [ X cana-1090 13 6 31 31 NUM cana-1090 13 7 ] ] PUNCT cana-1090 13 8 , , PUNCT cana-1090 13 9 [ [ X cana-1090 13 10 32 32 NUM cana-1090 13 11 ] ] PUNCT cana-1090 13 12 ) ) PUNCT cana-1090 13 13 . . PUNCT cana-1090 14 1 unlike unlike ADP cana-1090 14 2 differential differential ADJ cana-1090 14 3 equations equation NOUN cana-1090 14 4 , , PUNCT cana-1090 14 5 which which PRON cana-1090 14 6 describe describe VERB cana-1090 14 7 continuous continuous ADJ cana-1090 14 8 change change NOUN cana-1090 14 9 , , PUNCT cana-1090 14 10 difference difference NOUN cana-1090 14 11 equations equation NOUN cana-1090 14 12 are be AUX cana-1090 14 13 discrete discrete ADJ cana-1090 14 14 analogues analogue NOUN cana-1090 14 15 that that PRON cana-1090 14 16 characterize characterize VERB cana-1090 14 17 systems system NOUN cana-1090 14 18 evolving evolve VERB cana-1090 14 19 in in ADP cana-1090 14 20 distinct distinct ADJ cana-1090 14 21 time time NOUN cana-1090 14 22 steps.(see steps.(see NOUN cana-1090 15 1 [ [ X cana-1090 15 2 6],[8],[13],[22],[23],[24 6],[8],[13],[22],[23],[24 NOUN cana-1090 15 3 ] ] PUNCT cana-1090 15 4 ) ) PUNCT cana-1090 15 5 most most ADJ cana-1090 15 6 of of ADP cana-1090 15 7 the the DET cana-1090 15 8 popular popular ADJ cana-1090 15 9 models model NOUN cana-1090 15 10 like like ADP cana-1090 15 11 sirs sir NOUN cana-1090 15 12 and and CCONJ cana-1090 15 13 seirs seir NOUN cana-1090 15 14 are be AUX cana-1090 15 15 mainly mainly ADV cana-1090 15 16 of of ADP cana-1090 15 17 order order NOUN cana-1090 15 18 one one NUM cana-1090 15 19 . . PUNCT cana-1090 16 1 two two NUM cana-1090 16 2 species species NOUN cana-1090 16 3 models model NOUN cana-1090 16 4 are be AUX cana-1090 16 5 examined examine VERB cana-1090 16 6 in in ADP cana-1090 16 7 [ [ X cana-1090 16 8 3 3 NUM cana-1090 16 9 ] ] PUNCT cana-1090 16 10 , , PUNCT cana-1090 16 11 [ [ X cana-1090 16 12 11],[29 11],[29 X cana-1090 16 13 ] ] X cana-1090 16 14 and and CCONJ cana-1090 16 15 [ [ X cana-1090 16 16 33 33 NUM cana-1090 16 17 ] ] PUNCT cana-1090 16 18 . . PUNCT cana-1090 17 1 competition competition NOUN cana-1090 17 2 models model NOUN cana-1090 17 3 of of ADP cana-1090 17 4 two two NUM cana-1090 17 5 species specie NOUN cana-1090 17 6 second second ADJ cana-1090 17 7 order order NOUN cana-1090 17 8 with with ADP cana-1090 17 9 exponents exponent NOUN cana-1090 17 10 are be AUX cana-1090 17 11 analyzed analyze VERB cana-1090 17 12 in in ADP cana-1090 17 13 [ [ X cana-1090 17 14 10],[12 10],[12 NUM cana-1090 17 15 ] ] PUNCT cana-1090 17 16 and and CCONJ cana-1090 17 17 [ [ X cana-1090 17 18 15]-[20 15]-[20 X cana-1090 17 19 ] ] PUNCT cana-1090 17 20 . . PUNCT cana-1090 18 1 in in ADP cana-1090 18 2 [ [ X cana-1090 18 3 5 5 NUM cana-1090 18 4 ] ] PUNCT cana-1090 18 5 , , PUNCT cana-1090 18 6 the the DET cana-1090 18 7 authors author NOUN cana-1090 18 8 analyzed analyze VERB cana-1090 18 9 a a DET cana-1090 18 10 food food NOUN cana-1090 18 11 - - PUNCT cana-1090 18 12 chain chain NOUN cana-1090 18 13 model model NOUN cana-1090 18 14 using use VERB cana-1090 18 15 a a DET cana-1090 18 16 first first ADJ cana-1090 18 17 order order NOUN cana-1090 18 18 system system NOUN cana-1090 18 19 with with ADP cana-1090 18 20 three three NUM cana-1090 18 21 variables variable NOUN cana-1090 18 22 . . PUNCT cana-1090 19 1 more more ADJ cana-1090 19 2 first first ADJ cana-1090 19 3 order order NOUN cana-1090 19 4 system system NOUN cana-1090 19 5 with with ADP cana-1090 19 6 three three NUM cana-1090 19 7 variables variable NOUN cana-1090 19 8 can can AUX cana-1090 19 9 be be AUX cana-1090 19 10 seen see VERB cana-1090 19 11 in in ADP cana-1090 19 12 [ [ X cana-1090 19 13 1],[2],[4 1],[2],[4 X cana-1090 19 14 ] ] X cana-1090 19 15 , , PUNCT cana-1090 19 16 [ [ X cana-1090 19 17 5 5 NUM cana-1090 19 18 ] ] PUNCT cana-1090 19 19 , , PUNCT cana-1090 19 20 [ [ X cana-1090 19 21 7 7 X cana-1090 19 22 ] ] PUNCT cana-1090 19 23 and and CCONJ cana-1090 19 24 [ [ X cana-1090 19 25 28 28 NUM cana-1090 19 26 ] ] PUNCT cana-1090 19 27 whereas whereas SCONJ cana-1090 19 28 [ [ X cana-1090 19 29 31 31 NUM cana-1090 19 30 ] ] PUNCT cana-1090 19 31 deals deal NOUN cana-1090 19 32 with with ADP cana-1090 19 33 second second ADJ cana-1090 19 34 order order NOUN cana-1090 19 35 systems system NOUN cana-1090 19 36 with with ADP cana-1090 19 37 three three NUM cana-1090 19 38 variables variable NOUN cana-1090 19 39 . . PUNCT cana-1090 20 1 the the DET cana-1090 20 2 system system NOUN cana-1090 20 3 ( ( PUNCT cana-1090 20 4 1 1 X cana-1090 20 5 ) ) PUNCT cana-1090 20 6 which which PRON cana-1090 20 7 we we PRON cana-1090 20 8 investigate investigate VERB cana-1090 20 9 is be AUX cana-1090 20 10 an an DET cana-1090 20 11 extension extension NOUN cana-1090 20 12 of of ADP cana-1090 20 13 [ [ X cana-1090 20 14 30 30 NUM cana-1090 20 15 ] ] PUNCT cana-1090 20 16 where where SCONJ cana-1090 20 17 we we PRON cana-1090 20 18 focus focus VERB cana-1090 20 19 on on ADP cana-1090 20 20 a a DET cana-1090 20 21 system system NOUN cana-1090 20 22 of of ADP cana-1090 20 23 three three NUM cana-1090 20 24 interdependent interdependent ADJ cana-1090 20 25 difference difference NOUN cana-1090 20 26 equations equation NOUN cana-1090 20 27 involving involve VERB cana-1090 20 28 twelve twelve NUM cana-1090 20 29 parameters parameter NOUN cana-1090 20 30 and and CCONJ cana-1090 20 31 three three NUM cana-1090 20 32 variables variable NOUN cana-1090 20 33 . . PUNCT cana-1090 21 1 we we PRON cana-1090 21 2 analyze analyze VERB cana-1090 21 3 the the DET cana-1090 21 4 boundedness boundedness NOUN cana-1090 21 5 , , PUNCT cana-1090 21 6 persistence persistence NOUN cana-1090 21 7 , , PUNCT cana-1090 21 8 invariance invariance NOUN cana-1090 21 9 and and CCONJ cana-1090 21 10 convergence convergence NOUN cana-1090 21 11 of of ADP cana-1090 21 12 the the DET cana-1090 21 13 solutions solution NOUN cana-1090 21 14 of of ADP cana-1090 21 15 ( ( PUNCT cana-1090 21 16 1 1 NUM cana-1090 21 17 ) ) PUNCT cana-1090 21 18 . . PUNCT cana-1090 22 1 we we PRON cana-1090 22 2 then then ADV cana-1090 22 3 plot plot VERB cana-1090 22 4 few few ADJ cana-1090 22 5 bifurcation bifurcation NOUN cana-1090 22 6 diagrams diagram NOUN cana-1090 22 7 to to PART cana-1090 22 8 observe observe VERB cana-1090 22 9 the the DET cana-1090 22 10 periodic periodic ADJ cana-1090 22 11 nature nature NOUN cana-1090 22 12 of of ADP cana-1090 22 13 the the DET cana-1090 22 14 system system NOUN cana-1090 22 15 . . PUNCT cana-1090 23 1 by by ADP cana-1090 23 2 exploring explore VERB cana-1090 23 3 a a DET cana-1090 23 4 three three NUM cana-1090 23 5 variable variable ADJ cana-1090 23 6 second second ADJ cana-1090 23 7 order order NOUN cana-1090 23 8 system system NOUN cana-1090 23 9 , , PUNCT cana-1090 23 10 this this DET cana-1090 23 11 work work NOUN cana-1090 23 12 aims aim VERB cana-1090 23 13 to to PART cana-1090 23 14 contribute contribute VERB cana-1090 23 15 to to ADP cana-1090 23 16 the the DET cana-1090 23 17 broader broad ADJ cana-1090 23 18 understanding understanding NOUN cana-1090 23 19 of of ADP cana-1090 23 20 multi multi ADJ cana-1090 23 21 - - ADJ cana-1090 23 22 parameter parameter ADJ cana-1090 23 23 , , PUNCT cana-1090 23 24 multivariable multivariable ADJ cana-1090 23 25 difference difference NOUN cana-1090 23 26 system system NOUN cana-1090 23 27 . . PUNCT cana-1090 24 1 the the DET cana-1090 24 2 increased increase VERB cana-1090 24 3 number number NOUN cana-1090 24 4 of of ADP cana-1090 24 5 parameters parameter NOUN cana-1090 24 6 provide provide VERB cana-1090 24 7 a a DET cana-1090 24 8 high high ADJ cana-1090 24 9 level level NOUN cana-1090 24 10 of of ADP cana-1090 24 11 flexibility flexibility NOUN cana-1090 24 12 to to PART cana-1090 24 13 model model VERB cana-1090 24 14 real real ADJ cana-1090 24 15 world world NOUN cana-1090 24 16 scenarios scenario NOUN cana-1090 24 17 , , PUNCT cana-1090 24 18 which which PRON cana-1090 24 19 is be AUX cana-1090 24 20 also also ADV cana-1090 24 21 cructial cructial ADJ cana-1090 24 22 for for ADP cana-1090 24 23 system system NOUN cana-1090 24 24 control control NOUN cana-1090 24 25 dynamics dynamic NOUN cana-1090 24 26 . . PUNCT cana-1090 25 1 mailto:smathew11@gmail.com mailto:smathew11@gmail.com PROPN cana-1090 25 2 mailto:dilip@stjohns.ac.in mailto:dilip@stjohns.ac.in VERB cana-1090 25 3 mailto:sibicbabu@stjohns.ac.in mailto:sibicbabu@stjohns.ac.in NOUN cana-1090 25 4 communications communication NOUN cana-1090 25 5 on on ADP cana-1090 25 6 applied apply VERB cana-1090 25 7 nonlinear nonlinear ADJ cana-1090 25 8 analysis analysis NOUN cana-1090 25 9 issn issn NOUN cana-1090 25 10 : : PUNCT cana-1090 25 11 1074 1074 NUM cana-1090 25 12 - - PUNCT cana-1090 25 13 133x 133x NUM cana-1090 25 14 vol vol NOUN cana-1090 25 15 31 31 NUM cana-1090 25 16 no no NOUN cana-1090 25 17 . . PUNCT cana-1090 26 1 5s 5s NUM cana-1090 26 2 ( ( PUNCT cana-1090 26 3 2024 2024 NUM cana-1090 26 4 ) ) PUNCT cana-1090 26 5 553 553 NUM cana-1090 26 6 https://internationalpubls.com https://internationalpubls.com SYM cana-1090 26 7 2 2 NUM cana-1090 26 8 . . X cana-1090 26 9 main main ADJ cana-1090 26 10 results result NOUN cana-1090 26 11 theorem theorem VERB cana-1090 26 12 2.1 2.1 NUM cana-1090 26 13 the the DET cana-1090 26 14 positive positive ADJ cana-1090 26 15 solution solution NOUN cana-1090 26 16 ( ( PUNCT cana-1090 26 17 π‘₯𝑛 π‘₯𝑛 NOUN cana-1090 26 18 , , PUNCT cana-1090 26 19 𝑦𝑛 𝑦𝑛 NOUN cana-1090 26 20 , , PUNCT cana-1090 26 21 𝑧𝑛 𝑧𝑛 INTJ cana-1090 26 22 ) ) PUNCT cana-1090 26 23 of of ADP cana-1090 26 24 ( ( PUNCT cana-1090 26 25 1 1 X cana-1090 26 26 ) ) PUNCT cana-1090 26 27 persists persist VERB cana-1090 26 28 . . PUNCT cana-1090 27 1 it it PRON cana-1090 27 2 is be AUX cana-1090 27 3 bounded bound VERB cana-1090 27 4 whenever whenever SCONJ cana-1090 27 5 𝐡 𝐡 NOUN cana-1090 27 6 = = NOUN cana-1090 27 7 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 27 8 < < X cana-1090 27 9 1 1 X cana-1090 27 10 . . PUNCT cana-1090 27 11 ( ( PUNCT cana-1090 27 12 2 2 X cana-1090 27 13 ) ) PUNCT cana-1090 27 14 proof proof NOUN cana-1090 27 15 : : PUNCT cana-1090 27 16 clearly clearly ADV cana-1090 27 17 , , PUNCT cana-1090 27 18 the the DET cana-1090 27 19 system system NOUN cana-1090 27 20 persists persist VERB cana-1090 27 21 because because SCONJ cana-1090 27 22 of of ADP cana-1090 27 23 the the DET cana-1090 27 24 presence presence NOUN cana-1090 27 25 of of ADP cana-1090 27 26 𝛼𝑖 𝛼𝑖 PROPN cana-1090 27 27 . . PUNCT cana-1090 28 1 for for ADP cana-1090 28 2 𝑛 𝑛 PRON cana-1090 28 3 = = SYM cana-1090 28 4 4,5 4,5 NUM cana-1090 28 5 , , PUNCT cana-1090 28 6 … … PUNCT cana-1090 28 7 , , PUNCT cana-1090 28 8 ( ( PUNCT cana-1090 28 9 1 1 X cana-1090 28 10 ) ) PUNCT cana-1090 28 11 becomes become VERB cana-1090 28 12 π‘₯𝑛+1 π‘₯𝑛+1 PROPN cana-1090 28 13 ≀ ≀ NUM cana-1090 28 14 𝛼1 𝛼1 NOUN cana-1090 28 15 + + CCONJ cana-1090 28 16 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 NOUN cana-1090 28 17 + + CCONJ cana-1090 28 18 𝑐1π‘’βˆ’π›Ό3 𝑐1π‘’βˆ’π›Ό3 NOUN cana-1090 28 19 + + CCONJ cana-1090 28 20 𝑏1π‘’βˆ’π›Ό2[𝛼2 𝑏1π‘’βˆ’π›Ό2[𝛼2 ADJ cana-1090 29 1 + + CCONJ cana-1090 29 2 π‘Ž2π‘’βˆ’π›Ό2 π‘Ž2π‘’βˆ’π›Ό2 NOUN cana-1090 29 3 + + NOUN cana-1090 29 4 𝑐2π‘’βˆ’π›Ό1 𝑐2π‘’βˆ’π›Ό1 NOUN cana-1090 29 5 + + ADJ cana-1090 29 6 𝑏2π‘Ž2π‘’βˆ’π›Ό3π‘§π‘›βˆ’1 𝑏2π‘Ž2π‘’βˆ’π›Ό3π‘§π‘›βˆ’1 NOUN cana-1090 29 7 ] ] X cana-1090 29 8 substituting substitute VERB cana-1090 29 9 for for ADP cana-1090 29 10 π‘§π‘›βˆ’1 π‘§π‘›βˆ’1 NOUN cana-1090 29 11 , , PUNCT cana-1090 29 12 we we PRON cana-1090 29 13 get get VERB cana-1090 29 14 ≀ ≀ PUNCT cana-1090 29 15 𝐴1 𝐴1 PROPN cana-1090 29 16 + + CCONJ cana-1090 29 17 𝐡π‘₯π‘›βˆ’2 𝐡π‘₯π‘›βˆ’2 NOUN cana-1090 29 18 , , PUNCT cana-1090 29 19 ( ( PUNCT cana-1090 29 20 3 3 X cana-1090 29 21 ) ) PUNCT cana-1090 29 22 where where SCONJ cana-1090 29 23 𝐴1 𝐴1 PROPN cana-1090 29 24 = = PUNCT cana-1090 29 25 𝛼1 𝛼1 NOUN cana-1090 29 26 + + CCONJ cana-1090 29 27 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 NOUN cana-1090 29 28 + + CCONJ cana-1090 29 29 𝑐1π‘’βˆ’π›Ό3 𝑐1π‘’βˆ’π›Ό3 NOUN cana-1090 29 30 + + CCONJ cana-1090 29 31 𝑏1𝛼2π‘’βˆ’π›Ό2 𝑏1𝛼2π‘’βˆ’π›Ό2 NOUN cana-1090 29 32 + + CCONJ cana-1090 29 33 𝑏1π‘Ž2π‘’βˆ’π›Ό2βˆ’π›Ό2 𝑏1π‘Ž2π‘’βˆ’π›Ό2βˆ’π›Ό2 NOUN cana-1090 29 34 + + CCONJ cana-1090 29 35 𝑏1𝑐2π‘’βˆ’π›Ό2βˆ’π›Ό1 𝑏1𝑐2π‘’βˆ’π›Ό2βˆ’π›Ό1 NOUN cana-1090 29 36 + + CCONJ cana-1090 29 37 𝑏1𝑏2𝛼3π‘’βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝛼3π‘’βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 29 38 + + CCONJ cana-1090 29 39 𝑏1𝑏2π‘Ž3π‘’βˆ’π›Ό2βˆ’π›Ό3βˆ’π›Ό3 𝑏1𝑏2π‘Ž3π‘’βˆ’π›Ό2βˆ’π›Ό3βˆ’π›Ό3 VERB cana-1090 29 40 + + CCONJ cana-1090 29 41 𝑏1𝑏2𝑐3π‘’βˆ’π›Ό2βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑐3π‘’βˆ’π›Ό2βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 29 42 and and CCONJ cana-1090 29 43 𝐡 𝐡 NOUN cana-1090 29 44 = = NOUN cana-1090 29 45 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 29 46 . . PUNCT cana-1090 30 1 similarly similarly ADV cana-1090 30 2 , , PUNCT cana-1090 30 3 𝑦𝑛+1 𝑦𝑛+1 PROPN cana-1090 30 4 ≀ ≀ PROPN cana-1090 30 5 𝐴2 𝐴2 PROPN cana-1090 30 6 + + CCONJ cana-1090 30 7 π΅π‘¦π‘›βˆ’2 π΅π‘¦π‘›βˆ’2 NOUN cana-1090 30 8 , , PUNCT cana-1090 30 9 ( ( PUNCT cana-1090 30 10 4 4 X cana-1090 30 11 ) ) PUNCT cana-1090 30 12 where where SCONJ cana-1090 30 13 𝐴2 𝐴2 PROPN cana-1090 30 14 = = SYM cana-1090 30 15 𝛼2 𝛼2 PROPN cana-1090 30 16 + + CCONJ cana-1090 30 17 π‘Ž2π‘’βˆ’π›Ό2 π‘Ž2π‘’βˆ’π›Ό2 NOUN cana-1090 30 18 + + NOUN cana-1090 30 19 𝑐2π‘’βˆ’π›Ό1 𝑐2π‘’βˆ’π›Ό1 NOUN cana-1090 30 20 + + CCONJ cana-1090 30 21 𝑏2𝛼3π‘’βˆ’π›Ό3 𝑏2𝛼3π‘’βˆ’π›Ό3 ADJ cana-1090 30 22 + + CCONJ cana-1090 30 23 𝑏2π‘Ž3π‘’βˆ’π›Ό3βˆ’π›Ό3 𝑏2π‘Ž3π‘’βˆ’π›Ό3βˆ’π›Ό3 ADJ cana-1090 30 24 + + CCONJ cana-1090 30 25 𝑏2𝑐3π‘’βˆ’π›Ό2βˆ’π›Ό3 𝑏2𝑐3π‘’βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 30 26 + + CCONJ cana-1090 30 27 𝑏2𝑏3𝛼1π‘’βˆ’π›Ό1βˆ’π›Ό3 𝑏2𝑏3𝛼1π‘’βˆ’π›Ό1βˆ’π›Ό3 NOUN cana-1090 30 28 + + CCONJ cana-1090 30 29 𝑏2𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’π›Ό1βˆ’π›Ό3 𝑏2𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’π›Ό1βˆ’π›Ό3 NOUN cana-1090 30 30 + + CCONJ cana-1090 30 31 𝑏2𝑏3𝑐1π‘’βˆ’π›Ό3βˆ’π›Ό3βˆ’π›Ό1 𝑏2𝑏3𝑐1π‘’βˆ’π›Ό3βˆ’π›Ό3βˆ’π›Ό1 NOUN cana-1090 30 32 . . PUNCT cana-1090 31 1 also also ADV cana-1090 31 2 , , PUNCT cana-1090 31 3 𝑧𝑛+1 𝑧𝑛+1 NUM cana-1090 31 4 ≀ ≀ NUM cana-1090 31 5 𝐴3 𝐴3 PROPN cana-1090 31 6 + + SYM cana-1090 31 7 π΅π‘§π‘›βˆ’2 π΅π‘§π‘›βˆ’2 NOUN cana-1090 31 8 , , PUNCT cana-1090 31 9 ( ( PUNCT cana-1090 31 10 5 5 NUM cana-1090 31 11 ) ) PUNCT cana-1090 31 12 where where SCONJ cana-1090 31 13 𝐴3 𝐴3 PROPN cana-1090 31 14 = = PUNCT cana-1090 31 15 𝛼3 𝛼3 NOUN cana-1090 31 16 + + CCONJ cana-1090 31 17 π‘Ž3π‘’βˆ’π›Ό3 π‘Ž3π‘’βˆ’π›Ό3 X cana-1090 31 18 + + CCONJ cana-1090 31 19 𝑐3π‘’βˆ’π›Ό2 𝑐3π‘’βˆ’π›Ό2 NOUN cana-1090 31 20 + + PUNCT cana-1090 31 21 𝑏3𝛼1π‘’βˆ’π›Ό1 𝑏3𝛼1π‘’βˆ’π›Ό1 PROPN cana-1090 31 22 + + NUM cana-1090 31 23 𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’π›Ό1 𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’π›Ό1 NOUN cana-1090 31 24 + + CCONJ cana-1090 31 25 𝑏3𝑐1π‘’βˆ’π›Ό3βˆ’π›Ό1 𝑏3𝑐1π‘’βˆ’π›Ό3βˆ’π›Ό1 NOUN cana-1090 31 26 + + CCONJ cana-1090 31 27 𝑏3𝑏1𝛼2π‘’βˆ’π›Ό1βˆ’π›Ό2 𝑏3𝑏1𝛼2π‘’βˆ’π›Ό1βˆ’π›Ό2 PROPN cana-1090 31 28 + + CCONJ cana-1090 31 29 𝑏1𝑏3π‘Ž2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό2 𝑏1𝑏3π‘Ž2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό2 ADJ cana-1090 31 30 + + CCONJ cana-1090 31 31 𝑏3𝑏1𝑐2π‘’βˆ’π›Ό1βˆ’π›Ό1βˆ’π›Ό2 𝑏3𝑏1𝑐2π‘’βˆ’π›Ό1βˆ’π›Ό1βˆ’π›Ό2 NOUN cana-1090 31 32 . . PUNCT cana-1090 32 1 now now ADV cana-1090 32 2 , , PUNCT cana-1090 32 3 consider consider VERB cana-1090 32 4 the the DET cana-1090 32 5 difference difference NOUN cana-1090 32 6 equations equation NOUN cana-1090 32 7 𝑒𝑛+1 𝑒𝑛+1 NUM cana-1090 32 8 = = SYM cana-1090 32 9 𝐴1 𝐴1 PROPN cana-1090 32 10 + + CCONJ cana-1090 32 11 π΅π‘’π‘›βˆ’2 π΅π‘’π‘›βˆ’2 NOUN cana-1090 32 12 , , PUNCT cana-1090 32 13 𝑣𝑛+1 𝑣𝑛+1 NUM cana-1090 32 14 = = SYM cana-1090 32 15 𝐴2 𝐴2 PROPN cana-1090 32 16 + + CCONJ cana-1090 32 17 π΅π‘£π‘›βˆ’2 π΅π‘£π‘›βˆ’2 NOUN cana-1090 32 18 , , PUNCT cana-1090 32 19 𝑀𝑛+1 𝑀𝑛+1 X cana-1090 32 20 = = PUNCT cana-1090 32 21 𝐴3 𝐴3 PROPN cana-1090 32 22 + + SYM cana-1090 32 23 π΅π‘€π‘›βˆ’2 π΅π‘€π‘›βˆ’2 NOUN cana-1090 32 24 , , PUNCT cana-1090 32 25 𝑛 𝑛 PROPN cana-1090 32 26 = = SYM cana-1090 32 27 4,5 4,5 NUM cana-1090 32 28 , , PUNCT cana-1090 32 29 … … PUNCT cana-1090 32 30 ( ( PUNCT cana-1090 32 31 6 6 NUM cana-1090 32 32 ) ) PUNCT cana-1090 32 33 solution solution NOUN cana-1090 32 34 ( ( PUNCT cana-1090 32 35 𝑒𝑛 𝑒𝑛 NOUN cana-1090 32 36 , , PUNCT cana-1090 32 37 𝑣𝑛 𝑣𝑛 NOUN cana-1090 32 38 , , PUNCT cana-1090 32 39 𝑀𝑛 𝑀𝑛 NOUN cana-1090 32 40 ) ) PUNCT cana-1090 32 41 of of ADP cana-1090 32 42 ( ( PUNCT cana-1090 32 43 6 6 NUM cana-1090 32 44 ) ) PUNCT cana-1090 32 45 is be AUX cana-1090 32 46 of of ADP cana-1090 32 47 the the DET cana-1090 32 48 form form NOUN cana-1090 32 49 𝑒𝑛 𝑒𝑛 NOUN cana-1090 32 50 = = SYM cana-1090 32 51 π‘Ÿ1𝐡𝑛/3 π‘Ÿ1𝐡𝑛/3 PROPN cana-1090 33 1 + + CCONJ cana-1090 33 2 π‘Ÿ2𝐡𝑛/3cos π‘Ÿ2𝐡𝑛/3co NOUN cana-1090 33 3 ( ( PUNCT cana-1090 33 4 π‘›πœ‹ π‘›πœ‹ ADP cana-1090 33 5 2 2 NUM cana-1090 33 6 ) ) PUNCT cana-1090 34 1 + + CCONJ cana-1090 34 2 π‘Ÿ3𝐡𝑛/3sin π‘Ÿ3𝐡𝑛/3sin PROPN cana-1090 34 3 ( ( PUNCT cana-1090 34 4 π‘›πœ‹ π‘›πœ‹ ADP cana-1090 34 5 2 2 NUM cana-1090 34 6 ) ) PUNCT cana-1090 34 7 + + CCONJ cana-1090 34 8 𝐴1 𝐴1 PROPN cana-1090 34 9 1βˆ’π΅ 1βˆ’π΅ NUM cana-1090 34 10 , , PUNCT cana-1090 34 11 𝑛 𝑛 PROPN cana-1090 34 12 = = SYM cana-1090 34 13 5,6 5,6 NUM cana-1090 34 14 , , PUNCT cana-1090 34 15 … … PUNCT cana-1090 34 16 , , PUNCT cana-1090 34 17 ( ( PUNCT cana-1090 34 18 7 7 NUM cana-1090 34 19 ) ) PUNCT cana-1090 34 20 𝑣𝑛 𝑣𝑛 NOUN cana-1090 34 21 = = SYM cana-1090 34 22 𝑠1𝐡𝑛/3 𝑠1𝐡𝑛/3 PROPN cana-1090 34 23 + + CCONJ cana-1090 34 24 𝑠2𝐡𝑛/3cos 𝑠2𝐡𝑛/3cos PROPN cana-1090 34 25 ( ( PUNCT cana-1090 34 26 π‘›πœ‹ π‘›πœ‹ ADP cana-1090 34 27 2 2 NUM cana-1090 34 28 ) ) PUNCT cana-1090 34 29 + + CCONJ cana-1090 34 30 𝑠3𝐡𝑛/3sin 𝑠3𝐡𝑛/3sin PROPN cana-1090 34 31 ( ( PUNCT cana-1090 34 32 π‘›πœ‹ π‘›πœ‹ ADP cana-1090 34 33 2 2 NUM cana-1090 34 34 ) ) PUNCT cana-1090 35 1 + + CCONJ cana-1090 35 2 𝐴2 𝐴2 PROPN cana-1090 35 3 1βˆ’π΅ 1βˆ’π΅ NUM cana-1090 35 4 , , PUNCT cana-1090 35 5 𝑛 𝑛 PROPN cana-1090 35 6 = = SYM cana-1090 35 7 5,6 5,6 NUM cana-1090 35 8 … … PUNCT cana-1090 35 9 , , PUNCT cana-1090 35 10 ( ( PUNCT cana-1090 35 11 8) 8) NUM cana-1090 35 12 𝑀𝑛 𝑀𝑛 NOUN cana-1090 35 13 = = SYM cana-1090 35 14 𝑝1𝐡𝑛/3 𝑝1𝐡𝑛/3 PROPN cana-1090 35 15 + + CCONJ cana-1090 35 16 𝑝2𝐡𝑛/3cos 𝑝2𝐡𝑛/3cos PROPN cana-1090 35 17 ( ( PUNCT cana-1090 35 18 π‘›πœ‹ π‘›πœ‹ ADP cana-1090 35 19 2 2 NUM cana-1090 35 20 ) ) PUNCT cana-1090 36 1 + + CCONJ cana-1090 36 2 𝑝3𝐡𝑛/3sin 𝑝3𝐡𝑛/3sin PROPN cana-1090 36 3 ( ( PUNCT cana-1090 36 4 π‘›πœ‹ π‘›πœ‹ ADP cana-1090 36 5 2 2 NUM cana-1090 36 6 ) ) PUNCT cana-1090 36 7 + + CCONJ cana-1090 36 8 𝐴3 𝐴3 PROPN cana-1090 36 9 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 36 10 , , PUNCT cana-1090 36 11 𝑛 𝑛 PROPN cana-1090 36 12 = = SYM cana-1090 36 13 5,6 5,6 NUM cana-1090 36 14 … … PUNCT cana-1090 36 15 , , PUNCT cana-1090 36 16 ( ( PUNCT cana-1090 36 17 9 9 X cana-1090 36 18 ) ) PUNCT cana-1090 36 19 where where SCONJ cana-1090 36 20 𝑝𝑖 𝑝𝑖 NOUN cana-1090 36 21 , , PUNCT cana-1090 36 22 π‘Ÿπ‘– π‘Ÿπ‘– AUX cana-1090 36 23 , , PUNCT cana-1090 36 24 𝑠𝑖 𝑠𝑖 NOUN cana-1090 36 25 , , PUNCT cana-1090 36 26 𝑖 𝑖 SYM cana-1090 36 27 = = NOUN cana-1090 36 28 1,2,3 1,2,3 NUM cana-1090 36 29 depend depend VERB cana-1090 36 30 on on ADP cana-1090 36 31 𝑀4 𝑀4 NOUN cana-1090 36 32 , , PUNCT cana-1090 36 33 𝑒4 𝑒4 NOUN cana-1090 36 34 , , PUNCT cana-1090 36 35 𝑣4 𝑣4 NOUN cana-1090 36 36 respectively respectively ADV cana-1090 36 37 . . PUNCT cana-1090 37 1 communications communication NOUN cana-1090 37 2 on on ADP cana-1090 37 3 applied apply VERB cana-1090 37 4 nonlinear nonlinear ADJ cana-1090 37 5 analysis analysis NOUN cana-1090 37 6 issn issn NOUN cana-1090 37 7 : : PUNCT cana-1090 37 8 1074 1074 NUM cana-1090 37 9 - - PUNCT cana-1090 37 10 133x 133x NUM cana-1090 37 11 vol vol NOUN cana-1090 37 12 31 31 NUM cana-1090 37 13 no no NOUN cana-1090 37 14 . . PUNCT cana-1090 38 1 5s 5s NUM cana-1090 38 2 ( ( PUNCT cana-1090 38 3 2024 2024 NUM cana-1090 38 4 ) ) PUNCT cana-1090 38 5 554 554 NUM cana-1090 38 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 38 7 hence hence ADV cana-1090 38 8 , , PUNCT cana-1090 38 9 ( ( PUNCT cana-1090 38 10 𝑒𝑛 𝑒𝑛 NOUN cana-1090 38 11 , , PUNCT cana-1090 38 12 𝑣𝑛 𝑣𝑛 NOUN cana-1090 38 13 , , PUNCT cana-1090 38 14 𝑀𝑛 𝑀𝑛 NOUN cana-1090 38 15 ) ) PUNCT cana-1090 38 16 is be AUX cana-1090 38 17 bounded bound VERB cana-1090 38 18 . . PUNCT cana-1090 39 1 now now ADV cana-1090 39 2 we we PRON cana-1090 39 3 examine examine VERB cana-1090 39 4 ( ( PUNCT cana-1090 39 5 𝑒𝑛 𝑒𝑛 NOUN cana-1090 39 6 , , PUNCT cana-1090 39 7 𝑣𝑛 𝑣𝑛 PROPN cana-1090 39 8 , , PUNCT cana-1090 39 9 𝑀𝑛 𝑀𝑛 NOUN cana-1090 39 10 ) ) PUNCT cana-1090 39 11 such such ADJ cana-1090 39 12 that that SCONJ cana-1090 39 13 the the DET cana-1090 39 14 initial initial ADJ cana-1090 39 15 conditions condition NOUN cana-1090 39 16 of of ADP cana-1090 39 17 6 6 NUM cana-1090 39 18 and and CCONJ cana-1090 39 19 1 1 NUM cana-1090 39 20 are be AUX cana-1090 39 21 same same ADJ cana-1090 39 22 . . PUNCT cana-1090 40 1 clearly clearly ADV cana-1090 40 2 we we PRON cana-1090 40 3 can can AUX cana-1090 40 4 conclude conclude VERB cana-1090 40 5 that that PRON cana-1090 40 6 ( ( PUNCT cana-1090 40 7 π‘₯𝑛 π‘₯𝑛 NOUN cana-1090 40 8 , , PUNCT cana-1090 40 9 𝑦𝑛 𝑦𝑛 NOUN cana-1090 40 10 , , PUNCT cana-1090 40 11 𝑧𝑛 𝑧𝑛 INTJ cana-1090 40 12 ) ) PUNCT cana-1090 40 13 is be AUX cana-1090 40 14 bounded bound VERB cana-1090 40 15 . . PUNCT cana-1090 41 1 theorem theorem VERB cana-1090 41 2 2.2 2.2 NUM cana-1090 41 3 let let VERB cana-1090 41 4 ( ( PUNCT cana-1090 41 5 2 2 X cana-1090 41 6 ) ) PUNCT cana-1090 41 7 hold hold NOUN cana-1090 41 8 . . PUNCT cana-1090 42 1 let let VERB cana-1090 42 2 𝐴1 𝐴1 PROPN cana-1090 42 3 , , PUNCT cana-1090 42 4 𝐴2 𝐴2 PROPN cana-1090 42 5 , , PUNCT cana-1090 42 6 𝐴3 𝐴3 PROPN cana-1090 42 7 be be AUX cana-1090 42 8 defined define VERB cana-1090 42 9 as as ADP cana-1090 42 10 in in ADP cana-1090 42 11 theorem theorem ADJ cana-1090 42 12 2.1 2.1 NUM cana-1090 42 13 . . PUNCT cana-1090 43 1 then then ADV cana-1090 43 2 [ [ X cana-1090 43 3 𝛼1 𝛼1 NOUN cana-1090 43 4 , , PUNCT cana-1090 43 5 𝐴1 𝐴1 PROPN cana-1090 43 6 1βˆ’π΅ 1βˆ’π΅ NUM cana-1090 43 7 ] ] PUNCT cana-1090 43 8 Γ— Γ— PROPN cana-1090 44 1 [ [ X cana-1090 44 2 𝛼2 𝛼2 PROPN cana-1090 44 3 , , PUNCT cana-1090 44 4 𝐴2 𝐴2 PROPN cana-1090 44 5 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 44 6 ] ] PUNCT cana-1090 44 7 Γ— Γ— NOUN cana-1090 45 1 [ [ X cana-1090 45 2 𝛼3 𝛼3 X cana-1090 45 3 , , PUNCT cana-1090 45 4 𝐴3 𝐴3 PROPN cana-1090 45 5 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 45 6 ] ] PUNCT cana-1090 45 7 is be AUX cana-1090 45 8 an an DET cana-1090 45 9 invariant invariant ADJ cana-1090 45 10 set set NOUN cana-1090 45 11 for for ADP cana-1090 45 12 the the DET cana-1090 45 13 system system NOUN cana-1090 45 14 ( ( PUNCT cana-1090 45 15 1 1 NUM cana-1090 45 16 ) ) PUNCT cana-1090 45 17 . . PUNCT cana-1090 46 1 proof proof NOUN cana-1090 46 2 : : PUNCT cana-1090 46 3 take take VERB cana-1090 46 4 𝐼1 𝐼1 NOUN cana-1090 47 1 = = PUNCT cana-1090 48 1 [ [ X cana-1090 48 2 𝛼1 𝛼1 NOUN cana-1090 48 3 , , PUNCT cana-1090 48 4 𝐴1 𝐴1 PROPN cana-1090 48 5 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 48 6 ] ] PUNCT cana-1090 48 7 , , PUNCT cana-1090 48 8 𝐼2 𝐼2 X cana-1090 48 9 = = PUNCT cana-1090 49 1 [ [ X cana-1090 49 2 𝛼2 𝛼2 PROPN cana-1090 49 3 , , PUNCT cana-1090 49 4 𝐴2 𝐴2 PROPN cana-1090 49 5 1βˆ’π΅ 1βˆ’π΅ NUM cana-1090 49 6 ] ] PUNCT cana-1090 49 7 and and CCONJ cana-1090 49 8 𝐼3 𝐼3 PROPN cana-1090 49 9 = = PUNCT cana-1090 50 1 [ [ X cana-1090 50 2 𝛼3 𝛼3 X cana-1090 50 3 , , PUNCT cana-1090 50 4 𝐴3 𝐴3 PROPN cana-1090 50 5 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 50 6 ] ] PUNCT cana-1090 50 7 . . PUNCT cana-1090 51 1 let let VERB cana-1090 51 2 π‘₯βˆ’1 π‘₯βˆ’1 NOUN cana-1090 51 3 , , PUNCT cana-1090 51 4 π‘₯0 π‘₯0 PROPN cana-1090 51 5 ∈ ∈ PROPN cana-1090 51 6 𝐼1 𝐼1 NOUN cana-1090 51 7 , , PUNCT cana-1090 51 8 π‘¦βˆ’1 π‘¦βˆ’1 PROPN cana-1090 51 9 , , PUNCT cana-1090 51 10 𝑦0 𝑦0 PROPN cana-1090 51 11 ∈ ∈ PROPN cana-1090 51 12 𝐼2 𝐼2 NOUN cana-1090 51 13 and and CCONJ cana-1090 51 14 π‘§βˆ’1 π‘§βˆ’1 PROPN cana-1090 51 15 , , PUNCT cana-1090 51 16 𝑧0 𝑧0 PROPN cana-1090 51 17 ∈ ∈ PROPN cana-1090 51 18 𝐼3 𝐼3 PROPN cana-1090 51 19 . . PUNCT cana-1090 52 1 then then ADV cana-1090 52 2 π‘₯1 π‘₯1 NOUN cana-1090 52 3 ≀ ≀ NOUN cana-1090 52 4 𝛼1 𝛼1 NOUN cana-1090 52 5 + + CCONJ cana-1090 52 6 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 NOUN cana-1090 52 7 + + CCONJ cana-1090 52 8 𝑐1π‘’βˆ’π›Ό3 𝑐1π‘’βˆ’π›Ό3 NOUN cana-1090 52 9 + + CCONJ cana-1090 52 10 𝑏1π‘’βˆ’π›Ό2𝑦0 𝑏1π‘’βˆ’π›Ό2𝑦0 X cana-1090 52 11 since since SCONJ cana-1090 52 12 𝑦0 𝑦0 NOUN cana-1090 52 13 ≀ ≀ NUM cana-1090 52 14 𝐴2 𝐴2 PROPN cana-1090 52 15 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 52 16 , , PUNCT cana-1090 52 17 we we PRON cana-1090 52 18 get get VERB cana-1090 52 19 π‘₯1 π‘₯1 NOUN cana-1090 52 20 ≀ ≀ X cana-1090 52 21 [ [ PUNCT cana-1090 52 22 𝛼1 𝛼1 NOUN cana-1090 52 23 + + CCONJ cana-1090 52 24 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 NOUN cana-1090 52 25 + + CCONJ cana-1090 52 26 𝑐1π‘’βˆ’π›Ό3 𝑐1π‘’βˆ’π›Ό3 NOUN cana-1090 52 27 βˆ’ βˆ’ PROPN cana-1090 52 28 𝑏1𝑏2𝑏3𝛼1π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3𝛼1π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 ADJ cana-1090 52 29 βˆ’ βˆ’ NOUN cana-1090 52 30 𝑏1𝑏2𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 ADJ cana-1090 52 31 1 1 NUM cana-1090 52 32 βˆ’ βˆ’ NOUN cana-1090 52 33 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 52 34 ] ] PUNCT cana-1090 53 1 + + PUNCT cana-1090 53 2 [ [ PUNCT cana-1090 53 3 βˆ’π‘1𝑏2𝑏3𝑐1π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3βˆ’π›Ό3 βˆ’π‘1𝑏2𝑏3𝑐1π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3βˆ’π›Ό3 ADJ cana-1090 53 4 + + X cana-1090 53 5 𝑏1𝛼2π‘’βˆ’π›Ό2 𝑏1𝛼2π‘’βˆ’π›Ό2 NOUN cana-1090 53 6 + + CCONJ cana-1090 53 7 𝑏1π‘Ž2π‘’βˆ’π›Ό2βˆ’π›Ό2 𝑏1π‘Ž2π‘’βˆ’π›Ό2βˆ’π›Ό2 NOUN cana-1090 53 8 + + CCONJ cana-1090 53 9 𝑏1𝑐2π‘’βˆ’π›Ό1βˆ’π›Ό2 𝑏1𝑐2π‘’βˆ’π›Ό1βˆ’π›Ό2 NOUN cana-1090 53 10 1 1 NUM cana-1090 53 11 βˆ’ βˆ’ NOUN cana-1090 53 12 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 53 13 ] ] PUNCT cana-1090 54 1 + + PUNCT cana-1090 54 2 [ [ PUNCT cana-1090 54 3 𝑏1𝑏2𝛼3π‘’βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝛼3π‘’βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 54 4 + + CCONJ cana-1090 54 5 𝑏1𝑏2π‘Ž3𝑒𝛼3βˆ’π›Ό3 𝑏1𝑏2π‘Ž3𝑒𝛼3βˆ’π›Ό3 NOUN cana-1090 54 6 + + CCONJ cana-1090 54 7 𝑏1𝑏2𝑐3π‘’βˆ’π›Ό2βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑐3π‘’βˆ’π›Ό2βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 54 8 + + CCONJ cana-1090 54 9 𝑏1𝑏2𝑏3𝛼1π‘’βˆ’π›Ό2βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3𝛼1π‘’βˆ’π›Ό2βˆ’π›Ό2βˆ’π›Ό3 ADJ cana-1090 54 10 1 1 NUM cana-1090 54 11 βˆ’ βˆ’ NOUN cana-1090 54 12 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 54 13 ] ] PUNCT cana-1090 55 1 + + PROPN cana-1090 55 2 [ [ X cana-1090 55 3 𝑏1𝑏2𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘Ž1π‘’βˆ’π›Ό1βˆ’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 ADJ cana-1090 55 4 + + CCONJ cana-1090 55 5 𝑏1𝑏2𝑏3𝑐1π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3βˆ’π›Ό3 𝑏1𝑏2𝑏3𝑐1π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3βˆ’π›Ό3 ADJ cana-1090 55 6 1 1 NUM cana-1090 55 7 βˆ’ βˆ’ NOUN cana-1090 55 8 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 55 9 ] ] PUNCT cana-1090 55 10 i.e. i.e. X cana-1090 55 11 , , PUNCT cana-1090 55 12 π‘₯1 π‘₯1 PROPN cana-1090 55 13 ∈ ∈ PROPN cana-1090 55 14 𝐼1 𝐼1 NOUN cana-1090 55 15 . . PUNCT cana-1090 56 1 similarly similarly ADV cana-1090 56 2 we we PRON cana-1090 56 3 get get VERB cana-1090 56 4 𝑦1 𝑦1 PROPN cana-1090 56 5 ∈ ∈ PROPN cana-1090 56 6 𝐼2 𝐼2 NOUN cana-1090 56 7 and and CCONJ cana-1090 56 8 𝑧1 𝑧1 NOUN cana-1090 56 9 ∈ ∈ PROPN cana-1090 56 10 𝐼3 𝐼3 NOUN cana-1090 56 11 . . PUNCT cana-1090 57 1 hence hence ADV cana-1090 57 2 the the DET cana-1090 57 3 induction induction NOUN cana-1090 57 4 the the DET cana-1090 57 5 proof proof NOUN cana-1090 57 6 follows follow VERB cana-1090 57 7 . . PUNCT cana-1090 58 1 theorem theorem VERB cana-1090 58 2 2.3 2.3 NUM cana-1090 58 3 assume assume NOUN cana-1090 58 4 ( ( PUNCT cana-1090 58 5 2 2 NUM cana-1090 58 6 ) ) PUNCT cana-1090 58 7 . . PUNCT cana-1090 59 1 let let VERB cana-1090 59 2 𝐴1 𝐴1 PROPN cana-1090 59 3 , , PUNCT cana-1090 59 4 𝐴2 𝐴2 PROPN cana-1090 59 5 , , PUNCT cana-1090 59 6 𝐴3 𝐴3 PROPN cana-1090 59 7 be be VERB cana-1090 59 8 as as ADP cana-1090 59 9 in in ADP cana-1090 59 10 the the DET cana-1090 59 11 theorem theorem NOUN cana-1090 59 12 2.1 2.1 NUM cana-1090 59 13 . . PUNCT cana-1090 60 1 let let VERB cana-1090 60 2 𝐼4 𝐼4 NOUN cana-1090 60 3 = = PUNCT cana-1090 61 1 [ [ X cana-1090 61 2 𝛼1 𝛼1 NOUN cana-1090 61 3 , , PUNCT cana-1090 61 4 𝐴1+πœ– 𝐴1+πœ– PROPN cana-1090 61 5 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 61 6 ] ] PUNCT cana-1090 61 7 , , PUNCT cana-1090 61 8 𝐼5 𝐼5 X cana-1090 61 9 = = PUNCT cana-1090 62 1 [ [ X cana-1090 62 2 𝛼2 𝛼2 ADJ cana-1090 62 3 , , PUNCT cana-1090 62 4 𝐴2+πœ– 𝐴2+πœ– ADJ cana-1090 62 5 1βˆ’π΅ 1βˆ’π΅ NOUN cana-1090 62 6 ] ] PUNCT cana-1090 62 7 and and CCONJ cana-1090 62 8 𝐼6 𝐼6 NOUN cana-1090 62 9 = = SYM cana-1090 63 1 [ [ X cana-1090 63 2 𝛼3 𝛼3 X cana-1090 63 3 , , PUNCT cana-1090 63 4 𝐴3+πœ– 𝐴3+πœ– NOUN cana-1090 63 5 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 63 6 ] ] PUNCT cana-1090 63 7 where where SCONJ cana-1090 63 8 πœ– πœ– PROPN cana-1090 63 9 is be AUX cana-1090 63 10 arbitrary arbitrary ADJ cana-1090 63 11 . . PUNCT cana-1090 64 1 then then ADV cana-1090 64 2 π‘₯𝑛 π‘₯𝑛 VERB cana-1090 64 3 ∈ ∈ PROPN cana-1090 64 4 𝐼4 𝐼4 NOUN cana-1090 64 5 , , PUNCT cana-1090 64 6 𝑦𝑛 𝑦𝑛 PROPN cana-1090 64 7 ∈ ∈ PROPN cana-1090 64 8 𝐼5 𝐼5 NOUN cana-1090 64 9 and and CCONJ cana-1090 64 10 𝑧𝑛 𝑧𝑛 ADP cana-1090 64 11 ∈ ∈ PROPN cana-1090 64 12 𝐼6 𝐼6 NOUN cana-1090 64 13 , , PUNCT cana-1090 64 14 for for ADP cana-1090 64 15 every every DET cana-1090 64 16 𝑛 𝑛 DET cana-1090 64 17 β‰₯ β‰₯ NOUN cana-1090 64 18 𝑁 𝑁 PROPN cana-1090 64 19 , , PUNCT cana-1090 64 20 𝑁 𝑁 PROPN cana-1090 64 21 ∈ ∈ PROPN cana-1090 64 22 β„•. β„•. PROPN cana-1090 64 23 proof proof NOUN cana-1090 64 24 : : PUNCT cana-1090 64 25 given give VERB cana-1090 64 26 ( ( PUNCT cana-1090 64 27 π‘₯𝑛 π‘₯𝑛 PROPN cana-1090 64 28 , , PUNCT cana-1090 64 29 𝑦𝑛 𝑦𝑛 NOUN cana-1090 64 30 , , PUNCT cana-1090 64 31 𝑧𝑛 𝑧𝑛 VERB cana-1090 64 32 ) ) PUNCT cana-1090 64 33 be be AUX cana-1090 64 34 a a DET cana-1090 64 35 positive positive ADJ cana-1090 64 36 solution solution NOUN cana-1090 64 37 of of ADP cana-1090 64 38 ( ( PUNCT cana-1090 64 39 1 1 NUM cana-1090 64 40 ) ) PUNCT cana-1090 64 41 . . PUNCT cana-1090 65 1 theorem theorem VERB cana-1090 65 2 2.1 2.1 NUM cana-1090 65 3 implies implie NOUN cana-1090 65 4 , , PUNCT cana-1090 65 5 limsupπ‘›β†’βˆžπ‘₯𝑛 limsupπ‘›β†’βˆžπ‘₯𝑛 ADJ cana-1090 65 6 = = SYM cana-1090 65 7 𝑃 𝑃 NOUN cana-1090 65 8 < < X cana-1090 65 9 ∞ ∞ PROPN cana-1090 65 10 , , PUNCT cana-1090 65 11 limsupπ‘›β†’βˆžπ‘¦π‘› limsupπ‘›β†’βˆžπ‘¦π‘› ADJ cana-1090 65 12 = = SYM cana-1090 65 13 𝑄 𝑄 PROPN cana-1090 65 14 < < X cana-1090 65 15 ∞ ∞ PROPN cana-1090 65 16 and and CCONJ cana-1090 65 17 limsupπ‘›β†’βˆžπ‘§π‘› limsupπ‘›β†’βˆžπ‘§π‘› NOUN cana-1090 65 18 = = SYM cana-1090 65 19 𝑅 𝑅 PROPN cana-1090 65 20 < < X cana-1090 65 21 ∞. ∞. PROPN cana-1090 65 22 theorem theorem VERB cana-1090 65 23 2.1 2.1 NUM cana-1090 65 24 implies implie NOUN cana-1090 65 25 , , PUNCT cana-1090 65 26 π‘₯𝑛+1 π‘₯𝑛+1 PROPN cana-1090 65 27 ≀ ≀ NUM cana-1090 65 28 𝐴1 𝐴1 PROPN cana-1090 65 29 + + CCONJ cana-1090 65 30 𝑏1𝑏2𝑏3π‘₯π‘›βˆ’2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘₯π‘›βˆ’2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 65 31 , , PUNCT cana-1090 65 32 𝑦𝑛+1 𝑦𝑛+1 PROPN cana-1090 65 33 ≀ ≀ NOUN cana-1090 65 34 𝐴2 𝐴2 PROPN cana-1090 65 35 + + CCONJ cana-1090 65 36 𝑏1𝑏2𝑏3π‘¦π‘›βˆ’2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘¦π‘›βˆ’2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 65 37 and and CCONJ cana-1090 65 38 𝑧𝑛+1 𝑧𝑛+1 NUM cana-1090 65 39 ≀ ≀ NUM cana-1090 65 40 𝐴3 𝐴3 PROPN cana-1090 65 41 + + X cana-1090 65 42 𝑏1𝑏2𝑏3π‘§π‘›βˆ’2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 𝑏1𝑏2𝑏3π‘§π‘›βˆ’2π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 X cana-1090 65 43 . . PUNCT cana-1090 66 1 hence hence ADV cana-1090 66 2 , , PUNCT cana-1090 66 3 𝑃 𝑃 VERB cana-1090 66 4 ≀ ≀ NUM cana-1090 66 5 𝐴1 𝐴1 PROPN cana-1090 66 6 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 66 7 , , PUNCT cana-1090 66 8 𝑄 𝑄 PROPN cana-1090 66 9 ≀ ≀ NOUN cana-1090 66 10 𝐴2 𝐴2 PROPN cana-1090 66 11 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 66 12 and and CCONJ cana-1090 66 13 𝑅 𝑅 PROPN cana-1090 66 14 ≀ ≀ PROPN cana-1090 66 15 𝐴3 𝐴3 PROPN cana-1090 66 16 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 66 17 . . PUNCT cana-1090 67 1 communications communication NOUN cana-1090 67 2 on on ADP cana-1090 67 3 applied apply VERB cana-1090 67 4 nonlinear nonlinear ADJ cana-1090 67 5 analysis analysis NOUN cana-1090 67 6 issn issn NOUN cana-1090 67 7 : : PUNCT cana-1090 67 8 1074 1074 NUM cana-1090 67 9 - - PUNCT cana-1090 67 10 133x 133x NUM cana-1090 67 11 vol vol NOUN cana-1090 67 12 31 31 NUM cana-1090 67 13 no no NOUN cana-1090 67 14 . . PUNCT cana-1090 68 1 5s 5s NUM cana-1090 68 2 ( ( PUNCT cana-1090 68 3 2024 2024 NUM cana-1090 68 4 ) ) PUNCT cana-1090 68 5 555 555 NUM cana-1090 68 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 68 7 hence hence ADV cana-1090 68 8 , , PUNCT cana-1090 68 9 the the DET cana-1090 68 10 result result NOUN cana-1090 68 11 . . PUNCT cana-1090 69 1 here here ADV cana-1090 69 2 we we PRON cana-1090 69 3 state state VERB cana-1090 69 4 a a DET cana-1090 69 5 lemma lemma PROPN cana-1090 69 6 which which PRON cana-1090 69 7 is be AUX cana-1090 69 8 an an DET cana-1090 69 9 an an DET cana-1090 69 10 extension extension NOUN cana-1090 69 11 of of ADP cana-1090 69 12 lemma lemma PROPN cana-1090 69 13 5 5 NUM cana-1090 69 14 in in ADP cana-1090 69 15 [ [ X cana-1090 69 16 10 10 NUM cana-1090 69 17 ] ] PUNCT cana-1090 69 18 and and CCONJ cana-1090 69 19 a a DET cana-1090 69 20 variation variation NOUN cana-1090 69 21 of of ADP cana-1090 69 22 theorem theorem NOUN cana-1090 69 23 1.16 1.16 NUM cana-1090 69 24 in in ADP cana-1090 69 25 [ [ X cana-1090 69 26 26 26 NUM cana-1090 69 27 ] ] PUNCT cana-1090 69 28 . . PUNCT cana-1090 70 1 lemma lemma PROPN cana-1090 70 2 2.4 2.4 NUM cana-1090 70 3 assume assume PROPN cana-1090 70 4 𝐴 𝐴 PROPN cana-1090 70 5 , , PUNCT cana-1090 70 6 𝐡 𝐡 PROPN cana-1090 70 7 , , PUNCT cana-1090 70 8 𝐢 𝐢 PROPN cana-1090 70 9 , , PUNCT cana-1090 70 10 𝐷 𝐷 PROPN cana-1090 70 11 , , PUNCT cana-1090 70 12 𝐸 𝐸 PROPN cana-1090 70 13 , , PUNCT cana-1090 70 14 𝐹 𝐹 PROPN cana-1090 70 15 represent represent VERB cana-1090 70 16 reals real NOUN cana-1090 70 17 . . PUNCT cana-1090 71 1 let let VERB cana-1090 71 2 𝑓1 𝑓1 ADJ cana-1090 71 3 : : PUNCT cana-1090 72 1 [ [ X cana-1090 72 2 𝐴 𝐴 PROPN cana-1090 72 3 , , PUNCT cana-1090 72 4 𝐡 𝐡 PROPN cana-1090 72 5 ] ] X cana-1090 72 6 Γ— Γ— PROPN cana-1090 72 7 [ [ X cana-1090 72 8 𝐢 𝐢 PROPN cana-1090 72 9 , , PUNCT cana-1090 72 10 𝐷 𝐷 PROPN cana-1090 72 11 ] ] X cana-1090 72 12 Γ— Γ— NOUN cana-1090 72 13 [ [ X cana-1090 72 14 𝐢 𝐢 PROPN cana-1090 72 15 , , PUNCT cana-1090 72 16 𝐷 𝐷 PROPN cana-1090 72 17 ] ] X cana-1090 72 18 Γ— Γ— NOUN cana-1090 73 1 [ [ X cana-1090 73 2 𝐸 𝐸 PROPN cana-1090 73 3 , , PUNCT cana-1090 73 4 𝐹 𝐹 PROPN cana-1090 73 5 ] ] PUNCT cana-1090 73 6 β†’ β†’ X cana-1090 73 7 [ [ X cana-1090 73 8 𝐴 𝐴 PROPN cana-1090 73 9 , , PUNCT cana-1090 73 10 𝐡 𝐡 PROPN cana-1090 73 11 ] ] X cana-1090 73 12 , , PUNCT cana-1090 73 13 𝑓2 𝑓2 NOUN cana-1090 73 14 : : PUNCT cana-1090 73 15 [ [ X cana-1090 73 16 𝐢 𝐢 PROPN cana-1090 73 17 , , PUNCT cana-1090 73 18 𝐷 𝐷 PROPN cana-1090 73 19 ] ] X cana-1090 73 20 Γ— Γ— NOUN cana-1090 73 21 [ [ X cana-1090 73 22 𝐸 𝐸 PROPN cana-1090 73 23 , , PUNCT cana-1090 73 24 𝐹 𝐹 PROPN cana-1090 73 25 ] ] X cana-1090 73 26 Γ— Γ— NOUN cana-1090 73 27 [ [ X cana-1090 73 28 𝐸 𝐸 PROPN cana-1090 73 29 , , PUNCT cana-1090 73 30 𝐹 𝐹 PROPN cana-1090 73 31 ] ] X cana-1090 73 32 Γ— Γ— PROPN cana-1090 73 33 [ [ X cana-1090 73 34 𝐴 𝐴 PROPN cana-1090 73 35 , , PUNCT cana-1090 73 36 𝐡 𝐡 PROPN cana-1090 73 37 ] ] PUNCT cana-1090 73 38 β†’ β†’ X cana-1090 73 39 [ [ X cana-1090 73 40 𝐢 𝐢 PROPN cana-1090 73 41 , , PUNCT cana-1090 73 42 𝐷 𝐷 PROPN cana-1090 73 43 ] ] PUNCT cana-1090 73 44 and and CCONJ cana-1090 73 45 𝑓3 𝑓3 PROPN cana-1090 73 46 : : PUNCT cana-1090 73 47 [ [ X cana-1090 73 48 𝐸 𝐸 PROPN cana-1090 73 49 , , PUNCT cana-1090 73 50 𝐹 𝐹 PROPN cana-1090 73 51 ] ] X cana-1090 73 52 Γ— Γ— PROPN cana-1090 73 53 [ [ X cana-1090 73 54 𝐴 𝐴 PROPN cana-1090 73 55 , , PUNCT cana-1090 73 56 𝐡 𝐡 PROPN cana-1090 73 57 ] ] X cana-1090 73 58 Γ— Γ— PROPN cana-1090 73 59 [ [ X cana-1090 73 60 𝐴 𝐴 PROPN cana-1090 73 61 , , PUNCT cana-1090 73 62 𝐡 𝐡 PROPN cana-1090 73 63 ] ] X cana-1090 73 64 Γ— Γ— PROPN cana-1090 73 65 [ [ X cana-1090 73 66 𝐢 𝐢 PROPN cana-1090 73 67 , , PUNCT cana-1090 73 68 𝐷 𝐷 PROPN cana-1090 73 69 ] ] PUNCT cana-1090 73 70 β†’ β†’ X cana-1090 73 71 [ [ X cana-1090 73 72 𝐸 𝐸 PROPN cana-1090 73 73 , , PUNCT cana-1090 73 74 𝐹 𝐹 PROPN cana-1090 73 75 ] ] PUNCT cana-1090 73 76 be be AUX cana-1090 73 77 continuous continuous ADJ cana-1090 73 78 . . PUNCT cana-1090 74 1 examine examine VERB cana-1090 74 2 π‘₯𝑛+1 π‘₯𝑛+1 X cana-1090 74 3 = = SYM cana-1090 74 4 𝑓1(π‘₯π‘›βˆ’1 𝑓1(π‘₯π‘›βˆ’1 NOUN cana-1090 74 5 , , PUNCT cana-1090 74 6 𝑦𝑛 𝑦𝑛 NOUN cana-1090 74 7 , , PUNCT cana-1090 74 8 π‘¦π‘›βˆ’1 π‘¦π‘›βˆ’1 PROPN cana-1090 74 9 , , PUNCT cana-1090 74 10 π‘§π‘›βˆ’1 π‘§π‘›βˆ’1 NOUN cana-1090 74 11 ) ) PUNCT cana-1090 74 12 , , PUNCT cana-1090 74 13 𝑦𝑛+1 𝑦𝑛+1 PROPN cana-1090 74 14 = = SYM cana-1090 74 15 𝑓2(π‘¦π‘›βˆ’1 𝑓2(π‘¦π‘›βˆ’1 PROPN cana-1090 74 16 , , PUNCT cana-1090 74 17 𝑧𝑛 𝑧𝑛 PROPN cana-1090 74 18 , , PUNCT cana-1090 74 19 π‘§π‘›βˆ’1 π‘§π‘›βˆ’1 NOUN cana-1090 74 20 , , PUNCT cana-1090 74 21 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-1090 74 22 ) ) PUNCT cana-1090 74 23 , , PUNCT cana-1090 74 24 ( ( PUNCT cana-1090 74 25 10 10 NUM cana-1090 74 26 ) ) SYM cana-1090 74 27 𝑧𝑛+1 𝑧𝑛+1 NUM cana-1090 74 28 = = SYM cana-1090 74 29 𝑓3(π‘§π‘›βˆ’1 𝑓3(π‘§π‘›βˆ’1 PROPN cana-1090 74 30 , , PUNCT cana-1090 74 31 π‘₯𝑛 π‘₯𝑛 PROPN cana-1090 74 32 , , PUNCT cana-1090 74 33 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-1090 74 34 , , PUNCT cana-1090 74 35 π‘¦π‘›βˆ’1 π‘¦π‘›βˆ’1 PROPN cana-1090 74 36 ) ) PUNCT cana-1090 74 37 , , PUNCT cana-1090 74 38 𝑛 𝑛 PROPN cana-1090 74 39 = = SYM cana-1090 74 40 0,1,2 0,1,2 NUM cana-1090 74 41 , , PUNCT cana-1090 74 42 … … PUNCT cana-1090 74 43 where where SCONJ cana-1090 74 44 π‘₯βˆ’1 π‘₯βˆ’1 NOUN cana-1090 74 45 , , PUNCT cana-1090 74 46 π‘₯0 π‘₯0 PROPN cana-1090 74 47 ∈ ∈ PROPN cana-1090 75 1 [ [ X cana-1090 75 2 𝐴 𝐴 PROPN cana-1090 75 3 , , PUNCT cana-1090 75 4 𝐡 𝐡 PROPN cana-1090 75 5 ] ] X cana-1090 75 6 , , PUNCT cana-1090 75 7 π‘¦βˆ’1 π‘¦βˆ’1 PROPN cana-1090 75 8 , , PUNCT cana-1090 75 9 𝑦0 𝑦0 NOUN cana-1090 75 10 ∈ ∈ PROPN cana-1090 75 11 [ [ X cana-1090 75 12 𝐢 𝐢 PROPN cana-1090 75 13 , , PUNCT cana-1090 75 14 𝐷 𝐷 PROPN cana-1090 75 15 ] ] PUNCT cana-1090 75 16 and and CCONJ cana-1090 75 17 π‘§βˆ’1 π‘§βˆ’1 PROPN cana-1090 75 18 , , PUNCT cana-1090 75 19 𝑧0 𝑧0 PROPN cana-1090 75 20 ∈ ∈ PROPN cana-1090 76 1 [ [ X cana-1090 76 2 𝐸 𝐸 PROPN cana-1090 76 3 , , PUNCT cana-1090 76 4 𝐹 𝐹 PROPN cana-1090 76 5 ] ] PUNCT cana-1090 76 6 . . PUNCT cana-1090 77 1 ( ( PUNCT cana-1090 77 2 or or CCONJ cana-1090 77 3 π‘₯𝑛0 π‘₯𝑛0 INTJ cana-1090 77 4 , , PUNCT cana-1090 77 5 π‘₯𝑛0 π‘₯𝑛0 PROPN cana-1090 78 1 + + ADJ cana-1090 78 2 1 1 NUM cana-1090 78 3 ∈ ∈ PROPN cana-1090 78 4 [ [ X cana-1090 78 5 𝐴 𝐴 PROPN cana-1090 78 6 , , PUNCT cana-1090 78 7 𝐡 𝐡 PROPN cana-1090 78 8 ] ] PUNCT cana-1090 78 9 , , PUNCT cana-1090 78 10 𝑦𝑛0 𝑦𝑛0 NOUN cana-1090 78 11 , , PUNCT cana-1090 78 12 𝑦𝑛0 𝑦𝑛0 X cana-1090 78 13 + + ADJ cana-1090 78 14 1 1 NUM cana-1090 78 15 ∈ ∈ PROPN cana-1090 78 16 [ [ X cana-1090 78 17 𝐢 𝐢 PROPN cana-1090 78 18 , , PUNCT cana-1090 78 19 𝐷 𝐷 PROPN cana-1090 78 20 ] ] PUNCT cana-1090 78 21 , , PUNCT cana-1090 78 22 𝑧𝑛0 𝑧𝑛0 NOUN cana-1090 78 23 , , PUNCT cana-1090 78 24 𝑧𝑛0 𝑧𝑛0 NOUN cana-1090 78 25 + + ADJ cana-1090 78 26 1 1 NUM cana-1090 78 27 ∈ ∈ NOUN cana-1090 79 1 [ [ X cana-1090 79 2 𝐸 𝐸 PROPN cana-1090 79 3 , , PUNCT cana-1090 79 4 𝐹 𝐹 PROPN cana-1090 79 5 ] ] PUNCT cana-1090 79 6 , , PUNCT cana-1090 79 7 𝑛0 𝑛0 VERB cana-1090 79 8 ∈ ∈ PROPN cana-1090 79 9 β„• β„• PROPN cana-1090 79 10 ) ) PUNCT cana-1090 79 11 . . PUNCT cana-1090 80 1 assume assume VERB cana-1090 80 2 the the DET cana-1090 80 3 conditions condition NOUN cana-1090 80 4 given give VERB cana-1090 80 5 below below ADP cana-1090 80 6 holds hold NOUN cana-1090 80 7 . . PUNCT cana-1090 81 1 1 1 X cana-1090 81 2 . . X cana-1090 82 1 if if SCONJ cana-1090 82 2 𝑓1(π‘₯ 𝑓1(π‘₯ NUM cana-1090 82 3 , , PUNCT cana-1090 82 4 𝑦 𝑦 NOUN cana-1090 82 5 , , PUNCT cana-1090 82 6 𝑧 𝑧 NOUN cana-1090 82 7 , , PUNCT cana-1090 82 8 𝑒 𝑒 NOUN cana-1090 82 9 ) ) PUNCT cana-1090 82 10 , , PUNCT cana-1090 82 11 𝑓2(π‘₯ 𝑓2(π‘₯ PROPN cana-1090 82 12 , , PUNCT cana-1090 82 13 𝑦 𝑦 NOUN cana-1090 82 14 , , PUNCT cana-1090 82 15 𝑧 𝑧 NOUN cana-1090 82 16 , , PUNCT cana-1090 82 17 𝑒 𝑒 X cana-1090 82 18 ) ) PUNCT cana-1090 82 19 and and CCONJ cana-1090 82 20 𝑓3(π‘₯ 𝑓3(π‘₯ PROPN cana-1090 82 21 , , PUNCT cana-1090 82 22 𝑦 𝑦 PROPN cana-1090 82 23 , , PUNCT cana-1090 82 24 𝑧 𝑧 NOUN cana-1090 82 25 , , PUNCT cana-1090 82 26 𝑒 𝑒 X cana-1090 82 27 ) ) PUNCT cana-1090 82 28 are be AUX cana-1090 82 29 nonincreasing nonincrease VERB cana-1090 82 30 in in ADP cana-1090 82 31 x x PRON cana-1090 82 32 , , PUNCT cana-1090 82 33 nondecreasing nondecrease VERB cana-1090 82 34 in in ADP cana-1090 82 35 y y PROPN cana-1090 82 36 , , PUNCT cana-1090 82 37 nonincreasing nonincrease VERB cana-1090 82 38 in in ADP cana-1090 82 39 z z NOUN cana-1090 82 40 and and CCONJ cana-1090 82 41 nonincreasing nonincrease VERB cana-1090 82 42 in in ADP cana-1090 82 43 u. u. PROPN cana-1090 82 44 2 2 NUM cana-1090 82 45 . . PUNCT cana-1090 83 1 if if SCONJ cana-1090 83 2 ( ( PUNCT cana-1090 83 3 π‘š1 π‘š1 PROPN cana-1090 83 4 , , PUNCT cana-1090 83 5 𝑀1 𝑀1 PROPN cana-1090 83 6 , , PUNCT cana-1090 83 7 π‘š2 π‘š2 PROPN cana-1090 83 8 , , PUNCT cana-1090 83 9 𝑀2 𝑀2 PROPN cana-1090 83 10 , , PUNCT cana-1090 83 11 π‘š3 π‘š3 NOUN cana-1090 83 12 , , PUNCT cana-1090 83 13 𝑀3 𝑀3 NOUN cana-1090 83 14 ) ) PUNCT cana-1090 83 15 ∈ ∈ PROPN cana-1090 84 1 [ [ X cana-1090 84 2 𝐴 𝐴 PROPN cana-1090 84 3 , , PUNCT cana-1090 84 4 𝐡]2 𝐡]2 X cana-1090 84 5 Γ— Γ— PROPN cana-1090 85 1 [ [ X cana-1090 85 2 𝐢 𝐢 PROPN cana-1090 85 3 , , PUNCT cana-1090 85 4 𝐷]2 𝐷]2 NOUN cana-1090 85 5 Γ— Γ— NOUN cana-1090 85 6 [ [ X cana-1090 85 7 𝐸 𝐸 PROPN cana-1090 85 8 , , PUNCT cana-1090 85 9 𝐹]2 𝐹]2 PRON cana-1090 85 10 satisfies satisfy VERB cana-1090 85 11 the the DET cana-1090 85 12 systems system NOUN cana-1090 85 13 π‘š1 π‘š1 NOUN cana-1090 85 14 = = SYM cana-1090 85 15 𝑓1(𝑀1 𝑓1(𝑀1 PROPN cana-1090 85 16 , , PUNCT cana-1090 85 17 π‘š2 π‘š2 PROPN cana-1090 85 18 , , PUNCT cana-1090 85 19 𝑀2 𝑀2 PROPN cana-1090 85 20 , , PUNCT cana-1090 85 21 𝑀3 𝑀3 NOUN cana-1090 85 22 ) ) PUNCT cana-1090 85 23 ; ; PUNCT cana-1090 85 24 𝑀1 𝑀1 NOUN cana-1090 85 25 = = SYM cana-1090 85 26 𝑓1(π‘š1 𝑓1(π‘š1 PROPN cana-1090 85 27 , , PUNCT cana-1090 85 28 𝑀2 𝑀2 PROPN cana-1090 85 29 , , PUNCT cana-1090 85 30 π‘š2 π‘š2 NOUN cana-1090 85 31 , , PUNCT cana-1090 85 32 π‘š3 π‘š3 NOUN cana-1090 85 33 ) ) PUNCT cana-1090 85 34 , , PUNCT cana-1090 85 35 π‘š2 π‘š2 NOUN cana-1090 85 36 = = SYM cana-1090 85 37 𝑓2(𝑀2 𝑓2(𝑀2 PROPN cana-1090 85 38 , , PUNCT cana-1090 85 39 π‘š3 π‘š3 NOUN cana-1090 85 40 , , PUNCT cana-1090 85 41 𝑀3 𝑀3 NOUN cana-1090 85 42 , , PUNCT cana-1090 85 43 𝑀1 𝑀1 PROPN cana-1090 85 44 ) ) PUNCT cana-1090 85 45 ; ; PUNCT cana-1090 85 46 𝑀2 𝑀2 PROPN cana-1090 85 47 = = SYM cana-1090 85 48 𝑓2(π‘š2 𝑓2(π‘š2 NOUN cana-1090 85 49 , , PUNCT cana-1090 85 50 𝑀3 𝑀3 NOUN cana-1090 85 51 , , PUNCT cana-1090 85 52 π‘š3 π‘š3 NOUN cana-1090 85 53 , , PUNCT cana-1090 85 54 π‘š1 π‘š1 NOUN cana-1090 85 55 ) ) PUNCT cana-1090 85 56 and and CCONJ cana-1090 85 57 π‘š3 π‘š3 NOUN cana-1090 85 58 = = SYM cana-1090 85 59 𝑓3(π‘š1 𝑓3(π‘š1 PROPN cana-1090 85 60 , , PUNCT cana-1090 85 61 𝑀1 𝑀1 PROPN cana-1090 85 62 , , PUNCT cana-1090 85 63 𝑀3 𝑀3 NOUN cana-1090 85 64 , , PUNCT cana-1090 85 65 𝑀2 𝑀2 PROPN cana-1090 85 66 ) ) PUNCT cana-1090 85 67 ; ; PUNCT cana-1090 85 68 𝑀3 𝑀3 PROPN cana-1090 85 69 = = SYM cana-1090 85 70 𝑓3(𝑀1 𝑓3(𝑀1 PROPN cana-1090 85 71 , , PUNCT cana-1090 85 72 π‘š1 π‘š1 PROPN cana-1090 85 73 , , PUNCT cana-1090 85 74 π‘š3 π‘š3 NOUN cana-1090 85 75 , , PUNCT cana-1090 85 76 π‘š2 π‘š2 NOUN cana-1090 85 77 ) ) PUNCT cana-1090 85 78 then then ADV cana-1090 85 79 π‘š1 π‘š1 NOUN cana-1090 85 80 = = SYM cana-1090 85 81 𝑀1 𝑀1 PROPN cana-1090 85 82 , , PUNCT cana-1090 85 83 π‘š2 π‘š2 NOUN cana-1090 85 84 = = SYM cana-1090 85 85 𝑀2 𝑀2 PROPN cana-1090 85 86 and and CCONJ cana-1090 85 87 π‘š3 π‘š3 NOUN cana-1090 85 88 = = SYM cana-1090 85 89 𝑀3 𝑀3 NOUN cana-1090 85 90 , , PUNCT cana-1090 85 91 then then ADV cana-1090 85 92 ( ( PUNCT cana-1090 85 93 οΏ½ οΏ½ NOUN cana-1090 85 94 Μ… Μ… NOUN cana-1090 85 95 οΏ½ οΏ½ PROPN cana-1090 85 96 , , PUNCT cana-1090 85 97 οΏ½ οΏ½ PROPN cana-1090 85 98 Μ… Μ… NOUN cana-1090 85 99 οΏ½ οΏ½ NOUN cana-1090 85 100 , , PUNCT cana-1090 85 101 𝑧̅ 𝑧̅ NOUN cana-1090 85 102 ) ) PUNCT cana-1090 85 103 is be AUX cana-1090 85 104 the the DET cana-1090 85 105 unique unique ADJ cana-1090 85 106 equilibrium equilibrium NOUN cana-1090 85 107 point point NOUN cana-1090 85 108 of of ADP cana-1090 85 109 ( ( PUNCT cana-1090 85 110 10 10 NUM cana-1090 85 111 ) ) PUNCT cana-1090 85 112 where where SCONJ cana-1090 85 113 οΏ½ οΏ½ NOUN cana-1090 85 114 Μ… Μ… VERB cana-1090 85 115 οΏ½ οΏ½ PROPN cana-1090 85 116 ∈ ∈ PROPN cana-1090 85 117 [ [ X cana-1090 85 118 𝐴 𝐴 PROPN cana-1090 85 119 , , PUNCT cana-1090 85 120 𝐡], 𝐡], PROPN cana-1090 85 121 οΏ½ οΏ½ NOUN cana-1090 85 122 Μ… Μ… NOUN cana-1090 85 123 οΏ½ οΏ½ NOUN cana-1090 85 124 ∈ ∈ PROPN cana-1090 85 125 [ [ X cana-1090 85 126 𝐢 𝐢 PROPN cana-1090 85 127 , , PUNCT cana-1090 85 128 𝐷 𝐷 PROPN cana-1090 85 129 ] ] PUNCT cana-1090 85 130 and and CCONJ cana-1090 85 131 𝑧̅ 𝑧̅ PROPN cana-1090 85 132 ∈ ∈ PROPN cana-1090 85 133 [ [ X cana-1090 85 134 𝐸 𝐸 PROPN cana-1090 85 135 , , PUNCT cana-1090 85 136 𝐹 𝐹 PROPN cana-1090 85 137 ] ] PUNCT cana-1090 85 138 . . PUNCT cana-1090 86 1 and and CCONJ cana-1090 86 2 any any DET cana-1090 86 3 other other ADJ cana-1090 86 4 solution solution NOUN cana-1090 86 5 of of ADP cana-1090 86 6 ( ( PUNCT cana-1090 86 7 10 10 NUM cana-1090 86 8 ) ) PUNCT cana-1090 86 9 converges converge VERB cana-1090 86 10 to to ADP cana-1090 86 11 ( ( PUNCT cana-1090 86 12 οΏ½ οΏ½ NOUN cana-1090 86 13 Μ… Μ… NOUN cana-1090 86 14 οΏ½ οΏ½ PROPN cana-1090 86 15 , , PUNCT cana-1090 86 16 οΏ½ οΏ½ PROPN cana-1090 86 17 Μ… Μ… NOUN cana-1090 86 18 οΏ½ οΏ½ NOUN cana-1090 86 19 , , PUNCT cana-1090 86 20 𝑧̅ 𝑧̅ PROPN cana-1090 86 21 ) ) PUNCT cana-1090 86 22 . . PUNCT cana-1090 87 1 theorem theorem VERB cana-1090 87 2 2.5 2.5 NUM cana-1090 87 3 let let VERB cana-1090 87 4 ( ( PUNCT cana-1090 87 5 2 2 X cana-1090 87 6 ) ) PUNCT cana-1090 87 7 hold hold NOUN cana-1090 87 8 . . PUNCT cana-1090 88 1 suppose suppose VERB cana-1090 88 2 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 VERB cana-1090 88 3 < < X cana-1090 88 4 1 1 NUM cana-1090 88 5 , , PUNCT cana-1090 88 6 π‘Ž2π‘’βˆ’π›Ό2 π‘Ž2π‘’βˆ’π›Ό2 X cana-1090 88 7 < < X cana-1090 88 8 1 1 NUM cana-1090 88 9 , , PUNCT cana-1090 88 10 π‘Ž3π‘’βˆ’π›Ό3 π‘Ž3π‘’βˆ’π›Ό3 X cana-1090 88 11 < < X cana-1090 88 12 1 1 NUM cana-1090 88 13 ( ( PUNCT cana-1090 88 14 11 11 NUM cana-1090 88 15 ) ) PUNCT cana-1090 88 16 and and CCONJ cana-1090 89 1 [ [ X cana-1090 89 2 𝐷2𝐷3+𝐡3𝐿2][𝐷1𝐷2+𝐡2𝐿1][𝐷3𝐷1+𝐡1𝐿3 𝐷2𝐷3+𝐡3𝐿2][𝐷1𝐷2+𝐡2𝐿1][𝐷3𝐷1+𝐡1𝐿3 X cana-1090 89 3 ] ] X cana-1090 90 1 [ [ X cana-1090 90 2 𝐡2𝐡3βˆ’π·2𝐿3][𝐡1𝐡2βˆ’π·1𝐿2][𝐡3𝐡1βˆ’π·3𝐿1 𝐡2𝐡3βˆ’π·2𝐿3][𝐡1𝐡2βˆ’π·1𝐿2][𝐡3𝐡1βˆ’π·3𝐿1 X cana-1090 90 3 ] ] X cana-1090 90 4 < < X cana-1090 90 5 1 1 NUM cana-1090 90 6 , , PUNCT cana-1090 90 7 ( ( PUNCT cana-1090 90 8 12 12 NUM cana-1090 90 9 ) ) PUNCT cana-1090 90 10 where where SCONJ cana-1090 90 11 𝐡1 𝐡1 NOUN cana-1090 90 12 = = NOUN cana-1090 90 13 1 1 NUM cana-1090 90 14 βˆ’ βˆ’ NOUN cana-1090 90 15 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 NOUN cana-1090 90 16 , , PUNCT cana-1090 90 17 𝐡2 𝐡2 NOUN cana-1090 90 18 = = NOUN cana-1090 90 19 1 1 NUM cana-1090 90 20 βˆ’ βˆ’ NOUN cana-1090 90 21 π‘Ž2π‘’βˆ’π›Ό2 π‘Ž2π‘’βˆ’π›Ό2 NOUN cana-1090 90 22 , , PUNCT cana-1090 90 23 𝐡3 𝐡3 PROPN cana-1090 90 24 = = SYM cana-1090 90 25 1 1 NUM cana-1090 90 26 βˆ’ βˆ’ NOUN cana-1090 90 27 π‘Ž3π‘’βˆ’π›Ό3 π‘Ž3π‘’βˆ’π›Ό3 NOUN cana-1090 90 28 , , PUNCT cana-1090 90 29 𝐷1 𝐷1 NOUN cana-1090 90 30 = = PROPN cana-1090 90 31 𝑏1π‘’βˆ’π›Ό2(1 𝑏1π‘’βˆ’π›Ό2(1 PROPN cana-1090 90 32 + + CCONJ cana-1090 90 33 𝐴2 𝐴2 PROPN cana-1090 90 34 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 90 35 ) ) PUNCT cana-1090 90 36 , , PUNCT cana-1090 90 37 𝐷2 𝐷2 PROPN cana-1090 90 38 = = SYM cana-1090 91 1 𝑏2π‘’βˆ’π›Ό3(1 𝑏2π‘’βˆ’π›Ό3(1 PROPN cana-1090 91 2 + + NUM cana-1090 91 3 𝐴3 𝐴3 PROPN cana-1090 91 4 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 91 5 ) ) PUNCT cana-1090 91 6 , , PUNCT cana-1090 91 7 𝐷3 𝐷3 X cana-1090 91 8 = = PRON cana-1090 91 9 𝑏3π‘’βˆ’π›Ό1(1 𝑏3π‘’βˆ’π›Ό1(1 ADJ cana-1090 91 10 + + CCONJ cana-1090 91 11 𝐴1 𝐴1 PROPN cana-1090 91 12 1βˆ’π΅ 1βˆ’π΅ NUM cana-1090 91 13 ) ) PUNCT cana-1090 91 14 , , PUNCT cana-1090 91 15 𝐿1 𝐿1 NOUN cana-1090 91 16 = = SYM cana-1090 91 17 𝑐1π‘’βˆ’π›Ό3 𝑐1π‘’βˆ’π›Ό3 NOUN cana-1090 91 18 , , PUNCT cana-1090 91 19 𝐿2 𝐿2 NUM cana-1090 91 20 = = NOUN cana-1090 91 21 𝑐2π‘’βˆ’π›Ό1 𝑐2π‘’βˆ’π›Ό1 NOUN cana-1090 91 22 , , PUNCT cana-1090 91 23 𝐿3 𝐿3 NOUN cana-1090 91 24 = = X cana-1090 91 25 𝑐3π‘’βˆ’π›Ό2 𝑐3π‘’βˆ’π›Ό2 X cana-1090 91 26 . . PUNCT cana-1090 92 1 then then ADV cana-1090 92 2 𝐸( 𝐸( NOUN cana-1090 92 3 οΏ½ οΏ½ PROPN cana-1090 92 4 Μ… Μ… PROPN cana-1090 92 5 οΏ½ οΏ½ PROPN cana-1090 92 6 , , PUNCT cana-1090 92 7 οΏ½ οΏ½ PROPN cana-1090 92 8 Μ… Μ… NOUN cana-1090 92 9 οΏ½ οΏ½ NOUN cana-1090 92 10 , , PUNCT cana-1090 92 11 𝑧̅ 𝑧̅ NOUN cana-1090 92 12 ) ) PUNCT cana-1090 92 13 is be AUX cana-1090 92 14 the the DET cana-1090 92 15 unique unique ADJ cana-1090 92 16 positive positive ADJ cana-1090 92 17 equilibrium equilibrium NOUN cana-1090 92 18 of of ADP cana-1090 92 19 ( ( PUNCT cana-1090 92 20 1 1 NUM cana-1090 92 21 ) ) PUNCT cana-1090 92 22 . . PUNCT cana-1090 93 1 and and CCONJ cana-1090 93 2 any any DET cana-1090 93 3 solution solution NOUN cana-1090 93 4 of of ADP cana-1090 93 5 ( ( PUNCT cana-1090 93 6 1 1 X cana-1090 93 7 ) ) PUNCT cana-1090 93 8 converges converge NOUN cana-1090 93 9 to to ADP cana-1090 93 10 𝐸( 𝐸( NOUN cana-1090 93 11 οΏ½ οΏ½ NOUN cana-1090 93 12 Μ… Μ… PROPN cana-1090 93 13 οΏ½ οΏ½ PROPN cana-1090 93 14 , , PUNCT cana-1090 93 15 οΏ½ οΏ½ PROPN cana-1090 93 16 Μ… Μ… NOUN cana-1090 93 17 οΏ½ οΏ½ NOUN cana-1090 93 18 , , PUNCT cana-1090 93 19 𝑧̅ 𝑧̅ NOUN cana-1090 93 20 ) ) PUNCT cana-1090 93 21 . . PUNCT cana-1090 94 1 proof proof NOUN cana-1090 94 2 : : PUNCT cana-1090 94 3 define define VERB cana-1090 94 4 𝑓1(π‘₯ 𝑓1(π‘₯ NUM cana-1090 94 5 , , PUNCT cana-1090 94 6 𝑦 𝑦 NOUN cana-1090 94 7 , , PUNCT cana-1090 94 8 𝑧 𝑧 NOUN cana-1090 94 9 ) ) PUNCT cana-1090 94 10 = = SYM cana-1090 94 11 𝛼1 𝛼1 NOUN cana-1090 94 12 + + CCONJ cana-1090 94 13 π‘Ž1π‘’βˆ’π‘₯ π‘Ž1π‘’βˆ’π‘₯ NOUN cana-1090 94 14 + + X cana-1090 94 15 𝑏1π‘¦π‘’βˆ’π‘¦ 𝑏1π‘¦π‘’βˆ’π‘¦ ADV cana-1090 95 1 + + NOUN cana-1090 95 2 𝑐1π‘’βˆ’π‘§ 𝑐1π‘’βˆ’π‘§ ADV cana-1090 95 3 , , PUNCT cana-1090 95 4 𝑓2(π‘₯ 𝑓2(π‘₯ PROPN cana-1090 95 5 , , PUNCT cana-1090 95 6 𝑦 𝑦 NOUN cana-1090 95 7 , , PUNCT cana-1090 95 8 𝑧 𝑧 NOUN cana-1090 95 9 ) ) PUNCT cana-1090 95 10 = = SYM cana-1090 95 11 𝛼2 𝛼2 ADJ cana-1090 95 12 + + CCONJ cana-1090 95 13 π‘Ž2π‘’βˆ’π‘¦ π‘Ž2π‘’βˆ’π‘¦ ADJ cana-1090 95 14 + + CCONJ cana-1090 95 15 𝑏2π‘§π‘’βˆ’π‘§ 𝑏2π‘§π‘’βˆ’π‘§ NOUN cana-1090 95 16 + + NUM cana-1090 95 17 𝑐2π‘’βˆ’π‘₯ 𝑐2π‘’βˆ’π‘₯ NOUN cana-1090 95 18 , , PUNCT cana-1090 95 19 𝑓3(π‘₯ 𝑓3(π‘₯ PROPN cana-1090 95 20 , , PUNCT cana-1090 95 21 𝑦 𝑦 NOUN cana-1090 95 22 , , PUNCT cana-1090 95 23 𝑧 𝑧 NOUN cana-1090 95 24 ) ) PUNCT cana-1090 95 25 = = SYM cana-1090 96 1 𝛼3 𝛼3 NOUN cana-1090 96 2 + + CCONJ cana-1090 96 3 π‘Ž3π‘’βˆ’π‘§ π‘Ž3π‘’βˆ’π‘§ NOUN cana-1090 96 4 + + NUM cana-1090 96 5 𝑏3π‘₯π‘’βˆ’π‘₯ 𝑏3π‘₯π‘’βˆ’π‘₯ NOUN cana-1090 97 1 + + CCONJ cana-1090 97 2 𝑐3π‘’βˆ’π‘¦π‘†. 𝑐3π‘’βˆ’π‘¦π‘†. NOUN cana-1090 97 3 take take NOUN cana-1090 97 4 π‘šπ‘– π‘šπ‘– PRON cana-1090 97 5 ≀ ≀ ADJ cana-1090 97 6 𝑀𝑖 𝑀𝑖 PROPN cana-1090 97 7 , , PUNCT cana-1090 97 8 𝑖 𝑖 SYM cana-1090 97 9 = = NOUN cana-1090 97 10 1,2,3 1,2,3 NUM cana-1090 97 11 to to PART cana-1090 97 12 denote denote VERB cana-1090 97 13 positive positive ADJ cana-1090 97 14 reals real NOUN cana-1090 97 15 where where SCONJ cana-1090 97 16 and and CCONJ cana-1090 97 17 π‘š1 π‘š1 NOUN cana-1090 97 18 = = PROPN cana-1090 97 19 𝛼1 𝛼1 PROPN cana-1090 97 20 + + CCONJ cana-1090 97 21 π‘Ž1π‘’βˆ’π‘€1 π‘Ž1π‘’βˆ’π‘€1 PROPN cana-1090 97 22 + + NOUN cana-1090 97 23 𝑏1π‘š2π‘’βˆ’π‘€2 𝑏1π‘š2π‘’βˆ’π‘€2 NOUN cana-1090 97 24 + + X cana-1090 97 25 𝑐1π‘’βˆ’π‘€3 𝑐1π‘’βˆ’π‘€3 PROPN cana-1090 97 26 , , PUNCT cana-1090 97 27 𝑀1 𝑀1 NOUN cana-1090 97 28 = = NOUN cana-1090 97 29 𝛼1 𝛼1 NOUN cana-1090 97 30 + + CCONJ cana-1090 97 31 π‘Ž1π‘’βˆ’π‘š1 π‘Ž1π‘’βˆ’π‘š1 ADJ cana-1090 97 32 + + NUM cana-1090 97 33 𝑏1𝑀2π‘’βˆ’π‘š2 𝑏1𝑀2π‘’βˆ’π‘š2 NOUN cana-1090 97 34 + + CCONJ cana-1090 97 35 𝑐1π‘’βˆ’π‘š3 𝑐1π‘’βˆ’π‘š3 NOUN cana-1090 97 36 , , PUNCT cana-1090 97 37 π‘š2 π‘š2 NOUN cana-1090 97 38 = = SYM cana-1090 97 39 𝛼2 𝛼2 PROPN cana-1090 97 40 + + PUNCT cana-1090 97 41 π‘Ž2π‘’βˆ’π‘€2 π‘Ž2π‘’βˆ’π‘€2 NOUN cana-1090 97 42 + + CCONJ cana-1090 97 43 𝑏2π‘š3π‘’βˆ’π‘€3 𝑏2π‘š3π‘’βˆ’π‘€3 PROPN cana-1090 98 1 + + CCONJ cana-1090 98 2 𝑐1π‘’βˆ’π‘€1 𝑐1π‘’βˆ’π‘€1 NOUN cana-1090 98 3 , , PUNCT cana-1090 98 4 𝑀2 𝑀2 PROPN cana-1090 98 5 = = SYM cana-1090 98 6 𝛼2 𝛼2 PROPN cana-1090 98 7 + + NUM cana-1090 98 8 π‘Ž2π‘’βˆ’π‘š2 π‘Ž2π‘’βˆ’π‘š2 NOUN cana-1090 98 9 + + CCONJ cana-1090 98 10 𝑏2𝑀3π‘’βˆ’π‘š3 𝑏2𝑀3π‘’βˆ’π‘š3 X cana-1090 98 11 + + CCONJ cana-1090 98 12 𝑐1π‘’βˆ’π‘š1 𝑐1π‘’βˆ’π‘š1 NOUN cana-1090 98 13 communications communication NOUN cana-1090 98 14 on on ADP cana-1090 98 15 applied apply VERB cana-1090 98 16 nonlinear nonlinear ADJ cana-1090 98 17 analysis analysis NOUN cana-1090 98 18 issn issn NOUN cana-1090 98 19 : : PUNCT cana-1090 98 20 1074 1074 NUM cana-1090 98 21 - - PUNCT cana-1090 98 22 133x 133x NUM cana-1090 98 23 vol vol NOUN cana-1090 98 24 31 31 NUM cana-1090 98 25 no no NOUN cana-1090 98 26 . . PUNCT cana-1090 99 1 5s 5s NUM cana-1090 99 2 ( ( PUNCT cana-1090 99 3 2024 2024 NUM cana-1090 99 4 ) ) PUNCT cana-1090 99 5 556 556 NUM cana-1090 99 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 99 7 and and CCONJ cana-1090 99 8 π‘š3 π‘š3 NOUN cana-1090 99 9 = = SYM cana-1090 100 1 𝛼3 𝛼3 PROPN cana-1090 100 2 + + CCONJ cana-1090 100 3 π‘Ž3π‘’βˆ’π‘€3 π‘Ž3π‘’βˆ’π‘€3 PROPN cana-1090 101 1 + + CCONJ cana-1090 101 2 𝑏3π‘š1π‘’βˆ’π‘€3 𝑏3π‘š1π‘’βˆ’π‘€3 PROPN cana-1090 102 1 + + PUNCT cana-1090 102 2 𝑐1π‘’βˆ’π‘€2 𝑐1π‘’βˆ’π‘€2 NOUN cana-1090 102 3 , , PUNCT cana-1090 102 4 𝑀3 𝑀3 NOUN cana-1090 102 5 = = PUNCT cana-1090 102 6 𝛼3 𝛼3 NOUN cana-1090 102 7 + + CCONJ cana-1090 102 8 π‘Ž3π‘’βˆ’π‘š3 π‘Ž3π‘’βˆ’π‘š3 NOUN cana-1090 102 9 + + CCONJ cana-1090 102 10 𝑏3𝑀1π‘’βˆ’π‘š1 𝑏3𝑀1π‘’βˆ’π‘š1 PRON cana-1090 102 11 + + CCONJ cana-1090 102 12 𝑐1π‘’βˆ’π‘š2 𝑐1π‘’βˆ’π‘š2 NOUN cana-1090 102 13 . . PUNCT cana-1090 103 1 ( ( PUNCT cana-1090 103 2 13 13 NUM cana-1090 103 3 ) ) PUNCT cana-1090 103 4 therefore therefore ADV cana-1090 103 5 , , PUNCT cana-1090 103 6 𝑀1 𝑀1 PROPN cana-1090 103 7 βˆ’ βˆ’ PROPN cana-1090 103 8 π‘š1 π‘š1 NOUN cana-1090 103 9 = = SYM cana-1090 103 10 π‘Ž1[π‘’βˆ’π‘š1 π‘Ž1[π‘’βˆ’π‘š1 ADJ cana-1090 103 11 βˆ’ βˆ’ PROPN cana-1090 103 12 π‘’βˆ’π‘€1 π‘’βˆ’π‘€1 PROPN cana-1090 103 13 ] ] X cana-1090 103 14 + + NUM cana-1090 103 15 𝑏1[𝑀2π‘’βˆ’π‘š2 𝑏1[𝑀2π‘’βˆ’π‘š2 NOUN cana-1090 103 16 βˆ’ βˆ’ NOUN cana-1090 103 17 π‘š2π‘’βˆ’π‘€2 π‘š2π‘’βˆ’π‘€2 NOUN cana-1090 103 18 ] ] X cana-1090 103 19 + + CCONJ cana-1090 103 20 𝑐1[π‘’βˆ’π‘š3 𝑐1[π‘’βˆ’π‘š3 PROPN cana-1090 103 21 βˆ’ βˆ’ PROPN cana-1090 103 22 π‘’βˆ’π‘€3 π‘’βˆ’π‘€3 PROPN cana-1090 103 23 ] ] PUNCT cana-1090 103 24 . . PUNCT cana-1090 104 1 𝑀1 𝑀1 PROPN cana-1090 104 2 βˆ’ βˆ’ PROPN cana-1090 104 3 π‘š1 π‘š1 NOUN cana-1090 104 4 = = SYM cana-1090 104 5 π‘Ž1[π‘’βˆ’π‘š1 π‘Ž1[π‘’βˆ’π‘š1 ADJ cana-1090 104 6 βˆ’ βˆ’ PROPN cana-1090 104 7 π‘’βˆ’π‘€1 π‘’βˆ’π‘€1 PROPN cana-1090 104 8 ] ] X cana-1090 104 9 + + CCONJ cana-1090 104 10 𝑏1π‘’βˆ’π‘š2βˆ’π‘€2[𝑀2𝑒𝑀2 𝑏1π‘’βˆ’π‘š2βˆ’π‘€2[𝑀2𝑒𝑀2 ADJ cana-1090 104 11 βˆ’ βˆ’ PROPN cana-1090 104 12 π‘š2π‘’π‘š2 π‘š2π‘’π‘š2 NOUN cana-1090 104 13 ] ] PUNCT cana-1090 104 14 + + CCONJ cana-1090 104 15 𝑐1[π‘’βˆ’π‘š3 𝑐1[π‘’βˆ’π‘š3 PROPN cana-1090 104 16 βˆ’ βˆ’ PROPN cana-1090 104 17 π‘’βˆ’π‘€3 π‘’βˆ’π‘€3 PROPN cana-1090 104 18 ] ] PUNCT cana-1090 104 19 . . PUNCT cana-1090 105 1 ( ( PUNCT cana-1090 105 2 14 14 NUM cana-1090 105 3 ) ) PUNCT cana-1090 105 4 here here ADV cana-1090 105 5 , , PUNCT cana-1090 105 6 there there PRON cana-1090 105 7 exists exist VERB cana-1090 105 8 a a DET cana-1090 105 9 𝜁1 𝜁1 PROPN cana-1090 105 10 , , PUNCT cana-1090 105 11 𝑀2 𝑀2 PROPN cana-1090 105 12 β‰₯ β‰₯ PROPN cana-1090 105 13 𝜁1 𝜁1 PROPN cana-1090 105 14 β‰₯ β‰₯ NUM cana-1090 105 15 π‘š2 π‘š2 NOUN cana-1090 105 16 satisfying satisfying NOUN cana-1090 105 17 𝑀2𝑒𝑀2 𝑀2𝑒𝑀2 VERB cana-1090 105 18 βˆ’ βˆ’ NUM cana-1090 105 19 π‘š2π‘’π‘š2 π‘š2π‘’π‘š2 NOUN cana-1090 105 20 = = SYM cana-1090 105 21 ( ( PUNCT cana-1090 105 22 1 1 NUM cana-1090 106 1 + + NUM cana-1090 106 2 𝜁1)𝑒1 𝜁1)𝑒1 PROPN cana-1090 106 3 𝜁 𝜁 PROPN cana-1090 106 4 ( ( PUNCT cana-1090 106 5 𝑀2 𝑀2 PROPN cana-1090 106 6 βˆ’ βˆ’ PROPN cana-1090 106 7 π‘š2 π‘š2 PROPN cana-1090 106 8 ) ) PUNCT cana-1090 106 9 . . PUNCT cana-1090 107 1 ( ( PUNCT cana-1090 107 2 15 15 NUM cana-1090 107 3 ) ) PUNCT cana-1090 107 4 from from ADP cana-1090 107 5 ( ( PUNCT cana-1090 107 6 14 14 NUM cana-1090 107 7 ) ) PUNCT cana-1090 107 8 and and CCONJ cana-1090 107 9 ( ( PUNCT cana-1090 107 10 15 15 X cana-1090 107 11 ) ) PUNCT cana-1090 107 12 we we PRON cana-1090 107 13 get get VERB cana-1090 107 14 , , PUNCT cana-1090 107 15 𝑀1 𝑀1 ADV cana-1090 107 16 βˆ’ βˆ’ PROPN cana-1090 107 17 π‘š1 π‘š1 NOUN cana-1090 107 18 = = SYM cana-1090 107 19 π‘Ž1[π‘’βˆ’π‘š1 π‘Ž1[π‘’βˆ’π‘š1 ADJ cana-1090 107 20 βˆ’ βˆ’ PROPN cana-1090 107 21 π‘’βˆ’π‘€1 π‘’βˆ’π‘€1 PROPN cana-1090 107 22 ] ] X cana-1090 108 1 + + CCONJ cana-1090 108 2 𝑏1π‘’βˆ’π‘š2βˆ’π‘€2+𝜁1(1 𝑏1π‘’βˆ’π‘š2βˆ’π‘€2+𝜁1(1 ADJ cana-1090 108 3 + + CCONJ cana-1090 108 4 𝜁1)[𝑀2 𝜁1)[𝑀2 NOUN cana-1090 108 5 βˆ’ βˆ’ NOUN cana-1090 108 6 π‘š2 π‘š2 NOUN cana-1090 108 7 ] ] PUNCT cana-1090 109 1 + + CCONJ cana-1090 109 2 𝑐1[π‘’βˆ’π‘š3 𝑐1[π‘’βˆ’π‘š3 PROPN cana-1090 109 3 βˆ’ βˆ’ PROPN cana-1090 109 4 π‘’βˆ’π‘€3 π‘’βˆ’π‘€3 PROPN cana-1090 109 5 ] ] PUNCT cana-1090 109 6 . . PUNCT cana-1090 110 1 ( ( PUNCT cana-1090 110 2 16 16 NUM cana-1090 110 3 ) ) PUNCT cana-1090 110 4 now now ADV cana-1090 110 5 , , PUNCT cana-1090 110 6 π‘Ž1[π‘’βˆ’π‘š1 π‘Ž1[π‘’βˆ’π‘š1 ADJ cana-1090 110 7 βˆ’ βˆ’ PROPN cana-1090 110 8 π‘’βˆ’π‘€1 π‘’βˆ’π‘€1 PROPN cana-1090 110 9 ] ] X cana-1090 110 10 = = PUNCT cana-1090 110 11 π‘Ž1π‘’βˆ’π‘š1βˆ’π‘€1[𝑒𝑀1 π‘Ž1π‘’βˆ’π‘š1βˆ’π‘€1[𝑒𝑀1 VERB cana-1090 110 12 βˆ’ βˆ’ PROPN cana-1090 110 13 π‘’π‘š1 π‘’π‘š1 NOUN cana-1090 110 14 ] ] PUNCT cana-1090 110 15 . . PUNCT cana-1090 111 1 and and CCONJ cana-1090 111 2 , , PUNCT cana-1090 111 3 there there PRON cana-1090 111 4 exists exist VERB cana-1090 111 5 a a DET cana-1090 111 6 πœ† πœ† NOUN cana-1090 111 7 , , PUNCT cana-1090 111 8 𝑀1 𝑀1 ADV cana-1090 111 9 β‰₯ β‰₯ NOUN cana-1090 111 10 πœ† πœ† PRON cana-1090 111 11 β‰₯ β‰₯ NUM cana-1090 111 12 π‘š1 π‘š1 NOUN cana-1090 111 13 satisfying satisfy VERB cana-1090 111 14 π‘Ž1[π‘’βˆ’π‘š1 π‘Ž1[π‘’βˆ’π‘š1 NOUN cana-1090 111 15 βˆ’ βˆ’ PROPN cana-1090 111 16 π‘’βˆ’π‘€1 π‘’βˆ’π‘€1 PROPN cana-1090 111 17 ] ] X cana-1090 111 18 = = PUNCT cana-1090 111 19 π‘Ž1π‘’βˆ’π‘š1βˆ’π‘€1+πœ†[𝑀1 π‘Ž1π‘’βˆ’π‘š1βˆ’π‘€1+πœ†[𝑀1 PROPN cana-1090 111 20 βˆ’ βˆ’ NUM cana-1090 111 21 π‘š1 π‘š1 NOUN cana-1090 111 22 ] ] X cana-1090 111 23 . . PUNCT cana-1090 112 1 ( ( PUNCT cana-1090 112 2 17 17 NUM cana-1090 112 3 ) ) PUNCT cana-1090 112 4 since since SCONJ cana-1090 112 5 π‘š1 π‘š1 PROPN cana-1090 112 6 , , PUNCT cana-1090 112 7 𝑀1 𝑀1 PROPN cana-1090 112 8 β‰₯ β‰₯ NOUN cana-1090 112 9 𝛼1 𝛼1 NOUN cana-1090 112 10 , , PUNCT cana-1090 112 11 π‘Ž1[π‘’βˆ’π‘š1 π‘Ž1[π‘’βˆ’π‘š1 ADJ cana-1090 112 12 βˆ’ βˆ’ PROPN cana-1090 112 13 π‘’βˆ’π‘€1 π‘’βˆ’π‘€1 PROPN cana-1090 112 14 ] ] PUNCT cana-1090 112 15 ≀ ≀ ADV cana-1090 113 1 π‘Ž1π‘’βˆ’π›Ό1[𝑀1 π‘Ž1π‘’βˆ’π›Ό1[𝑀1 NOUN cana-1090 113 2 βˆ’ βˆ’ NUM cana-1090 113 3 π‘š1 π‘š1 NOUN cana-1090 113 4 ] ] X cana-1090 113 5 . . PUNCT cana-1090 114 1 ( ( PUNCT cana-1090 114 2 18 18 NUM cana-1090 114 3 ) ) PUNCT cana-1090 114 4 thus thus ADV cana-1090 114 5 from from ADP cana-1090 114 6 ( ( PUNCT cana-1090 114 7 16 16 NUM cana-1090 114 8 ) ) PUNCT cana-1090 114 9 and and CCONJ cana-1090 114 10 ( ( PUNCT cana-1090 114 11 18 18 NUM cana-1090 114 12 ) ) PUNCT cana-1090 114 13 we we PRON cana-1090 114 14 get get VERB cana-1090 114 15 , , PUNCT cana-1090 114 16 𝑀1 𝑀1 PROPN cana-1090 114 17 βˆ’ βˆ’ PROPN cana-1090 114 18 π‘š1 π‘š1 NOUN cana-1090 114 19 ≀ ≀ PUNCT cana-1090 115 1 π‘Ž1π‘’βˆ’π›Ό1[𝑀1 π‘Ž1π‘’βˆ’π›Ό1[𝑀1 NOUN cana-1090 115 2 βˆ’ βˆ’ NUM cana-1090 115 3 π‘š1 π‘š1 NOUN cana-1090 115 4 ] ] X cana-1090 116 1 + + CCONJ cana-1090 116 2 𝑏1π‘’βˆ’π‘š2βˆ’π‘€2+𝜁1(1 𝑏1π‘’βˆ’π‘š2βˆ’π‘€2+𝜁1(1 ADJ cana-1090 116 3 + + CCONJ cana-1090 116 4 𝜁1)[𝑀2 𝜁1)[𝑀2 NOUN cana-1090 116 5 βˆ’ βˆ’ NOUN cana-1090 116 6 π‘š2 π‘š2 NOUN cana-1090 116 7 ] ] PUNCT cana-1090 117 1 + + CCONJ cana-1090 117 2 𝑐1π‘’βˆ’π›Ό3[𝑀3 𝑐1π‘’βˆ’π›Ό3[𝑀3 PROPN cana-1090 117 3 βˆ’ βˆ’ ADP cana-1090 117 4 π‘š3 π‘š3 NOUN cana-1090 117 5 ] ] X cana-1090 117 6 . . PUNCT cana-1090 118 1 ( ( PUNCT cana-1090 118 2 19 19 NUM cana-1090 118 3 ) ) PUNCT cana-1090 118 4 since since SCONJ cana-1090 118 5 π‘š2 π‘š2 NOUN cana-1090 118 6 , , PUNCT cana-1090 118 7 𝑀2 𝑀2 PROPN cana-1090 118 8 β‰₯ β‰₯ PROPN cana-1090 118 9 𝛼2 𝛼2 PROPN cana-1090 118 10 , , PUNCT cana-1090 118 11 ( ( PUNCT cana-1090 118 12 19 19 NUM cana-1090 118 13 ) ) PUNCT cana-1090 118 14 becomes become VERB cana-1090 118 15 𝑀1 𝑀1 PROPN cana-1090 118 16 βˆ’ βˆ’ PROPN cana-1090 118 17 π‘š1 π‘š1 NOUN cana-1090 118 18 ≀ ≀ PUNCT cana-1090 119 1 π‘Ž1π‘’βˆ’π›Ό1[𝑀1 π‘Ž1π‘’βˆ’π›Ό1[𝑀1 NOUN cana-1090 119 2 βˆ’ βˆ’ NUM cana-1090 119 3 π‘š1 π‘š1 NOUN cana-1090 119 4 ] ] X cana-1090 120 1 + + CCONJ cana-1090 120 2 𝑏1π‘’βˆ’π›Ό2(1 𝑏1π‘’βˆ’π›Ό2(1 NOUN cana-1090 120 3 + + CCONJ cana-1090 120 4 𝜁1)[𝑀2 𝜁1)[𝑀2 NOUN cana-1090 120 5 βˆ’ βˆ’ NOUN cana-1090 120 6 π‘š2 π‘š2 NOUN cana-1090 120 7 ] ] PUNCT cana-1090 121 1 + + CCONJ cana-1090 121 2 𝑐1π‘’βˆ’π›Ό3[𝑀3 𝑐1π‘’βˆ’π›Ό3[𝑀3 PROPN cana-1090 121 3 βˆ’ βˆ’ ADP cana-1090 121 4 π‘š3 π‘š3 NOUN cana-1090 121 5 ] ] X cana-1090 121 6 . . PUNCT cana-1090 122 1 ( ( PUNCT cana-1090 122 2 20 20 NUM cana-1090 122 3 ) ) PUNCT cana-1090 122 4 i.e. i.e. X cana-1090 122 5 , , PUNCT cana-1090 122 6 [ [ X cana-1090 122 7 1 1 NUM cana-1090 122 8 βˆ’ βˆ’ PRON cana-1090 122 9 π‘Ž1π‘’βˆ’π›Ό1][𝑀1 π‘Ž1π‘’βˆ’π›Ό1][𝑀1 NOUN cana-1090 122 10 βˆ’ βˆ’ PROPN cana-1090 122 11 π‘š1 π‘š1 NOUN cana-1090 122 12 ] ] PUNCT cana-1090 122 13 ≀ ≀ NUM cana-1090 122 14 𝑏1π‘’βˆ’π›Ό2(1 𝑏1π‘’βˆ’π›Ό2(1 NOUN cana-1090 122 15 + + CCONJ cana-1090 122 16 𝜁1)[𝑀2 𝜁1)[𝑀2 NOUN cana-1090 122 17 βˆ’ βˆ’ NOUN cana-1090 122 18 π‘š2 π‘š2 NOUN cana-1090 122 19 ] ] PUNCT cana-1090 123 1 + + CCONJ cana-1090 123 2 𝑐1π‘’βˆ’π›Ό3[𝑀3 𝑐1π‘’βˆ’π›Ό3[𝑀3 PROPN cana-1090 123 3 βˆ’ βˆ’ ADP cana-1090 123 4 π‘š3 π‘š3 NOUN cana-1090 123 5 ] ] X cana-1090 123 6 . . PUNCT cana-1090 124 1 ( ( PUNCT cana-1090 124 2 21 21 NUM cana-1090 124 3 ) ) PUNCT cana-1090 124 4 also also ADV cana-1090 124 5 , , PUNCT cana-1090 124 6 ( ( PUNCT cana-1090 124 7 13 13 NUM cana-1090 124 8 ) ) PUNCT cana-1090 124 9 can can AUX cana-1090 124 10 be be AUX cana-1090 124 11 written write VERB cana-1090 124 12 as as ADP cana-1090 124 13 𝑀2 𝑀2 PROPN cana-1090 124 14 = = SYM cana-1090 124 15 𝛼2 𝛼2 PROPN cana-1090 124 16 + + NUM cana-1090 124 17 π‘Ž2π‘’βˆ’π‘š2 π‘Ž2π‘’βˆ’π‘š2 NOUN cana-1090 124 18 + + CCONJ cana-1090 124 19 𝑏2[𝛼3 𝑏2[𝛼3 PROPN cana-1090 124 20 + + CCONJ cana-1090 124 21 π‘Ž3π‘’βˆ’π‘š3 π‘Ž3π‘’βˆ’π‘š3 X cana-1090 124 22 + + CCONJ cana-1090 124 23 𝑏3𝑀1π‘’βˆ’π‘š1 𝑏3𝑀1π‘’βˆ’π‘š1 NOUN cana-1090 124 24 + + X cana-1090 124 25 𝑐3π‘’βˆ’π‘š2]π‘’βˆ’π‘š3 𝑐3π‘’βˆ’π‘š2]π‘’βˆ’π‘š3 X cana-1090 124 26 + + NOUN cana-1090 124 27 𝑐2π‘’βˆ’π‘š1 𝑐2π‘’βˆ’π‘š1 X cana-1090 124 28 . . PUNCT cana-1090 125 1 ( ( PUNCT cana-1090 125 2 22 22 X cana-1090 125 3 ) ) PUNCT cana-1090 125 4 substituting substitute VERB cana-1090 125 5 again again ADV cana-1090 125 6 for for ADP cana-1090 125 7 𝑀1 𝑀1 NOUN cana-1090 125 8 and and CCONJ cana-1090 125 9 simplifying simplify VERB cana-1090 125 10 we we PRON cana-1090 125 11 get get VERB cana-1090 125 12 𝑀2 𝑀2 PROPN cana-1090 125 13 ≀ ≀ NUM cana-1090 125 14 𝐴2 𝐴2 PROPN cana-1090 125 15 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 125 16 . . PUNCT cana-1090 126 1 ( ( PUNCT cana-1090 126 2 23 23 NUM cana-1090 126 3 ) ) PUNCT cana-1090 126 4 since since SCONJ cana-1090 126 5 𝜁1 𝜁1 PROPN cana-1090 126 6 ≀ ≀ PROPN cana-1090 126 7 𝑀2 𝑀2 NOUN cana-1090 126 8 we we PRON cana-1090 126 9 get get VERB cana-1090 126 10 , , PUNCT cana-1090 126 11 𝜁1 𝜁1 PROPN cana-1090 126 12 ≀ ≀ PROPN cana-1090 126 13 𝐴2 𝐴2 PROPN cana-1090 126 14 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 126 15 . . PUNCT cana-1090 127 1 ( ( PUNCT cana-1090 127 2 24 24 NUM cana-1090 127 3 ) ) PUNCT cana-1090 127 4 therefore therefore ADV cana-1090 127 5 , , PUNCT cana-1090 127 6 ( ( PUNCT cana-1090 127 7 21 21 NUM cana-1090 127 8 ) ) PUNCT cana-1090 127 9 becomes become VERB cana-1090 127 10 [ [ X cana-1090 127 11 1 1 NUM cana-1090 127 12 βˆ’ βˆ’ NOUN cana-1090 127 13 π‘Ž1π‘’βˆ’π›Ό1][𝑀1 π‘Ž1π‘’βˆ’π›Ό1][𝑀1 NOUN cana-1090 127 14 βˆ’ βˆ’ NUM cana-1090 127 15 π‘š1 π‘š1 NOUN cana-1090 127 16 ] ] X cana-1090 127 17 ≀ ≀ X cana-1090 127 18 𝑏1π‘’βˆ’π›Ό2[1 𝑏1π‘’βˆ’π›Ό2[1 PUNCT cana-1090 128 1 + + CCONJ cana-1090 128 2 𝐴2 𝐴2 PROPN cana-1090 128 3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 128 4 ] ] PUNCT cana-1090 128 5 [ [ X cana-1090 128 6 𝑀2 𝑀2 X cana-1090 128 7 βˆ’ βˆ’ PROPN cana-1090 128 8 π‘š2 π‘š2 PROPN cana-1090 128 9 ] ] X cana-1090 129 1 + + CCONJ cana-1090 129 2 𝑐1π‘’βˆ’π›Ό3[𝑀3 𝑐1π‘’βˆ’π›Ό3[𝑀3 PROPN cana-1090 129 3 βˆ’ βˆ’ PUNCT cana-1090 129 4 π‘š3].(25 π‘š3].(25 NOUN cana-1090 129 5 ) ) PUNCT cana-1090 129 6 similarly similarly ADV cana-1090 129 7 we we PRON cana-1090 129 8 get get VERB cana-1090 129 9 , , PUNCT cana-1090 129 10 [ [ X cana-1090 129 11 1 1 NUM cana-1090 129 12 βˆ’ βˆ’ NOUN cana-1090 129 13 π‘Ž2π‘’βˆ’π›Ό2][𝑀2 π‘Ž2π‘’βˆ’π›Ό2][𝑀2 NOUN cana-1090 129 14 βˆ’ βˆ’ PROPN cana-1090 129 15 π‘š2 π‘š2 PROPN cana-1090 129 16 ] ] PUNCT cana-1090 129 17 ≀ ≀ PROPN cana-1090 129 18 𝑏2π‘’βˆ’π›Ό3[1 𝑏2π‘’βˆ’π›Ό3[1 PUNCT cana-1090 130 1 + + NUM cana-1090 130 2 𝐴3 𝐴3 PROPN cana-1090 130 3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 PROPN cana-1090 130 4 ] ] PUNCT cana-1090 130 5 [ [ X cana-1090 130 6 𝑀3 𝑀3 NOUN cana-1090 130 7 βˆ’ βˆ’ PROPN cana-1090 130 8 π‘š3 π‘š3 NOUN cana-1090 130 9 ] ] X cana-1090 130 10 + + CCONJ cana-1090 130 11 𝑐2π‘’βˆ’π›Ό1[𝑀1 𝑐2π‘’βˆ’π›Ό1[𝑀1 PROPN cana-1090 130 12 βˆ’ βˆ’ NUM cana-1090 130 13 π‘š1](26 π‘š1](26 NOUN cana-1090 130 14 ) ) PUNCT cana-1090 130 15 communications communication NOUN cana-1090 130 16 on on ADP cana-1090 130 17 applied apply VERB cana-1090 130 18 nonlinear nonlinear ADJ cana-1090 130 19 analysis analysis NOUN cana-1090 130 20 issn issn NOUN cana-1090 130 21 : : PUNCT cana-1090 130 22 1074 1074 NUM cana-1090 130 23 - - PUNCT cana-1090 130 24 133x 133x NUM cana-1090 130 25 vol vol NOUN cana-1090 130 26 31 31 NUM cana-1090 130 27 no no NOUN cana-1090 130 28 . . PUNCT cana-1090 131 1 5s 5s NUM cana-1090 131 2 ( ( PUNCT cana-1090 131 3 2024 2024 NUM cana-1090 131 4 ) ) PUNCT cana-1090 131 5 557 557 NUM cana-1090 131 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 131 7 and and CCONJ cana-1090 131 8 [ [ X cana-1090 131 9 1 1 NUM cana-1090 131 10 βˆ’ βˆ’ NUM cana-1090 131 11 π‘Ž3π‘’βˆ’π›Ό3][𝑀3 π‘Ž3π‘’βˆ’π›Ό3][𝑀3 NOUN cana-1090 131 12 βˆ’ βˆ’ PROPN cana-1090 131 13 π‘š3 π‘š3 NOUN cana-1090 131 14 ] ] PUNCT cana-1090 131 15 ≀ ≀ X cana-1090 131 16 𝑏3π‘’βˆ’π›Ό1[1 𝑏3π‘’βˆ’π›Ό1[1 PUNCT cana-1090 132 1 + + NUM cana-1090 132 2 𝐴1 𝐴1 PROPN cana-1090 132 3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 1βˆ’π‘1𝑏2𝑏3π‘’βˆ’π›Ό1βˆ’π›Ό2βˆ’π›Ό3 NOUN cana-1090 132 4 ] ] PUNCT cana-1090 132 5 [ [ X cana-1090 132 6 𝑀1 𝑀1 ADV cana-1090 132 7 βˆ’ βˆ’ PROPN cana-1090 132 8 π‘š1 π‘š1 NOUN cana-1090 132 9 ] ] X cana-1090 132 10 + + CCONJ cana-1090 132 11 𝑐3π‘’βˆ’π›Ό2[𝑀2 𝑐3π‘’βˆ’π›Ό2[𝑀2 NOUN cana-1090 132 12 βˆ’ βˆ’ PROPN cana-1090 132 13 π‘š2](27 π‘š2](27 NUM cana-1090 132 14 ) ) PUNCT cana-1090 132 15 from from ADP cana-1090 132 16 ( ( PUNCT cana-1090 132 17 25 25 NUM cana-1090 132 18 ) ) PUNCT cana-1090 132 19 , , PUNCT cana-1090 132 20 ( ( PUNCT cana-1090 132 21 26 26 NUM cana-1090 132 22 ) ) PUNCT cana-1090 132 23 and and CCONJ cana-1090 132 24 ( ( PUNCT cana-1090 132 25 27 27 NUM cana-1090 132 26 ) ) PUNCT cana-1090 132 27 we we PRON cana-1090 132 28 get get VERB cana-1090 132 29 , , PUNCT cana-1090 132 30 [ [ X cana-1090 132 31 𝐡1𝐡2 𝐡1𝐡2 NOUN cana-1090 132 32 βˆ’ βˆ’ PROPN cana-1090 132 33 𝐷1𝐿2 𝐷1𝐿2 PROPN cana-1090 132 34 ] ] PUNCT cana-1090 132 35 𝐡2 𝐡2 NOUN cana-1090 133 1 [ [ X cana-1090 133 2 𝑀1 𝑀1 ADV cana-1090 133 3 βˆ’ βˆ’ PROPN cana-1090 133 4 π‘š1 π‘š1 NOUN cana-1090 133 5 ] ] X cana-1090 133 6 ≀ ≀ NOUN cana-1090 134 1 [ [ X cana-1090 134 2 𝐷1𝐷2 𝐷1𝐷2 NOUN cana-1090 134 3 + + CCONJ cana-1090 134 4 𝐡2𝐿1 𝐡2𝐿1 NOUN cana-1090 134 5 ] ] X cana-1090 134 6 𝐡2 𝐡2 NOUN cana-1090 134 7 [ [ X cana-1090 134 8 𝑀3 𝑀3 NOUN cana-1090 134 9 βˆ’ βˆ’ PROPN cana-1090 134 10 π‘š3 π‘š3 NOUN cana-1090 134 11 ] ] PUNCT cana-1090 134 12 . . PUNCT cana-1090 135 1 similarly similarly ADV cana-1090 135 2 , , PUNCT cana-1090 135 3 [ [ X cana-1090 135 4 𝐡2𝐡3βˆ’π·2𝐿3 𝐡2𝐡3βˆ’π·2𝐿3 X cana-1090 135 5 ] ] PUNCT cana-1090 135 6 𝐡3 𝐡3 NOUN cana-1090 136 1 [ [ X cana-1090 136 2 𝑀2 𝑀2 X cana-1090 136 3 βˆ’ βˆ’ PROPN cana-1090 136 4 π‘š2 π‘š2 PROPN cana-1090 136 5 ] ] PUNCT cana-1090 136 6 ≀ ≀ NOUN cana-1090 137 1 [ [ X cana-1090 137 2 𝐷2𝐷3+𝐡3𝐿2 𝐷2𝐷3+𝐡3𝐿2 X cana-1090 137 3 ] ] X cana-1090 137 4 𝐡3 𝐡3 NOUN cana-1090 138 1 [ [ X cana-1090 138 2 𝑀1 𝑀1 ADV cana-1090 138 3 βˆ’ βˆ’ PROPN cana-1090 138 4 π‘š1 π‘š1 NOUN cana-1090 138 5 ] ] X cana-1090 138 6 ( ( PUNCT cana-1090 138 7 28 28 NUM cana-1090 138 8 ) ) PUNCT cana-1090 138 9 and and CCONJ cana-1090 139 1 [ [ X cana-1090 139 2 𝐡3𝐡1βˆ’π·3𝐿1 𝐡3𝐡1βˆ’π·3𝐿1 X cana-1090 139 3 ] ] PUNCT cana-1090 139 4 𝐡1 𝐡1 NOUN cana-1090 140 1 [ [ X cana-1090 140 2 𝑀3 𝑀3 NOUN cana-1090 140 3 βˆ’ βˆ’ PROPN cana-1090 140 4 π‘š3 π‘š3 NOUN cana-1090 140 5 ] ] PUNCT cana-1090 140 6 ≀ ≀ NOUN cana-1090 140 7 [ [ X cana-1090 140 8 𝐷3𝐷1+𝐡1𝐿3 𝐷3𝐷1+𝐡1𝐿3 X cana-1090 140 9 ] ] PUNCT cana-1090 140 10 𝐡1 𝐡1 NOUN cana-1090 141 1 [ [ X cana-1090 141 2 𝑀2 𝑀2 PROPN cana-1090 141 3 βˆ’ βˆ’ PROPN cana-1090 141 4 π‘š2 π‘š2 PROPN cana-1090 141 5 ] ] PUNCT cana-1090 141 6 ( ( PUNCT cana-1090 141 7 29 29 NUM cana-1090 141 8 ) ) PUNCT cana-1090 141 9 hence hence ADV cana-1090 141 10 from from ADP cana-1090 141 11 ( ( PUNCT cana-1090 141 12 12 12 NUM cana-1090 141 13 ) ) PUNCT cana-1090 141 14 and and CCONJ cana-1090 141 15 ( ( PUNCT cana-1090 141 16 28 28 NUM cana-1090 141 17 ) ) PUNCT cana-1090 141 18 , , PUNCT cana-1090 141 19 we we PRON cana-1090 141 20 get get VERB cana-1090 141 21 𝑀1 𝑀1 NOUN cana-1090 141 22 = = SYM cana-1090 141 23 π‘š1 π‘š1 NOUN cana-1090 141 24 . . PUNCT cana-1090 142 1 similarly similarly ADV cana-1090 142 2 we we PRON cana-1090 142 3 get get VERB cana-1090 142 4 𝑀2 𝑀2 NOUN cana-1090 142 5 = = SYM cana-1090 142 6 π‘š2 π‘š2 NOUN cana-1090 142 7 and and CCONJ cana-1090 142 8 𝑀3 𝑀3 NOUN cana-1090 142 9 = = SYM cana-1090 142 10 π‘š3 π‘š3 NOUN cana-1090 142 11 . . PUNCT cana-1090 143 1 hence hence ADV cana-1090 143 2 by by ADP cana-1090 143 3 lemma lemma PROPN cana-1090 143 4 2.4 2.4 NUM cana-1090 143 5 , , PUNCT cana-1090 143 6 we we PRON cana-1090 143 7 get get VERB cana-1090 143 8 the the DET cana-1090 143 9 required require VERB cana-1090 143 10 result result NOUN cana-1090 143 11 . . PUNCT cana-1090 144 1 theorem theorem VERB cana-1090 144 2 2.6 2.6 NUM cana-1090 144 3 assume assume NOUN cana-1090 144 4 ( ( PUNCT cana-1090 144 5 2 2 NUM cana-1090 144 6 ) ) PUNCT cana-1090 144 7 , , PUNCT cana-1090 144 8 ( ( PUNCT cana-1090 144 9 11 11 NUM cana-1090 144 10 ) ) PUNCT cana-1090 144 11 and and CCONJ cana-1090 144 12 ( ( PUNCT cana-1090 144 13 12 12 NUM cana-1090 144 14 ) ) PUNCT cana-1090 144 15 hold hold NOUN cana-1090 144 16 . . PUNCT cana-1090 145 1 if if SCONJ cana-1090 145 2 π‘Ž1π‘’βˆ’π›Ό1[1 π‘Ž1π‘’βˆ’π›Ό1[1 X cana-1090 145 3 + + NOUN cana-1090 145 4 π‘Ž2π‘’βˆ’π›Ό2 π‘Ž2π‘’βˆ’π›Ό2 X cana-1090 145 5 ] ] X cana-1090 145 6 + + CCONJ cana-1090 145 7 π‘Ž2π‘’βˆ’π›Ό2[1 π‘Ž2π‘’βˆ’π›Ό2[1 X cana-1090 146 1 + + NOUN cana-1090 146 2 π‘Ž3π‘’βˆ’π›Ό3 π‘Ž3π‘’βˆ’π›Ό3 NOUN cana-1090 146 3 ] ] X cana-1090 147 1 + + CCONJ cana-1090 147 2 π‘Ž3π‘’βˆ’π›Ό3[1 π‘Ž3π‘’βˆ’π›Ό3[1 PUNCT cana-1090 148 1 + + NUM cana-1090 148 2 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 VERB cana-1090 148 3 ] ] X cana-1090 148 4 + + CCONJ cana-1090 148 5 π‘Ž1π‘Ž2π‘Ž3π‘’βˆ’π›Ό1π‘’βˆ’π›Ό2π‘’βˆ’π›Ό3 π‘Ž1π‘Ž2π‘Ž3π‘’βˆ’π›Ό1π‘’βˆ’π›Ό2π‘’βˆ’π›Ό3 PROPN cana-1090 148 6 + + CCONJ cana-1090 148 7 𝐡 𝐡 NOUN cana-1090 148 8 ( ( PUNCT cana-1090 148 9 1βˆ’π΅)3 1βˆ’π΅)3 NUM cana-1090 148 10 [ [ X cana-1090 148 11 ( ( PUNCT cana-1090 148 12 1 1 NUM cana-1090 148 13 βˆ’ βˆ’ PROPN cana-1090 148 14 𝐡)3 𝐡)3 NUM cana-1090 148 15 + + CCONJ cana-1090 148 16 ( ( PUNCT cana-1090 148 17 1 1 NUM cana-1090 148 18 βˆ’ βˆ’ NOUN cana-1090 148 19 𝐡)2(𝐴1 𝐡)2(𝐴1 PROPN cana-1090 148 20 + + CCONJ cana-1090 148 21 𝐴2 𝐴2 PROPN cana-1090 148 22 + + SYM cana-1090 148 23 𝐴3 𝐴3 PROPN cana-1090 148 24 ) ) PUNCT cana-1090 149 1 + + NOUN cana-1090 149 2 ( ( PUNCT cana-1090 149 3 1 1 NUM cana-1090 149 4 βˆ’ βˆ’ PROPN cana-1090 149 5 𝐡)(𝐴1𝐴2 𝐡)(𝐴1𝐴2 NOUN cana-1090 149 6 + + X cana-1090 149 7 𝐴2𝐴3 𝐴2𝐴3 PROPN cana-1090 149 8 + + CCONJ cana-1090 149 9 𝐴1𝐴3 𝐴1𝐴3 PROPN cana-1090 149 10 ) ) PUNCT cana-1090 150 1 + + CCONJ cana-1090 151 1 𝐴1𝐴2𝐴3 𝐴1𝐴2𝐴3 PROPN cana-1090 151 2 ] ] X cana-1090 151 3 < < X cana-1090 151 4 1 1 NUM cana-1090 151 5 , , PUNCT cana-1090 151 6 ( ( PUNCT cana-1090 151 7 30 30 NUM cana-1090 151 8 ) ) PUNCT cana-1090 151 9 where where SCONJ cana-1090 151 10 𝐡 𝐡 NOUN cana-1090 151 11 , , PUNCT cana-1090 151 12 𝐴1 𝐴1 PROPN cana-1090 151 13 , , PUNCT cana-1090 151 14 𝐴2 𝐴2 PROPN cana-1090 151 15 , , PUNCT cana-1090 151 16 𝐴3 𝐴3 PROPN cana-1090 151 17 are be AUX cana-1090 151 18 as as ADP cana-1090 151 19 in in ADP cana-1090 151 20 theorem theorem ADJ cana-1090 151 21 2.1 2.1 NUM cana-1090 151 22 , , PUNCT cana-1090 151 23 then then ADV cana-1090 151 24 𝐸( 𝐸( NOUN cana-1090 151 25 οΏ½ οΏ½ PROPN cana-1090 151 26 Μ… Μ… PROPN cana-1090 151 27 οΏ½ οΏ½ PROPN cana-1090 151 28 , , PUNCT cana-1090 151 29 οΏ½ οΏ½ PROPN cana-1090 151 30 Μ… Μ… NOUN cana-1090 151 31 οΏ½ οΏ½ NOUN cana-1090 151 32 , , PUNCT cana-1090 151 33 𝑧̅ 𝑧̅ NOUN cana-1090 151 34 ) ) PUNCT cana-1090 151 35 of of ADP cana-1090 151 36 ( ( PUNCT cana-1090 151 37 2.5 2.5 NUM cana-1090 151 38 ) ) PUNCT cana-1090 151 39 is be AUX cana-1090 151 40 globally globally ADV cana-1090 151 41 asymptotically asymptotically ADV cana-1090 151 42 stable stable ADJ cana-1090 151 43 . . PUNCT cana-1090 152 1 proof proof NOUN cana-1090 152 2 : : PUNCT cana-1090 152 3 we we PRON cana-1090 152 4 need need VERB cana-1090 152 5 to to PART cana-1090 152 6 illustrate illustrate VERB cana-1090 152 7 𝐸( 𝐸( NOUN cana-1090 152 8 οΏ½ οΏ½ NOUN cana-1090 152 9 Μ… Μ… NOUN cana-1090 152 10 οΏ½ οΏ½ PROPN cana-1090 152 11 , , PUNCT cana-1090 152 12 οΏ½ οΏ½ PROPN cana-1090 152 13 Μ… Μ… NOUN cana-1090 152 14 οΏ½ οΏ½ NOUN cana-1090 152 15 , , PUNCT cana-1090 152 16 𝑧̅ 𝑧̅ NOUN cana-1090 152 17 ) ) PUNCT cana-1090 152 18 is be AUX cana-1090 152 19 locally locally ADV cana-1090 152 20 asymptotic asymptotic ADJ cana-1090 152 21 . . PUNCT cana-1090 153 1 construct construct VERB cana-1090 153 2 the the DET cana-1090 153 3 jacobian jacobian PROPN cana-1090 153 4 𝐽𝐹( 𝐽𝐹( PROPN cana-1090 153 5 οΏ½ οΏ½ PROPN cana-1090 153 6 Μ… Μ… NOUN cana-1090 153 7 οΏ½ οΏ½ PROPN cana-1090 153 8 , , PUNCT cana-1090 153 9 οΏ½ οΏ½ PROPN cana-1090 153 10 Μ… Μ… NOUN cana-1090 153 11 οΏ½ οΏ½ NOUN cana-1090 153 12 , , PUNCT cana-1090 153 13 𝑧̅ 𝑧̅ NOUN cana-1090 153 14 ) ) PUNCT cana-1090 153 15 about about ADP cana-1090 153 16 𝐸( 𝐸( NOUN cana-1090 153 17 οΏ½ οΏ½ NOUN cana-1090 153 18 Μ… Μ… NOUN cana-1090 153 19 οΏ½ οΏ½ PROPN cana-1090 153 20 , , PUNCT cana-1090 153 21 οΏ½ οΏ½ PROPN cana-1090 153 22 Μ… Μ… NOUN cana-1090 153 23 οΏ½ οΏ½ NOUN cana-1090 153 24 , , PUNCT cana-1090 153 25 𝑧̅ 𝑧̅ PROPN cana-1090 153 26 ) ) PUNCT cana-1090 153 27 . . PUNCT cana-1090 154 1 its its PRON cana-1090 154 2 characteristic characteristic ADJ cana-1090 154 3 equation equation NOUN cana-1090 154 4 is be AUX cana-1090 154 5 given give VERB cana-1090 154 6 by by ADP cana-1090 154 7 πœ†6 πœ†6 PROPN cana-1090 154 8 + + CCONJ cana-1090 154 9 πœ†4(π‘Ž1π‘’βˆ’ πœ†4(π‘Ž1π‘’βˆ’ NUM cana-1090 154 10 οΏ½ οΏ½ NOUN cana-1090 154 11 Μ… Μ… NOUN cana-1090 154 12 οΏ½ οΏ½ NOUN cana-1090 154 13 + + CCONJ cana-1090 154 14 π‘Ž2π‘’βˆ’ π‘Ž2π‘’βˆ’ PROPN cana-1090 154 15 οΏ½ οΏ½ NOUN cana-1090 154 16 Μ… Μ… NOUN cana-1090 154 17 οΏ½ οΏ½ PROPN cana-1090 154 18 + + CCONJ cana-1090 154 19 π‘Ž3π‘’βˆ’ π‘Ž3π‘’βˆ’ PROPN cana-1090 154 20 οΏ½ οΏ½ PROPN cana-1090 154 21 Μ… Μ… NOUN cana-1090 154 22 οΏ½ οΏ½ NOUN cana-1090 154 23 ) ) PUNCT cana-1090 154 24 + + CCONJ cana-1090 154 25 πœ†3(𝑏2𝑐3π‘’βˆ’ πœ†3(𝑏2𝑐3π‘’βˆ’ PROPN cana-1090 154 26 οΏ½ οΏ½ NOUN cana-1090 154 27 Μ… Μ… NOUN cana-1090 154 28 οΏ½ οΏ½ NOUN cana-1090 154 29 βˆ’ βˆ’ NUM cana-1090 154 30 οΏ½ οΏ½ NOUN cana-1090 154 31 Μ… Μ… NOUN cana-1090 154 32 οΏ½ οΏ½ NOUN cana-1090 154 33 + + CCONJ cana-1090 154 34 𝑏1𝑐2π‘’βˆ’ 𝑏1𝑐2π‘’βˆ’ PROPN cana-1090 154 35 οΏ½ οΏ½ NOUN cana-1090 154 36 Μ… Μ… NOUN cana-1090 154 37 οΏ½ οΏ½ NOUN cana-1090 154 38 βˆ’ βˆ’ NUM cana-1090 154 39 οΏ½ οΏ½ NOUN cana-1090 154 40 Μ… Μ… NOUN cana-1090 154 41 οΏ½ οΏ½ NOUN cana-1090 154 42 + + CCONJ cana-1090 154 43 𝑏3𝑐1π‘’βˆ’ 𝑏3𝑐1π‘’βˆ’ PROPN cana-1090 154 44 οΏ½ οΏ½ NOUN cana-1090 154 45 Μ… Μ… NOUN cana-1090 154 46 οΏ½ οΏ½ NOUN cana-1090 154 47 βˆ’ βˆ’ NUM cana-1090 154 48 οΏ½ οΏ½ NOUN cana-1090 154 49 Μ… Μ… NOUN cana-1090 154 50 οΏ½ οΏ½ NOUN cana-1090 154 51 + + CCONJ cana-1090 154 52 𝑏1𝑏2𝑏3π‘’βˆ’ 𝑏1𝑏2𝑏3π‘’βˆ’ PROPN cana-1090 154 53 οΏ½ οΏ½ NOUN cana-1090 154 54 Μ… Μ… NOUN cana-1090 154 55 οΏ½ οΏ½ NOUN cana-1090 154 56 βˆ’ βˆ’ NUM cana-1090 154 57 οΏ½ οΏ½ NOUN cana-1090 154 58 Μ… Μ… NOUN cana-1090 154 59 οΏ½ οΏ½ NOUN cana-1090 154 60 βˆ’ βˆ’ NUM cana-1090 154 61 οΏ½ οΏ½ NOUN cana-1090 154 62 Μ… Μ… NOUN cana-1090 154 63 οΏ½ οΏ½ NOUN cana-1090 154 64 ) ) PUNCT cana-1090 154 65 + + CCONJ cana-1090 154 66 πœ†2(π‘Ž2π‘Ž3π‘’βˆ’ πœ†2(π‘Ž2π‘Ž3π‘’βˆ’ NOUN cana-1090 154 67 οΏ½ οΏ½ NOUN cana-1090 154 68 Μ… Μ… NOUN cana-1090 154 69 οΏ½ οΏ½ NOUN cana-1090 154 70 βˆ’ βˆ’ NUM cana-1090 154 71 οΏ½ οΏ½ NOUN cana-1090 154 72 Μ… Μ… NOUN cana-1090 154 73 οΏ½ οΏ½ NOUN cana-1090 154 74 βˆ’ βˆ’ ADP cana-1090 154 75 𝑏2𝑐3π‘§Μ…π‘’βˆ’ 𝑏2𝑐3π‘§Μ…π‘’βˆ’ PROPN cana-1090 154 76 οΏ½ οΏ½ NOUN cana-1090 154 77 Μ… Μ… NOUN cana-1090 154 78 οΏ½ οΏ½ NOUN cana-1090 154 79 βˆ’ βˆ’ NUM cana-1090 154 80 οΏ½ οΏ½ NOUN cana-1090 154 81 Μ… Μ… NOUN cana-1090 154 82 οΏ½ οΏ½ NOUN cana-1090 154 83 + + CCONJ cana-1090 154 84 π‘Ž1π‘Ž3π‘’βˆ’ π‘Ž1π‘Ž3π‘’βˆ’ PROPN cana-1090 154 85 οΏ½ οΏ½ PROPN cana-1090 154 86 Μ… Μ… NOUN cana-1090 154 87 οΏ½ οΏ½ NOUN cana-1090 154 88 βˆ’ βˆ’ NUM cana-1090 154 89 οΏ½ οΏ½ NOUN cana-1090 154 90 Μ… Μ… NOUN cana-1090 154 91 οΏ½ οΏ½ PROPN cana-1090 154 92 βˆ’ βˆ’ NUM cana-1090 154 93 𝑏3𝑐1 𝑏3𝑐1 PROPN cana-1090 154 94 οΏ½ οΏ½ PROPN cana-1090 154 95 Μ… Μ… NOUN cana-1090 154 96 οΏ½ οΏ½ NOUN cana-1090 154 97 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 154 98 οΏ½ οΏ½ NOUN cana-1090 154 99 Μ… Μ… NOUN cana-1090 154 100 οΏ½ οΏ½ NOUN cana-1090 154 101 βˆ’ βˆ’ NUM cana-1090 154 102 οΏ½ οΏ½ NOUN cana-1090 154 103 Μ… Μ… NOUN cana-1090 154 104 οΏ½ οΏ½ NOUN cana-1090 154 105 + + CCONJ cana-1090 154 106 π‘Ž1π‘Ž2π‘’βˆ’ π‘Ž1π‘Ž2π‘’βˆ’ NOUN cana-1090 154 107 οΏ½ οΏ½ NOUN cana-1090 154 108 Μ… Μ… NOUN cana-1090 154 109 οΏ½ οΏ½ NOUN cana-1090 154 110 π‘’βˆ’ π‘’βˆ’ SYM cana-1090 154 111 οΏ½ οΏ½ NOUN cana-1090 154 112 Μ… Μ… NOUN cana-1090 154 113 οΏ½ οΏ½ NOUN cana-1090 154 114 βˆ’ βˆ’ PROPN cana-1090 154 115 𝑏1𝑐2 𝑏1𝑐2 PRON cana-1090 154 116 οΏ½ οΏ½ PROPN cana-1090 154 117 Μ… Μ… NOUN cana-1090 154 118 οΏ½ οΏ½ NOUN cana-1090 154 119 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 154 120 οΏ½ οΏ½ NOUN cana-1090 154 121 Μ… Μ… NOUN cana-1090 154 122 οΏ½ οΏ½ NOUN cana-1090 154 123 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 154 124 οΏ½ οΏ½ NOUN cana-1090 154 125 Μ… Μ… NOUN cana-1090 154 126 οΏ½ οΏ½ NOUN cana-1090 154 127 + + CCONJ cana-1090 154 128 π‘’βˆ’ π‘’βˆ’ VERB cana-1090 154 129 οΏ½ οΏ½ NOUN cana-1090 154 130 Μ… Μ… NOUN cana-1090 154 131 οΏ½ οΏ½ NOUN cana-1090 154 132 βˆ’ βˆ’ NUM cana-1090 154 133 οΏ½ οΏ½ NOUN cana-1090 154 134 Μ… Μ… NOUN cana-1090 154 135 οΏ½ οΏ½ NOUN cana-1090 154 136 βˆ’ βˆ’ NUM cana-1090 154 137 οΏ½ οΏ½ NOUN cana-1090 154 138 Μ… Μ… NOUN cana-1090 154 139 οΏ½ οΏ½ NOUN cana-1090 154 140 [𝑏1𝑏2𝑏3 [𝑏1𝑏2𝑏3 NOUN cana-1090 154 141 οΏ½ οΏ½ NOUN cana-1090 154 142 Μ… Μ… NOUN cana-1090 154 143 οΏ½ οΏ½ NOUN cana-1090 154 144 + + CCONJ cana-1090 154 145 𝑏1𝑏2𝑏3 𝑏1𝑏2𝑏3 PROPN cana-1090 154 146 οΏ½ οΏ½ PROPN cana-1090 154 147 Μ… Μ… NOUN cana-1090 154 148 οΏ½ οΏ½ NOUN cana-1090 154 149 + + CCONJ cana-1090 154 150 𝑏1𝑏2𝑏3𝑧̅ 𝑏1𝑏2𝑏3𝑧̅ NOUN cana-1090 154 151 ] ] PUNCT cana-1090 154 152 ) ) PUNCT cana-1090 155 1 + + CCONJ cana-1090 155 2 πœ†π‘’βˆ’ πœ†π‘’βˆ’ NUM cana-1090 155 3 οΏ½ οΏ½ NOUN cana-1090 155 4 Μ… Μ… NOUN cana-1090 155 5 οΏ½ οΏ½ NOUN cana-1090 155 6 βˆ’ βˆ’ NUM cana-1090 155 7 οΏ½ οΏ½ NOUN cana-1090 155 8 Μ… Μ… NOUN cana-1090 155 9 οΏ½ οΏ½ NOUN cana-1090 155 10 βˆ’ βˆ’ NUM cana-1090 155 11 οΏ½ οΏ½ NOUN cana-1090 155 12 Μ… Μ… NOUN cana-1090 155 13 οΏ½ οΏ½ PROPN cana-1090 155 14 (βˆ’π‘1𝑏2𝑏3 (βˆ’π‘1𝑏2𝑏3 PROPN cana-1090 155 15 οΏ½ οΏ½ PROPN cana-1090 155 16 Μ… Μ… NOUN cana-1090 155 17 οΏ½ οΏ½ NOUN cana-1090 155 18 οΏ½ οΏ½ NOUN cana-1090 155 19 Μ… Μ… NOUN cana-1090 155 20 οΏ½ οΏ½ PROPN cana-1090 155 21 βˆ’ βˆ’ PROPN cana-1090 155 22 𝑏1𝑏2𝑏3 𝑏1𝑏2𝑏3 PROPN cana-1090 155 23 οΏ½ οΏ½ PROPN cana-1090 155 24 Μ… Μ… NOUN cana-1090 155 25 οΏ½ οΏ½ NOUN cana-1090 155 26 𝑧̅ 𝑧̅ NOUN cana-1090 155 27 βˆ’ βˆ’ NOUN cana-1090 155 28 𝑏1𝑏2𝑏3 𝑏1𝑏2𝑏3 PROPN cana-1090 155 29 οΏ½ οΏ½ PROPN cana-1090 155 30 Μ… Μ… NOUN cana-1090 155 31 οΏ½ οΏ½ NOUN cana-1090 155 32 𝑧̅ 𝑧̅ NOUN cana-1090 155 33 + + CCONJ cana-1090 155 34 π‘Ž3𝑏1𝑐2 π‘Ž3𝑏1𝑐2 ADJ cana-1090 155 35 + + CCONJ cana-1090 155 36 π‘Ž1𝑏2𝑐3 π‘Ž1𝑏2𝑐3 ADJ cana-1090 155 37 + + CCONJ cana-1090 155 38 π‘Ž2𝑏3𝑐1 π‘Ž2𝑏3𝑐1 NOUN cana-1090 155 39 ) ) PUNCT cana-1090 156 1 + + CCONJ cana-1090 156 2 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 156 3 οΏ½ οΏ½ NOUN cana-1090 156 4 Μ… Μ… NOUN cana-1090 156 5 οΏ½ οΏ½ NOUN cana-1090 156 6 βˆ’ βˆ’ NUM cana-1090 156 7 οΏ½ οΏ½ NOUN cana-1090 156 8 Μ… Μ… NOUN cana-1090 156 9 οΏ½ οΏ½ NOUN cana-1090 156 10 βˆ’ βˆ’ NUM cana-1090 156 11 οΏ½ οΏ½ NOUN cana-1090 156 12 Μ… Μ… NOUN cana-1090 156 13 οΏ½ οΏ½ NOUN cana-1090 156 14 (π‘Ž1π‘Ž2π‘Ž3 (π‘Ž1π‘Ž2π‘Ž3 NOUN cana-1090 156 15 + + CCONJ cana-1090 156 16 𝑏1𝑏2𝑏3 𝑏1𝑏2𝑏3 PROPN cana-1090 156 17 οΏ½ οΏ½ PROPN cana-1090 156 18 Μ… Μ… NOUN cana-1090 156 19 οΏ½ οΏ½ NOUN cana-1090 156 20 οΏ½ οΏ½ NOUN cana-1090 156 21 Μ… Μ… NOUN cana-1090 156 22 οΏ½ οΏ½ NOUN cana-1090 156 23 𝑧̅ 𝑧̅ NOUN cana-1090 156 24 + + CCONJ cana-1090 156 25 𝑐1𝑐2𝑐3 𝑐1𝑐2𝑐3 NOUN cana-1090 156 26 βˆ’ βˆ’ PROPN cana-1090 156 27 π‘Ž2𝑏3𝑐1 π‘Ž2𝑏3𝑐1 PROPN cana-1090 156 28 οΏ½ οΏ½ PROPN cana-1090 156 29 Μ… Μ… NOUN cana-1090 156 30 οΏ½ οΏ½ PROPN cana-1090 156 31 βˆ’ βˆ’ PROPN cana-1090 156 32 π‘Ž3𝑏1𝑐2 π‘Ž3𝑏1𝑐2 PROPN cana-1090 156 33 οΏ½ οΏ½ PROPN cana-1090 156 34 Μ… Μ… NOUN cana-1090 156 35 οΏ½ οΏ½ NOUN cana-1090 156 36 βˆ’ βˆ’ PROPN cana-1090 156 37 π‘Ž1𝑏2𝑐3𝑧̅ π‘Ž1𝑏2𝑐3𝑧̅ PROPN cana-1090 156 38 ) ) PUNCT cana-1090 156 39 = = SYM cana-1090 156 40 0 0 X cana-1090 156 41 . . X cana-1090 157 1 applying apply VERB cana-1090 157 2 remark remark NOUN cana-1090 157 3 1.3.1 1.3.1 NUM cana-1090 157 4 of of ADP cana-1090 157 5 [ [ X cana-1090 157 6 25 25 NUM cana-1090 157 7 ] ] PUNCT cana-1090 157 8 , , PUNCT cana-1090 157 9 |π‘Ž1π‘’βˆ’ |π‘Ž1π‘’βˆ’ ADJ cana-1090 157 10 οΏ½ οΏ½ NOUN cana-1090 157 11 Μ… Μ… NOUN cana-1090 157 12 οΏ½ οΏ½ NOUN cana-1090 157 13 | | NOUN cana-1090 157 14 + + CCONJ cana-1090 157 15 |π‘Ž2π‘’βˆ’ |π‘Ž2π‘’βˆ’ NOUN cana-1090 157 16 οΏ½ οΏ½ NOUN cana-1090 157 17 Μ… Μ… NOUN cana-1090 157 18 οΏ½ οΏ½ NOUN cana-1090 157 19 | | NOUN cana-1090 157 20 + + CCONJ cana-1090 157 21 |π‘Ž3π‘’βˆ’ |π‘Ž3π‘’βˆ’ NOUN cana-1090 157 22 οΏ½ οΏ½ NOUN cana-1090 157 23 Μ… Μ… NOUN cana-1090 157 24 οΏ½ οΏ½ NOUN cana-1090 157 25 | | NOUN cana-1090 157 26 + + SYM cana-1090 157 27 |𝑏1𝑏2𝑏3π‘’βˆ’ |𝑏1𝑏2𝑏3π‘’βˆ’ NOUN cana-1090 157 28 οΏ½ οΏ½ NOUN cana-1090 157 29 Μ… Μ… NOUN cana-1090 157 30 οΏ½ οΏ½ NOUN cana-1090 157 31 βˆ’ βˆ’ NUM cana-1090 157 32 οΏ½ οΏ½ NOUN cana-1090 157 33 Μ… Μ… NOUN cana-1090 157 34 οΏ½ οΏ½ NOUN cana-1090 157 35 βˆ’ βˆ’ NUM cana-1090 157 36 οΏ½ οΏ½ NOUN cana-1090 157 37 Μ… Μ… NOUN cana-1090 157 38 οΏ½ οΏ½ NOUN cana-1090 157 39 | | ADV cana-1090 157 40 + + CCONJ cana-1090 157 41 |𝑏1𝑏2𝑏3 |𝑏1𝑏2𝑏3 NUM cana-1090 157 42 οΏ½ οΏ½ NOUN cana-1090 157 43 Μ… Μ… NOUN cana-1090 157 44 οΏ½ οΏ½ NOUN cana-1090 157 45 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 157 46 οΏ½ οΏ½ NOUN cana-1090 157 47 Μ… Μ… NOUN cana-1090 157 48 οΏ½ οΏ½ NOUN cana-1090 157 49 βˆ’ βˆ’ NUM cana-1090 157 50 οΏ½ οΏ½ NOUN cana-1090 157 51 Μ… Μ… NOUN cana-1090 157 52 οΏ½ οΏ½ NOUN cana-1090 157 53 βˆ’ βˆ’ NUM cana-1090 157 54 οΏ½ οΏ½ NOUN cana-1090 157 55 Μ… Μ… NOUN cana-1090 157 56 οΏ½ οΏ½ NOUN cana-1090 157 57 | | ADV cana-1090 157 58 + + CCONJ cana-1090 158 1 |𝑏1𝑏2𝑏3 |𝑏1𝑏2𝑏3 NUM cana-1090 158 2 οΏ½ οΏ½ NOUN cana-1090 158 3 Μ… Μ… NOUN cana-1090 158 4 οΏ½ οΏ½ NOUN cana-1090 158 5 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 158 6 οΏ½ οΏ½ NOUN cana-1090 158 7 Μ… Μ… NOUN cana-1090 158 8 οΏ½ οΏ½ NOUN cana-1090 158 9 βˆ’ βˆ’ NUM cana-1090 158 10 οΏ½ οΏ½ NOUN cana-1090 158 11 Μ… Μ… NOUN cana-1090 158 12 οΏ½ οΏ½ NOUN cana-1090 158 13 βˆ’ βˆ’ NUM cana-1090 158 14 οΏ½ οΏ½ NOUN cana-1090 158 15 Μ… Μ… NOUN cana-1090 158 16 οΏ½ οΏ½ NOUN cana-1090 158 17 | | ADV cana-1090 158 18 + + CCONJ cana-1090 158 19 |𝑏1𝑏2𝑏3π‘§Μ…π‘’βˆ’ |𝑏1𝑏2𝑏3π‘§Μ…π‘’βˆ’ NOUN cana-1090 158 20 οΏ½ οΏ½ NOUN cana-1090 158 21 Μ… Μ… NOUN cana-1090 158 22 οΏ½ οΏ½ NOUN cana-1090 158 23 βˆ’ βˆ’ NUM cana-1090 158 24 οΏ½ οΏ½ NOUN cana-1090 158 25 Μ… Μ… NOUN cana-1090 158 26 οΏ½ οΏ½ NOUN cana-1090 158 27 βˆ’ βˆ’ NUM cana-1090 158 28 οΏ½ οΏ½ NOUN cana-1090 158 29 Μ… Μ… NOUN cana-1090 158 30 οΏ½ οΏ½ NOUN cana-1090 158 31 | | NOUN cana-1090 158 32 + + CCONJ cana-1090 158 33 |π‘Ž1π‘Ž2π‘’βˆ’ |π‘Ž1π‘Ž2π‘’βˆ’ PROPN cana-1090 158 34 οΏ½ οΏ½ NOUN cana-1090 158 35 Μ… Μ… NOUN cana-1090 158 36 οΏ½ οΏ½ NOUN cana-1090 158 37 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 158 38 οΏ½ οΏ½ NOUN cana-1090 158 39 Μ… Μ… NOUN cana-1090 158 40 οΏ½ οΏ½ NOUN cana-1090 158 41 | | NOUN cana-1090 158 42 + + CCONJ cana-1090 158 43 |π‘Ž1π‘Ž3π‘’βˆ’ |π‘Ž1π‘Ž3π‘’βˆ’ NOUN cana-1090 158 44 οΏ½ οΏ½ NOUN cana-1090 158 45 Μ… Μ… NOUN cana-1090 158 46 οΏ½ οΏ½ NOUN cana-1090 158 47 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 158 48 οΏ½ οΏ½ NOUN cana-1090 158 49 Μ… Μ… NOUN cana-1090 158 50 οΏ½ οΏ½ NOUN cana-1090 158 51 | | NOUN cana-1090 158 52 + + CCONJ cana-1090 158 53 |π‘Ž2π‘Ž3π‘’βˆ’ |π‘Ž2π‘Ž3π‘’βˆ’ PROPN cana-1090 158 54 οΏ½ οΏ½ PROPN cana-1090 158 55 Μ… Μ… NOUN cana-1090 158 56 οΏ½ οΏ½ NOUN cana-1090 158 57 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 158 58 οΏ½ οΏ½ NOUN cana-1090 158 59 Μ… Μ… NOUN cana-1090 158 60 οΏ½ οΏ½ NOUN cana-1090 158 61 | | ADV cana-1090 158 62 + + CCONJ cana-1090 158 63 |𝑏1𝑏2𝑏3 |𝑏1𝑏2𝑏3 NUM cana-1090 158 64 οΏ½ οΏ½ NOUN cana-1090 158 65 Μ… Μ… NOUN cana-1090 158 66 οΏ½ οΏ½ NOUN cana-1090 158 67 οΏ½ οΏ½ NOUN cana-1090 158 68 Μ… Μ… NOUN cana-1090 158 69 οΏ½ οΏ½ NOUN cana-1090 158 70 π‘’βˆ’ π‘’βˆ’ NOUN cana-1090 158 71 οΏ½ οΏ½ NOUN cana-1090 158 72 Μ… Μ… NOUN cana-1090 158 73 οΏ½ οΏ½ NOUN cana-1090 158 74 βˆ’ βˆ’ NUM cana-1090 158 75 οΏ½ οΏ½ NOUN cana-1090 158 76 Μ… Μ… NOUN cana-1090 158 77 οΏ½ οΏ½ NOUN cana-1090 158 78 βˆ’ βˆ’ NUM cana-1090 158 79 οΏ½ οΏ½ NOUN cana-1090 158 80 Μ… Μ… NOUN cana-1090 158 81 οΏ½ οΏ½ NOUN cana-1090 158 82 | | ADV cana-1090 158 83 + + CCONJ cana-1090 158 84 |𝑏1𝑏2𝑏3 |𝑏1𝑏2𝑏3 NUM cana-1090 158 85 οΏ½ οΏ½ NOUN cana-1090 158 86 Μ… Μ… NOUN cana-1090 158 87 οΏ½ οΏ½ PROPN cana-1090 158 88 π‘§Μ…π‘’βˆ’ π‘§Μ…π‘’βˆ’ NOUN cana-1090 158 89 οΏ½ οΏ½ PROPN cana-1090 158 90 Μ… Μ… NOUN cana-1090 158 91 οΏ½ οΏ½ NOUN cana-1090 158 92 βˆ’ βˆ’ NUM cana-1090 158 93 οΏ½ οΏ½ NOUN cana-1090 158 94 Μ… Μ… NOUN cana-1090 158 95 οΏ½ οΏ½ NOUN cana-1090 158 96 βˆ’ βˆ’ NUM cana-1090 158 97 οΏ½ οΏ½ NOUN cana-1090 158 98 Μ… Μ… NOUN cana-1090 158 99 οΏ½ οΏ½ NOUN cana-1090 158 100 | | ADV cana-1090 158 101 + + CCONJ cana-1090 158 102 |𝑏1𝑏2𝑏3 |𝑏1𝑏2𝑏3 NUM cana-1090 158 103 οΏ½ οΏ½ NOUN cana-1090 158 104 Μ… Μ… NOUN cana-1090 158 105 οΏ½ οΏ½ PROPN cana-1090 158 106 π‘§Μ…π‘’βˆ’ π‘§Μ…π‘’βˆ’ NOUN cana-1090 158 107 οΏ½ οΏ½ PROPN cana-1090 158 108 Μ… Μ… NOUN cana-1090 158 109 οΏ½ οΏ½ NOUN cana-1090 158 110 βˆ’ βˆ’ NUM cana-1090 158 111 οΏ½ οΏ½ NOUN cana-1090 158 112 Μ… Μ… NOUN cana-1090 158 113 οΏ½ οΏ½ NOUN cana-1090 158 114 βˆ’ βˆ’ NUM cana-1090 158 115 οΏ½ οΏ½ NOUN cana-1090 158 116 Μ… Μ… NOUN cana-1090 158 117 οΏ½ οΏ½ NOUN cana-1090 158 118 | | NOUN cana-1090 158 119 + + NUM cana-1090 158 120 |π‘Ž1π‘Ž2π‘Ž3π‘’βˆ’ |π‘Ž1π‘Ž2π‘Ž3π‘’βˆ’ PROPN cana-1090 158 121 οΏ½ οΏ½ PROPN cana-1090 158 122 Μ… Μ… NOUN cana-1090 158 123 οΏ½ οΏ½ NOUN cana-1090 158 124 βˆ’ βˆ’ NUM cana-1090 158 125 οΏ½ οΏ½ NOUN cana-1090 158 126 Μ… Μ… NOUN cana-1090 158 127 οΏ½ οΏ½ NOUN cana-1090 158 128 βˆ’ βˆ’ NUM cana-1090 158 129 οΏ½ οΏ½ NOUN cana-1090 158 130 Μ… Μ… NOUN cana-1090 158 131 οΏ½ οΏ½ NOUN cana-1090 158 132 | | ADV cana-1090 158 133 + + CCONJ cana-1090 158 134 |𝑏1𝑏2𝑏3 |𝑏1𝑏2𝑏3 NUM cana-1090 158 135 οΏ½ οΏ½ NOUN cana-1090 158 136 Μ… Μ… NOUN cana-1090 158 137 οΏ½ οΏ½ NOUN cana-1090 158 138 οΏ½ οΏ½ NOUN cana-1090 158 139 Μ… Μ… NOUN cana-1090 158 140 οΏ½ οΏ½ PROPN cana-1090 158 141 π‘§Μ…π‘’βˆ’ π‘§Μ…π‘’βˆ’ NOUN cana-1090 158 142 οΏ½ οΏ½ PROPN cana-1090 158 143 Μ… Μ… NOUN cana-1090 158 144 οΏ½ οΏ½ NOUN cana-1090 158 145 βˆ’ βˆ’ NUM cana-1090 158 146 οΏ½ οΏ½ NOUN cana-1090 158 147 Μ… Μ… NOUN cana-1090 158 148 οΏ½ οΏ½ NOUN cana-1090 158 149 βˆ’ βˆ’ NUM cana-1090 158 150 οΏ½ οΏ½ NOUN cana-1090 158 151 Μ… Μ… NOUN cana-1090 158 152 οΏ½ οΏ½ NOUN cana-1090 158 153 | | NOUN cana-1090 158 154 < < X cana-1090 158 155 1 1 NUM cana-1090 158 156 communications communication NOUN cana-1090 158 157 on on ADP cana-1090 158 158 applied apply VERB cana-1090 158 159 nonlinear nonlinear ADJ cana-1090 158 160 analysis analysis NOUN cana-1090 158 161 issn issn NOUN cana-1090 158 162 : : PUNCT cana-1090 158 163 1074 1074 NUM cana-1090 158 164 - - PUNCT cana-1090 158 165 133x 133x NUM cana-1090 158 166 vol vol NOUN cana-1090 158 167 31 31 NUM cana-1090 158 168 no no NOUN cana-1090 158 169 . . PUNCT cana-1090 159 1 5s 5s NUM cana-1090 159 2 ( ( PUNCT cana-1090 159 3 2024 2024 NUM cana-1090 159 4 ) ) PUNCT cana-1090 159 5 558 558 NUM cana-1090 159 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 159 7 is be AUX cana-1090 159 8 satisfied satisfied ADJ cana-1090 159 9 whenever whenever SCONJ cana-1090 159 10 π‘Ž1π‘’βˆ’π›Ό1[1 π‘Ž1π‘’βˆ’π›Ό1[1 X cana-1090 159 11 + + X cana-1090 159 12 π‘Ž2π‘’βˆ’π›Ό2 π‘Ž2π‘’βˆ’π›Ό2 X cana-1090 159 13 ] ] X cana-1090 160 1 + + CCONJ cana-1090 160 2 π‘Ž2π‘’βˆ’π›Ό2[1 π‘Ž2π‘’βˆ’π›Ό2[1 X cana-1090 161 1 + + NOUN cana-1090 161 2 π‘Ž3π‘’βˆ’π›Ό3 π‘Ž3π‘’βˆ’π›Ό3 NOUN cana-1090 161 3 ] ] X cana-1090 162 1 + + CCONJ cana-1090 162 2 π‘Ž3π‘’βˆ’π›Ό3[1 π‘Ž3π‘’βˆ’π›Ό3[1 PUNCT cana-1090 163 1 + + NUM cana-1090 163 2 π‘Ž1π‘’βˆ’π›Ό1 π‘Ž1π‘’βˆ’π›Ό1 VERB cana-1090 163 3 ] ] X cana-1090 164 1 + + CCONJ cana-1090 164 2 π‘Ž1π‘Ž2π‘Ž3π‘’βˆ’π›Ό1π‘’βˆ’π›Ό2π‘’βˆ’π›Ό3 π‘Ž1π‘Ž2π‘Ž3π‘’βˆ’π›Ό1π‘’βˆ’π›Ό2π‘’βˆ’π›Ό3 PROPN cana-1090 164 3 + + NUM cana-1090 164 4 𝐡[ 𝐡[ NUM cana-1090 164 5 οΏ½ οΏ½ NOUN cana-1090 164 6 Μ… Μ… NOUN cana-1090 164 7 οΏ½ οΏ½ NOUN cana-1090 164 8 οΏ½ οΏ½ NOUN cana-1090 164 9 Μ… Μ… NOUN cana-1090 164 10 οΏ½ οΏ½ NOUN cana-1090 164 11 𝑧̅ 𝑧̅ NOUN cana-1090 164 12 + + SYM cana-1090 164 13 οΏ½ οΏ½ NOUN cana-1090 164 14 Μ… Μ… NOUN cana-1090 164 15 οΏ½ οΏ½ NOUN cana-1090 164 16 οΏ½ οΏ½ NOUN cana-1090 164 17 Μ… Μ… NOUN cana-1090 164 18 οΏ½ οΏ½ PROPN cana-1090 164 19 + + CCONJ cana-1090 164 20 οΏ½ οΏ½ NOUN cana-1090 164 21 Μ… Μ… NOUN cana-1090 164 22 οΏ½ οΏ½ NOUN cana-1090 164 23 𝑧̅ 𝑧̅ NOUN cana-1090 164 24 + + SYM cana-1090 164 25 οΏ½ οΏ½ NOUN cana-1090 164 26 Μ… Μ… NOUN cana-1090 164 27 οΏ½ οΏ½ NOUN cana-1090 164 28 𝑧̅ 𝑧̅ NOUN cana-1090 164 29 + + CCONJ cana-1090 164 30 𝑧̅ 𝑧̅ NOUN cana-1090 164 31 + + CCONJ cana-1090 164 32 οΏ½ οΏ½ NOUN cana-1090 164 33 Μ… Μ… NOUN cana-1090 164 34 οΏ½ οΏ½ PROPN cana-1090 164 35 + + CCONJ cana-1090 164 36 οΏ½ οΏ½ PROPN cana-1090 164 37 Μ… Μ… NOUN cana-1090 164 38 οΏ½ οΏ½ NOUN cana-1090 164 39 + + CCONJ cana-1090 164 40 1 1 NUM cana-1090 164 41 ] ] PUNCT cana-1090 164 42 < < X cana-1090 164 43 1 1 X cana-1090 164 44 . . PUNCT cana-1090 165 1 ( ( PUNCT cana-1090 165 2 31 31 NUM cana-1090 165 3 ) ) PUNCT cana-1090 165 4 clearly clearly ADV cana-1090 165 5 from from ADP cana-1090 165 6 theorem theorem ADJ cana-1090 165 7 2.1 2.1 NUM cana-1090 165 8 , , PUNCT cana-1090 165 9 οΏ½ οΏ½ NOUN cana-1090 165 10 Μ… Μ… NOUN cana-1090 165 11 οΏ½ οΏ½ NOUN cana-1090 165 12 ≀ ≀ NUM cana-1090 165 13 𝐴1 𝐴1 PROPN cana-1090 165 14 1βˆ’π΅ 1βˆ’π΅ NUM cana-1090 165 15 , , PUNCT cana-1090 165 16 ( ( PUNCT cana-1090 165 17 32 32 NUM cana-1090 165 18 ) ) PUNCT cana-1090 165 19 οΏ½ οΏ½ NOUN cana-1090 165 20 Μ… Μ… NOUN cana-1090 165 21 οΏ½ οΏ½ PROPN cana-1090 165 22 ≀ ≀ NUM cana-1090 165 23 𝐴2 𝐴2 PROPN cana-1090 165 24 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 165 25 ( ( PUNCT cana-1090 165 26 33 33 NUM cana-1090 165 27 ) ) PUNCT cana-1090 165 28 and and CCONJ cana-1090 165 29 𝑧̅ 𝑧̅ VERB cana-1090 165 30 ≀ ≀ PROPN cana-1090 165 31 𝐴3 𝐴3 PROPN cana-1090 165 32 1βˆ’π΅ 1βˆ’π΅ PROPN cana-1090 165 33 . . PUNCT cana-1090 166 1 ( ( PUNCT cana-1090 166 2 34 34 NUM cana-1090 166 3 ) ) PUNCT cana-1090 166 4 substitute substitute NOUN cana-1090 166 5 ( ( PUNCT cana-1090 166 6 32 32 NUM cana-1090 166 7 ) ) PUNCT cana-1090 166 8 , , PUNCT cana-1090 166 9 ( ( PUNCT cana-1090 166 10 33 33 NUM cana-1090 166 11 ) ) PUNCT cana-1090 166 12 and and CCONJ cana-1090 166 13 ( ( PUNCT cana-1090 166 14 34 34 NUM cana-1090 166 15 ) ) PUNCT cana-1090 166 16 in in ADP cana-1090 166 17 ( ( PUNCT cana-1090 166 18 31 31 NUM cana-1090 166 19 ) ) PUNCT cana-1090 166 20 . . PUNCT cana-1090 167 1 use use NOUN cana-1090 167 2 remark remark NOUN cana-1090 167 3 1.3.1 1.3.1 NUM cana-1090 167 4 of of ADP cana-1090 167 5 [ [ X cana-1090 167 6 25 25 NUM cana-1090 167 7 ] ] PUNCT cana-1090 167 8 and and CCONJ cana-1090 167 9 theorem theorem VERB cana-1090 167 10 2.5 2.5 NUM cana-1090 167 11 to to PART cana-1090 167 12 get get VERB cana-1090 167 13 the the DET cana-1090 167 14 result result NOUN cana-1090 167 15 . . PUNCT cana-1090 168 1 3 3 X cana-1090 168 2 . . NOUN cana-1090 168 3 numerical numerical ADJ cana-1090 168 4 analysis analysis NOUN cana-1090 168 5 and and CCONJ cana-1090 168 6 open open ADJ cana-1090 168 7 problem problem NOUN cana-1090 168 8 in in ADP cana-1090 168 9 this this DET cana-1090 168 10 section section NOUN cana-1090 168 11 we we PRON cana-1090 168 12 observe observe VERB cana-1090 168 13 the the DET cana-1090 168 14 dynamics dynamic NOUN cana-1090 168 15 of of ADP cana-1090 168 16 the the DET cana-1090 168 17 discrete discrete ADJ cana-1090 168 18 model model NOUN cana-1090 168 19 ( ( PUNCT cana-1090 168 20 1 1 NUM cana-1090 168 21 ) ) PUNCT cana-1090 168 22 numerically numerically ADV cana-1090 168 23 and and CCONJ cana-1090 168 24 propose propose VERB cana-1090 168 25 an an DET cana-1090 168 26 open open ADJ cana-1090 168 27 problem problem NOUN cana-1090 168 28 . . PUNCT cana-1090 169 1 figure figure NOUN cana-1090 169 2 ( ( PUNCT cana-1090 169 3 1a 1a X cana-1090 169 4 ) ) PUNCT cana-1090 169 5 shows show VERB cana-1090 169 6 the the DET cana-1090 169 7 bifurcation bifurcation NOUN cana-1090 169 8 diagram diagram NOUN cana-1090 169 9 with with ADP cana-1090 169 10 π‘Ž1 π‘Ž1 PROPN cana-1090 169 11 as as ADP cana-1090 169 12 bifurcation bifurcation NOUN cana-1090 169 13 parameter parameter NOUN cana-1090 169 14 and and CCONJ cana-1090 169 15 figure figure NOUN cana-1090 169 16 ( ( PUNCT cana-1090 169 17 1b 1b NUM cana-1090 169 18 ) ) PUNCT cana-1090 169 19 shows show VERB cana-1090 169 20 the the DET cana-1090 169 21 plots plot NOUN cana-1090 169 22 of of ADP cana-1090 169 23 π‘₯𝑛 π‘₯𝑛 PROPN cana-1090 169 24 , , PUNCT cana-1090 169 25 𝑦𝑛 𝑦𝑛 ADP cana-1090 169 26 , , PUNCT cana-1090 170 1 𝑧𝑛 𝑧𝑛 ADP cana-1090 170 2 for for ADP cana-1090 170 3 a a DET cana-1090 170 4 particular particular ADJ cana-1090 170 5 value value NOUN cana-1090 170 6 , , PUNCT cana-1090 170 7 ie ie X cana-1090 170 8 . . PUNCT cana-1090 170 9 , , PUNCT cana-1090 170 10 π‘Ž1 π‘Ž1 NOUN cana-1090 170 11 = = PUNCT cana-1090 170 12 32.0 32.0 NUM cana-1090 170 13 . . PUNCT cana-1090 171 1 figure figure NOUN cana-1090 171 2 ( ( PUNCT cana-1090 171 3 1b 1b NUM cana-1090 171 4 ) ) PUNCT cana-1090 171 5 shows show VERB cana-1090 171 6 that that SCONJ cana-1090 171 7 the the DET cana-1090 171 8 plot plot NOUN cana-1090 171 9 is be AUX cana-1090 171 10 eventually eventually ADV cana-1090 171 11 4 4 NUM cana-1090 171 12 - - NOUN cana-1090 171 13 periodic periodic NOUN cana-1090 171 14 . . PUNCT cana-1090 172 1 similarly similarly ADV cana-1090 172 2 figures(2a figures(2a PROPN cana-1090 172 3 ) ) PUNCT cana-1090 172 4 and and CCONJ cana-1090 172 5 ( ( PUNCT cana-1090 172 6 3a 3a NUM cana-1090 172 7 ) ) PUNCT cana-1090 172 8 shows show VERB cana-1090 172 9 the the DET cana-1090 172 10 bifurcation bifurcation NOUN cana-1090 172 11 diagrams diagram NOUN cana-1090 172 12 with with ADP cana-1090 172 13 π‘Ž2 π‘Ž2 NOUN cana-1090 172 14 and and CCONJ cana-1090 172 15 π‘Ž3 π‘Ž3 NOUN cana-1090 172 16 as as ADP cana-1090 172 17 bifurcation bifurcation NOUN cana-1090 172 18 parameter parameter NOUN cana-1090 172 19 , , PUNCT cana-1090 172 20 whereas whereas SCONJ cana-1090 172 21 figures(2b figures(2b NOUN cana-1090 172 22 ) ) PUNCT cana-1090 172 23 and and CCONJ cana-1090 172 24 ( ( PUNCT cana-1090 172 25 3b 3b NUM cana-1090 172 26 ) ) PUNCT cana-1090 172 27 shows show VERB cana-1090 172 28 their their PRON cana-1090 172 29 corresponding correspond VERB cana-1090 172 30 plots plot NOUN cana-1090 172 31 for for ADP cana-1090 172 32 π‘Ž2 π‘Ž2 NOUN cana-1090 172 33 = = PROPN cana-1090 172 34 10.0 10.0 NUM cana-1090 172 35 and and CCONJ cana-1090 172 36 π‘Ž3 π‘Ž3 PROPN cana-1090 172 37 = = PROPN cana-1090 172 38 45.9 45.9 NUM cana-1090 172 39 respectively respectively ADV cana-1090 172 40 . . PUNCT cana-1090 173 1 figures figure NOUN cana-1090 173 2 ( ( PUNCT cana-1090 173 3 2b 2b NOUN cana-1090 173 4 ) ) PUNCT cana-1090 173 5 and and CCONJ cana-1090 173 6 ( ( PUNCT cana-1090 173 7 3b 3b NUM cana-1090 173 8 ) ) PUNCT cana-1090 173 9 shows show VERB cana-1090 173 10 that that SCONJ cana-1090 173 11 the the DET cana-1090 173 12 plots plot NOUN cana-1090 173 13 are be AUX cana-1090 173 14 eventually eventually ADV cana-1090 173 15 4 4 NUM cana-1090 173 16 - - NOUN cana-1090 173 17 periodic periodic ADJ cana-1090 173 18 . . PUNCT cana-1090 174 1 3.1 3.1 NUM cana-1090 174 2 open open ADJ cana-1090 174 3 problem problem NOUN cana-1090 174 4 derive derive VERB cana-1090 174 5 the the DET cana-1090 174 6 condition condition NOUN cana-1090 174 7 for for ADP cana-1090 174 8 ( ( PUNCT cana-1090 174 9 1 1 NUM cana-1090 174 10 ) ) PUNCT cana-1090 174 11 to to PART cana-1090 174 12 be be AUX cana-1090 174 13 eventually eventually ADV cana-1090 174 14 4 4 NUM cana-1090 174 15 - - NOUN cana-1090 174 16 periodic periodic NOUN cana-1090 174 17 . . PUNCT cana-1090 175 1 a a PRON cana-1090 175 2 ) ) PUNCT cana-1090 176 1 [ [ X cana-1090 176 2 bifurcation bifurcation NOUN cana-1090 176 3 diagrams diagram NOUN cana-1090 176 4 of of ADP cana-1090 176 5 ( ( PUNCT cana-1090 176 6 1 1 NUM cana-1090 176 7 ) ) PUNCT cana-1090 176 8 with with ADP cana-1090 176 9 π‘Ž1as π‘Ž1as PUNCT cana-1090 176 10 bifurcation bifurcation NOUN cana-1090 176 11 parameter parameter NOUN cana-1090 176 12 ] ] X cana-1090 176 13 b b X cana-1090 176 14 ) ) PUNCT cana-1090 177 1 [ [ X cana-1090 177 2 plots plot NOUN cana-1090 177 3 of of ADP cana-1090 177 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-1090 177 5 , , PUNCT cana-1090 177 6 𝑦𝑛 𝑦𝑛 ADP cana-1090 177 7 , , PUNCT cana-1090 177 8 𝑧𝑛 𝑧𝑛 ADP cana-1090 177 9 with with ADP cana-1090 177 10 π‘Ž1 π‘Ž1 PROPN cana-1090 177 11 = = SYM cana-1090 177 12 32.0 32.0 NUM cana-1090 177 13 ] ] PUNCT cana-1090 177 14 figure figure NOUN cana-1090 177 15 1 1 NUM cana-1090 177 16 : : PUNCT cana-1090 177 17 here here ADV cana-1090 177 18 𝛼1 𝛼1 VERB cana-1090 177 19 = = SYM cana-1090 177 20 2.3 2.3 NUM cana-1090 177 21 , , PUNCT cana-1090 177 22 𝛼2 𝛼2 PROPN cana-1090 177 23 = = SYM cana-1090 177 24 3.2 3.2 NUM cana-1090 177 25 , , PUNCT cana-1090 177 26 𝛼3 𝛼3 NOUN cana-1090 177 27 = = NOUN cana-1090 177 28 2.6 2.6 NUM cana-1090 177 29 , , PUNCT cana-1090 177 30 𝑏1 𝑏1 NOUN cana-1090 177 31 = = SYM cana-1090 177 32 .4 .4 NOUN cana-1090 177 33 , , PUNCT cana-1090 177 34 𝑐1 𝑐1 NOUN cana-1090 177 35 = = SYM cana-1090 177 36 .5 .5 NUM cana-1090 177 37 , , PUNCT cana-1090 177 38 π‘Ž2 π‘Ž2 NOUN cana-1090 177 39 = = SYM cana-1090 177 40 0.5 0.5 NUM cana-1090 177 41 , , PUNCT cana-1090 177 42 𝑏2 𝑏2 NOUN cana-1090 177 43 = = PROPN cana-1090 177 44 4.4 4.4 NUM cana-1090 177 45 , , PUNCT cana-1090 177 46 𝑐2 𝑐2 NOUN cana-1090 177 47 = = SYM cana-1090 177 48 3.5 3.5 NUM cana-1090 177 49 , , PUNCT cana-1090 177 50 π‘Ž3 π‘Ž3 NOUN cana-1090 177 51 = = SYM cana-1090 177 52 0.9 0.9 NUM cana-1090 177 53 , , PUNCT cana-1090 177 54 𝑏3 𝑏3 NOUN cana-1090 177 55 = = SYM cana-1090 177 56 0.6 0.6 NUM cana-1090 177 57 , , PUNCT cana-1090 177 58 𝑐3 𝑐3 NOUN cana-1090 177 59 = = SYM cana-1090 177 60 0.4 0.4 NUM cana-1090 177 61 . . PUNCT cana-1090 178 1 communications communication NOUN cana-1090 178 2 on on ADP cana-1090 178 3 applied apply VERB cana-1090 178 4 nonlinear nonlinear ADJ cana-1090 178 5 analysis analysis NOUN cana-1090 178 6 issn issn NOUN cana-1090 178 7 : : PUNCT cana-1090 178 8 1074 1074 NUM cana-1090 178 9 - - PUNCT cana-1090 178 10 133x 133x NUM cana-1090 178 11 vol vol NOUN cana-1090 178 12 31 31 NUM cana-1090 178 13 no no NOUN cana-1090 178 14 . . PUNCT cana-1090 179 1 5s 5s NUM cana-1090 179 2 ( ( PUNCT cana-1090 179 3 2024 2024 NUM cana-1090 179 4 ) ) PUNCT cana-1090 179 5 559 559 NUM cana-1090 179 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 179 7 a a X cana-1090 179 8 ) ) PUNCT cana-1090 180 1 [ [ X cana-1090 180 2 bifurcation bifurcation NOUN cana-1090 180 3 diagrams diagram NOUN cana-1090 180 4 of of ADP cana-1090 180 5 ( ( PUNCT cana-1090 180 6 1 1 NUM cana-1090 180 7 ) ) PUNCT cana-1090 180 8 with with ADP cana-1090 180 9 π‘Ž2 π‘Ž2 NOUN cana-1090 180 10 as as ADP cana-1090 180 11 bifurcation bifurcation NOUN cana-1090 180 12 parameter parameter NOUN cana-1090 180 13 ] ] X cana-1090 180 14 b b X cana-1090 180 15 ) ) PUNCT cana-1090 181 1 [ [ X cana-1090 181 2 plots plot NOUN cana-1090 181 3 of of ADP cana-1090 181 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-1090 181 5 , , PUNCT cana-1090 181 6 𝑦𝑛 𝑦𝑛 ADP cana-1090 181 7 , , PUNCT cana-1090 181 8 𝑧𝑛 𝑧𝑛 ADP cana-1090 181 9 with with ADP cana-1090 181 10 π‘Ž2 π‘Ž2 NOUN cana-1090 181 11 = = NOUN cana-1090 181 12 10.0 10.0 NUM cana-1090 181 13 ] ] PUNCT cana-1090 181 14 figure figure NOUN cana-1090 181 15 2 2 NUM cana-1090 181 16 : : PUNCT cana-1090 181 17 here here ADV cana-1090 181 18 𝛼1 𝛼1 VERB cana-1090 181 19 = = SYM cana-1090 181 20 5.3 5.3 NUM cana-1090 181 21 , , PUNCT cana-1090 181 22 𝛼2 𝛼2 PROPN cana-1090 181 23 = = NUM cana-1090 181 24 0.2 0.2 NUM cana-1090 181 25 , , PUNCT cana-1090 181 26 𝛼3 𝛼3 NOUN cana-1090 181 27 = = NOUN cana-1090 181 28 12.6 12.6 NUM cana-1090 181 29 , , PUNCT cana-1090 181 30 π‘Ž1 π‘Ž1 NOUN cana-1090 181 31 = = SYM cana-1090 181 32 0.5 0.5 NUM cana-1090 181 33 , , PUNCT cana-1090 181 34 𝑏1 𝑏1 NOUN cana-1090 181 35 = = SYM cana-1090 181 36 .4 .4 NOUN cana-1090 181 37 , , PUNCT cana-1090 181 38 𝑐1 𝑐1 NOUN cana-1090 181 39 = = SYM cana-1090 181 40 .5 .5 NUM cana-1090 181 41 , , PUNCT cana-1090 181 42 𝑏2 𝑏2 NOUN cana-1090 181 43 = = PROPN cana-1090 181 44 4.4 4.4 NUM cana-1090 181 45 , , PUNCT cana-1090 181 46 𝑐2 𝑐2 NOUN cana-1090 181 47 = = SYM cana-1090 181 48 1.5 1.5 NUM cana-1090 181 49 , , PUNCT cana-1090 181 50 π‘Ž3 π‘Ž3 PROPN cana-1090 181 51 = = SYM cana-1090 181 52 5.9 5.9 NUM cana-1090 181 53 , , PUNCT cana-1090 181 54 𝑏3 𝑏3 NOUN cana-1090 181 55 = = SYM cana-1090 181 56 8.6 8.6 NUM cana-1090 181 57 , , PUNCT cana-1090 181 58 𝑐3 𝑐3 NOUN cana-1090 181 59 = = PUNCT cana-1090 181 60 0.4 0.4 NUM cana-1090 181 61 . . PUNCT cana-1090 182 1 a a PRON cana-1090 182 2 ) ) PUNCT cana-1090 183 1 [ [ X cana-1090 183 2 bifurcation bifurcation NOUN cana-1090 183 3 diagrams diagram NOUN cana-1090 183 4 of of ADP cana-1090 183 5 ( ( PUNCT cana-1090 183 6 1 1 NUM cana-1090 183 7 ) ) PUNCT cana-1090 183 8 with with ADP cana-1090 183 9 π‘Ž3 π‘Ž3 PROPN cana-1090 183 10 as as ADP cana-1090 183 11 bifurcation bifurcation NOUN cana-1090 183 12 parameter parameter NOUN cana-1090 183 13 ] ] X cana-1090 183 14 b b X cana-1090 183 15 ) ) PUNCT cana-1090 184 1 [ [ X cana-1090 184 2 plots plot NOUN cana-1090 184 3 of of ADP cana-1090 184 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-1090 184 5 , , PUNCT cana-1090 184 6 𝑦𝑛 𝑦𝑛 ADP cana-1090 184 7 , , PUNCT cana-1090 184 8 𝑧𝑛 𝑧𝑛 ADP cana-1090 184 9 with with ADP cana-1090 184 10 π‘Ž3 π‘Ž3 PROPN cana-1090 184 11 = = NOUN cana-1090 184 12 45.0 45.0 NUM cana-1090 184 13 ] ] PUNCT cana-1090 184 14 figure figure NOUN cana-1090 184 15 3 3 NUM cana-1090 184 16 : : PUNCT cana-1090 184 17 here here ADV cana-1090 184 18 𝛼1 𝛼1 VERB cana-1090 184 19 = = SYM cana-1090 184 20 2.3 2.3 NUM cana-1090 184 21 , , PUNCT cana-1090 184 22 𝛼2 𝛼2 PROPN cana-1090 184 23 = = SYM cana-1090 184 24 3.2 3.2 NUM cana-1090 184 25 , , PUNCT cana-1090 184 26 𝛼3 𝛼3 NOUN cana-1090 184 27 = = NOUN cana-1090 184 28 2.6 2.6 NUM cana-1090 184 29 , , PUNCT cana-1090 184 30 π‘Ž1 π‘Ž1 NOUN cana-1090 184 31 = = NOUN cana-1090 184 32 0.9 0.9 NUM cana-1090 184 33 , , PUNCT cana-1090 184 34 𝑏1 𝑏1 NOUN cana-1090 184 35 = = SYM cana-1090 184 36 .4 .4 NOUN cana-1090 184 37 , , PUNCT cana-1090 184 38 𝑐1 𝑐1 NOUN cana-1090 184 39 = = SYM cana-1090 184 40 .5 .5 NUM cana-1090 184 41 , , PUNCT cana-1090 184 42 π‘Ž2 π‘Ž2 NOUN cana-1090 184 43 = = SYM cana-1090 184 44 0.5 0.5 NUM cana-1090 184 45 , , PUNCT cana-1090 184 46 𝑏2 𝑏2 NOUN cana-1090 184 47 = = PROPN cana-1090 184 48 4.4 4.4 NUM cana-1090 184 49 , , PUNCT cana-1090 184 50 𝑐2 𝑐2 NOUN cana-1090 184 51 = = SYM cana-1090 184 52 3.5 3.5 NUM cana-1090 184 53 , , PUNCT cana-1090 184 54 𝑏3 𝑏3 NOUN cana-1090 184 55 = = SYM cana-1090 184 56 0.6 0.6 NUM cana-1090 184 57 , , PUNCT cana-1090 184 58 𝑐3 𝑐3 NOUN cana-1090 184 59 = = PUNCT cana-1090 184 60 0.4 0.4 NUM cana-1090 184 61 . . PUNCT cana-1090 185 1 4 4 NUM cana-1090 185 2 . . X cana-1090 185 3 conclusion conclusion NOUN cana-1090 185 4 in in ADP cana-1090 185 5 this this DET cana-1090 185 6 paper paper NOUN cana-1090 185 7 , , PUNCT cana-1090 185 8 we we PRON cana-1090 185 9 studied study VERB cana-1090 185 10 the the DET cana-1090 185 11 dynamics dynamic NOUN cana-1090 185 12 of of ADP cana-1090 185 13 a a DET cana-1090 185 14 second second ADJ cana-1090 185 15 - - PUNCT cana-1090 185 16 order order NOUN cana-1090 185 17 system system NOUN cana-1090 185 18 defined define VERB cana-1090 185 19 by by ADP cana-1090 185 20 three three NUM cana-1090 185 21 variables variable NOUN cana-1090 185 22 , , PUNCT cana-1090 185 23 focusing focus VERB cana-1090 185 24 on on ADP cana-1090 185 25 the the DET cana-1090 185 26 existence existence NOUN cana-1090 185 27 of of ADP cana-1090 185 28 a a DET cana-1090 185 29 unique unique ADJ cana-1090 185 30 positive positive ADJ cana-1090 185 31 equilibrium equilibrium NOUN cana-1090 185 32 and and CCONJ cana-1090 185 33 its its PRON cana-1090 185 34 global global ADJ cana-1090 185 35 stability stability NOUN cana-1090 185 36 . . PUNCT cana-1090 186 1 this this DET cana-1090 186 2 study study NOUN cana-1090 186 3 is be AUX cana-1090 186 4 particularly particularly ADV cana-1090 186 5 relevant relevant ADJ cana-1090 186 6 in in ADP cana-1090 186 7 the the DET cana-1090 186 8 context context NOUN cana-1090 186 9 of of ADP cana-1090 186 10 population population NOUN cana-1090 186 11 biology biology NOUN cana-1090 186 12 , , PUNCT cana-1090 186 13 where where SCONJ cana-1090 186 14 understanding understand VERB cana-1090 186 15 the the DET cana-1090 186 16 conditions condition NOUN cana-1090 186 17 for for ADP cana-1090 186 18 local local ADJ cana-1090 186 19 asymptotic asymptotic ADJ cana-1090 186 20 stability stability NOUN cana-1090 186 21 and and CCONJ cana-1090 186 22 global global ADJ cana-1090 186 23 stability stability NOUN cana-1090 186 24 are be AUX cana-1090 186 25 crucial crucial ADJ cana-1090 186 26 . . PUNCT cana-1090 187 1 we we PRON cana-1090 187 2 successfully successfully ADV cana-1090 187 3 established establish VERB cana-1090 187 4 the the DET cana-1090 187 5 conditions condition NOUN cana-1090 187 6 which which PRON cana-1090 187 7 assure assure VERB cana-1090 187 8 the the DET cana-1090 187 9 global global ADJ cana-1090 187 10 asymptotic asymptotic ADJ cana-1090 187 11 stability stability NOUN cana-1090 187 12 of of ADP cana-1090 187 13 the the DET cana-1090 187 14 unique unique ADJ cana-1090 187 15 positive positive ADJ cana-1090 187 16 equilibrium.moreover equilibrium.moreover NOUN cana-1090 187 17 , , PUNCT cana-1090 187 18 we we PRON cana-1090 187 19 proposed propose VERB cana-1090 187 20 an an DET cana-1090 187 21 open open ADJ cana-1090 187 22 problem problem NOUN cana-1090 187 23 that that PRON cana-1090 187 24 invite invite VERB cana-1090 187 25 further further ADJ cana-1090 187 26 investigation investigation NOUN cana-1090 187 27 into into ADP cana-1090 187 28 the the DET cana-1090 187 29 conditions condition NOUN cana-1090 187 30 necessary necessary ADJ cana-1090 187 31 for for SCONJ cana-1090 187 32 the the DET cana-1090 187 33 system system NOUN cana-1090 187 34 to to PART cana-1090 187 35 exhibit exhibit VERB cana-1090 187 36 4 4 NUM cana-1090 187 37 - - PUNCT cana-1090 187 38 periodic periodic ADJ cana-1090 187 39 behavior behavior NOUN cana-1090 187 40 . . PUNCT cana-1090 188 1 addressing address VERB cana-1090 188 2 this this DET cana-1090 188 3 problem problem NOUN cana-1090 188 4 will will AUX cana-1090 188 5 provide provide VERB cana-1090 188 6 deeper deep ADJ cana-1090 188 7 insights insight NOUN cana-1090 188 8 into into ADP cana-1090 188 9 the the DET cana-1090 188 10 periodic periodic ADJ cana-1090 188 11 nature nature NOUN cana-1090 188 12 of of ADP cana-1090 188 13 the the DET cana-1090 188 14 system system NOUN cana-1090 188 15 and and CCONJ cana-1090 188 16 its its PRON cana-1090 188 17 long long ADJ cana-1090 188 18 - - PUNCT cana-1090 188 19 term term NOUN cana-1090 188 20 behavior behavior NOUN cana-1090 188 21 . . PUNCT cana-1090 189 1 communications communication NOUN cana-1090 189 2 on on ADP cana-1090 189 3 applied apply VERB cana-1090 189 4 nonlinear nonlinear ADJ cana-1090 189 5 analysis analysis NOUN cana-1090 189 6 issn issn NOUN cana-1090 189 7 : : PUNCT cana-1090 189 8 1074 1074 NUM cana-1090 189 9 - - PUNCT cana-1090 189 10 133x 133x NUM cana-1090 189 11 vol vol NOUN cana-1090 189 12 31 31 NUM cana-1090 189 13 no no NOUN cana-1090 189 14 . . PUNCT cana-1090 190 1 5s 5s NUM cana-1090 190 2 ( ( PUNCT cana-1090 190 3 2024 2024 NUM cana-1090 190 4 ) ) PUNCT cana-1090 190 5 560 560 NUM cana-1090 190 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 190 7 references reference NOUN cana-1090 190 8 [ [ X cana-1090 190 9 1 1 NUM cana-1090 190 10 ] ] PUNCT cana-1090 190 11 a. a. NOUN cana-1090 190 12 s. s. PROPN cana-1090 190 13 ackleh ackleh PROPN cana-1090 190 14 and and CCONJ cana-1090 190 15 p. p. PROPN cana-1090 190 16 zhang zhang PROPN cana-1090 190 17 , , PUNCT cana-1090 190 18 competitive competitive ADJ cana-1090 190 19 exclusion exclusion NOUN cana-1090 190 20 in in ADP cana-1090 190 21 a a DET cana-1090 190 22 discrete discrete ADJ cana-1090 190 23 stage stage NOUN cana-1090 190 24 - - PUNCT cana-1090 190 25 structured structure VERB cana-1090 190 26 two two NUM cana-1090 190 27 species specie NOUN cana-1090 190 28 model model NOUN cana-1090 190 29 , , PUNCT cana-1090 190 30 mathematical mathematical ADJ cana-1090 190 31 modelling modelling NOUN cana-1090 190 32 of of ADP cana-1090 190 33 natural natural ADJ cana-1090 190 34 phenomena phenomenon NOUN cana-1090 190 35 , , PUNCT cana-1090 190 36 4(6 4(6 NUM cana-1090 190 37 ) ) PUNCT cana-1090 190 38 , , PUNCT cana-1090 190 39 ( ( PUNCT cana-1090 190 40 2009 2009 NUM cana-1090 190 41 ) ) PUNCT cana-1090 190 42 , , PUNCT cana-1090 190 43 156 156 NUM cana-1090 190 44 175 175 NUM cana-1090 190 45 . . PUNCT cana-1090 191 1 [ [ X cana-1090 191 2 2 2 NUM cana-1090 191 3 ] ] PUNCT cana-1090 191 4 azmy azmy PROPN cana-1090 191 5 s. s. PROPN cana-1090 191 6 ackleh ackleh PROPN cana-1090 191 7 and and CCONJ cana-1090 191 8 patrick patrick PROPN cana-1090 191 9 de de PROPN cana-1090 191 10 leenheer leenheer PROPN cana-1090 191 11 , , PUNCT cana-1090 191 12 discrete discrete VERB cana-1090 191 13 three three NUM cana-1090 191 14 - - PUNCT cana-1090 191 15 stage stage NOUN cana-1090 191 16 population population NOUN cana-1090 191 17 model model NOUN cana-1090 191 18 : : PUNCT cana-1090 191 19 persistence persistence NOUN cana-1090 191 20 and and CCONJ cana-1090 191 21 global global ADJ cana-1090 191 22 stability stability NOUN cana-1090 191 23 results result NOUN cana-1090 191 24 , , PUNCT cana-1090 191 25 journal journal NOUN cana-1090 191 26 of of ADP cana-1090 191 27 biological biological ADJ cana-1090 191 28 dynamics dynamic NOUN cana-1090 191 29 , , PUNCT cana-1090 191 30 2(4 2(4 NUM cana-1090 191 31 ) ) PUNCT cana-1090 191 32 , , PUNCT cana-1090 191 33 ( ( PUNCT cana-1090 191 34 2008 2008 NUM cana-1090 191 35 ) ) PUNCT cana-1090 191 36 , , PUNCT cana-1090 191 37 415 415 NUM cana-1090 191 38 427 427 NUM cana-1090 191 39 . . PUNCT cana-1090 192 1 [ [ X cana-1090 192 2 3 3 NUM cana-1090 192 3 ] ] PUNCT cana-1090 192 4 a. a. NOUN cana-1090 192 5 t. t. PROPN cana-1090 192 6 ademola ademola PROPN cana-1090 192 7 , , PUNCT cana-1090 192 8 p. p. PROPN cana-1090 192 9 o. o. PROPN cana-1090 192 10 arawomo arawomo PROPN cana-1090 192 11 , , PUNCT cana-1090 192 12 and and CCONJ cana-1090 192 13 a. a. PROPN cana-1090 192 14 s. s. PROPN cana-1090 192 15 idowu idowu PROPN cana-1090 192 16 , , PUNCT cana-1090 192 17 stability stability NOUN cana-1090 192 18 , , PUNCT cana-1090 192 19 boundedness boundedness NOUN cana-1090 192 20 and and CCONJ cana-1090 192 21 periodic periodic ADJ cana-1090 192 22 solutions solution NOUN cana-1090 192 23 to to ADP cana-1090 192 24 certain certain ADJ cana-1090 192 25 second second ADJ cana-1090 192 26 order order NOUN cana-1090 192 27 delay delay NOUN cana-1090 192 28 differential differential ADJ cana-1090 192 29 equations equation NOUN cana-1090 192 30 , , PUNCT cana-1090 192 31 proyecciones proyeccione NOUN cana-1090 192 32 ( ( PUNCT cana-1090 192 33 antofagasta antofagasta PROPN cana-1090 192 34 , , PUNCT cana-1090 192 35 on on ADP cana-1090 192 36 line line NOUN cana-1090 192 37 ) ) PUNCT cana-1090 192 38 , , PUNCT cana-1090 192 39 vol vol NOUN cana-1090 192 40 . . PROPN cana-1090 193 1 36 36 NUM cana-1090 193 2 , , PUNCT cana-1090 193 3 no no INTJ cana-1090 193 4 . . NOUN cana-1090 193 5 2 2 NUM cana-1090 193 6 , , PUNCT cana-1090 193 7 pp pp ADJ cana-1090 193 8 . . PUNCT cana-1090 194 1 257 257 NUM cana-1090 194 2 - - SYM cana-1090 194 3 282 282 NUM cana-1090 194 4 , , PUNCT cana-1090 194 5 jun jun PROPN cana-1090 194 6 . . PROPN cana-1090 194 7 2017 2017 NUM cana-1090 194 8 . . PUNCT cana-1090 195 1 [ [ X cana-1090 195 2 4 4 NUM cana-1090 195 3 ] ] X cana-1090 195 4 irfan irfan PROPN cana-1090 195 5 ali ali PROPN cana-1090 195 6 , , PUNCT cana-1090 195 7 umer umer PROPN cana-1090 195 8 saeed saeed PROPN cana-1090 195 9 , , PUNCT cana-1090 195 10 qamar qamar PROPN cana-1090 195 11 din din PROPN cana-1090 195 12 , , PUNCT cana-1090 195 13 bifurcation bifurcation NOUN cana-1090 195 14 analysis analysis NOUN cana-1090 195 15 and and CCONJ cana-1090 195 16 chaos chaos NOUN cana-1090 195 17 control control NOUN cana-1090 195 18 in in ADP cana-1090 195 19 discrete discrete ADJ cana-1090 195 20 - - PUNCT cana-1090 195 21 time time NOUN cana-1090 195 22 system system NOUN cana-1090 195 23 of of ADP cana-1090 195 24 three three NUM cana-1090 195 25 competing compete VERB cana-1090 195 26 species specie NOUN cana-1090 195 27 , , PUNCT cana-1090 195 28 arabian arabian ADJ cana-1090 195 29 journal journal NOUN cana-1090 195 30 of of ADP cana-1090 195 31 mathematics mathematic NOUN cana-1090 195 32 , , PUNCT cana-1090 195 33 8 8 NUM cana-1090 195 34 , , PUNCT cana-1090 195 35 ( ( PUNCT cana-1090 195 36 2019 2019 NUM cana-1090 195 37 ) ) PUNCT cana-1090 195 38 , , PUNCT cana-1090 195 39 1 1 NUM cana-1090 195 40 - - SYM cana-1090 195 41 14 14 NUM cana-1090 195 42 . . PUNCT cana-1090 196 1 [ [ X cana-1090 196 2 5 5 NUM cana-1090 196 3 ] ] X cana-1090 196 4 ll ll NOUN cana-1090 196 5 . . PUNCT cana-1090 197 1 alseda alseda PROPN cana-1090 197 2 , , PUNCT cana-1090 197 3 b. b. PROPN cana-1090 197 4 vidiella vidiella PROPN cana-1090 197 5 , , PUNCT cana-1090 197 6 r. r. PROPN cana-1090 197 7 sole sole PROPN cana-1090 197 8 , , PUNCT cana-1090 197 9 j.t j.t PROPN cana-1090 197 10 . . PROPN cana-1090 197 11 lazaro lazaro PROPN cana-1090 197 12 , , PUNCT cana-1090 197 13 j. j. PROPN cana-1090 197 14 sardanyes sardanyes PROPN cana-1090 197 15 , , PUNCT cana-1090 197 16 dynamics dynamic NOUN cana-1090 197 17 in in ADP cana-1090 197 18 a a DET cana-1090 197 19 ́ ́ NOUN cana-1090 197 20 time time NOUN cana-1090 197 21 - - PUNCT cana-1090 197 22 discrete discrete ADJ cana-1090 197 23 food food NOUN cana-1090 197 24 - - PUNCT cana-1090 197 25 chain chain NOUN cana-1090 197 26 model model NOUN cana-1090 197 27 with with ADP cana-1090 197 28 strong strong ADJ cana-1090 197 29 pressure pressure NOUN cana-1090 197 30 on on ADP cana-1090 197 31 preys prey NOUN cana-1090 197 32 , , PUNCT cana-1090 197 33 communications communication NOUN cana-1090 197 34 in in ADP cana-1090 197 35 nonlinear nonlinear ADJ cana-1090 197 36 science science NOUN cana-1090 197 37 and and CCONJ cana-1090 197 38 numerical numerical PROPN cana-1090 197 39 simulation simulation PROPN cana-1090 197 40 , , PUNCT cana-1090 197 41 84 84 NUM cana-1090 197 42 , , PUNCT cana-1090 197 43 ( ( PUNCT cana-1090 197 44 2020 2020 NUM cana-1090 197 45 ) ) PUNCT cana-1090 197 46 , , PUNCT cana-1090 197 47 105187 105187 NUM cana-1090 197 48 . . PUNCT cana-1090 198 1 [ [ X cana-1090 198 2 6 6 NUM cana-1090 198 3 ] ] PUNCT cana-1090 198 4 j. j. PROPN cana-1090 198 5 leo leo PROPN cana-1090 198 6 amalraj amalraj PROPN cana-1090 198 7 , , PUNCT cana-1090 198 8 m.maria m.maria NOUN cana-1090 198 9 susai susai NOUN cana-1090 198 10 manuel manuel NOUN cana-1090 198 11 , , PUNCT cana-1090 198 12 dumitru dumitru PROPN cana-1090 198 13 baleanu baleanu NOUN cana-1090 198 14 and and CCONJ cana-1090 198 15 d.s.dilip d.s.dilip NOUN cana-1090 198 16 , , PUNCT cana-1090 198 17 global global ADJ cana-1090 198 18 stability stability NOUN cana-1090 198 19 , , PUNCT cana-1090 198 20 periodicity periodicity NOUN cana-1090 198 21 and and CCONJ cana-1090 198 22 bifurcation bifurcation NOUN cana-1090 198 23 analysis analysis NOUN cana-1090 198 24 of of ADP cana-1090 198 25 a a DET cana-1090 198 26 difference difference NOUN cana-1090 198 27 equation equation NOUN cana-1090 198 28 , , PUNCT cana-1090 198 29 aip aip PROPN cana-1090 198 30 advances advance NOUN cana-1090 198 31 , , PUNCT cana-1090 198 32 13 13 NUM cana-1090 198 33 , , PUNCT cana-1090 198 34 ( ( PUNCT cana-1090 198 35 2023 2023 NUM cana-1090 198 36 ) ) PUNCT cana-1090 198 37 , , PUNCT cana-1090 198 38 015116 015116 NUM cana-1090 198 39 . . PUNCT cana-1090 199 1 [ [ X cana-1090 199 2 7 7 X cana-1090 199 3 ] ] X cana-1090 199 4 ritwick ritwick NOUN cana-1090 199 5 banerjee banerjee PROPN cana-1090 199 6 , , PUNCT cana-1090 199 7 pritha pritha PROPN cana-1090 199 8 das das PROPN cana-1090 199 9 , , PUNCT cana-1090 199 10 debasis debasis NOUN cana-1090 199 11 mukherjee mukherjee NOUN cana-1090 199 12 , , PUNCT cana-1090 199 13 stability stability NOUN cana-1090 199 14 and and CCONJ cana-1090 199 15 permanence permanence NOUN cana-1090 199 16 of of ADP cana-1090 199 17 a a DET cana-1090 199 18 discrete discrete ADJ cana-1090 199 19 - - PUNCT cana-1090 199 20 time time NOUN cana-1090 199 21 two two NUM cana-1090 199 22 - - PUNCT cana-1090 199 23 prey prey NOUN cana-1090 199 24 one one NUM cana-1090 199 25 - - PUNCT cana-1090 199 26 predator predator NOUN cana-1090 199 27 system system NOUN cana-1090 199 28 with with ADP cana-1090 199 29 holling holle VERB cana-1090 199 30 type type NOUN cana-1090 199 31 - - PUNCT cana-1090 199 32 iii iii NOUN cana-1090 199 33 functional functional ADJ cana-1090 199 34 response response NOUN cana-1090 199 35 , , PUNCT cana-1090 199 36 chaos chaos NOUN cana-1090 199 37 , , PUNCT cana-1090 199 38 solitons soliton NOUN cana-1090 199 39 and and CCONJ cana-1090 199 40 fractals fractal NOUN cana-1090 199 41 , , PUNCT cana-1090 199 42 117 117 NUM cana-1090 199 43 , , PUNCT cana-1090 199 44 ( ( PUNCT cana-1090 199 45 2001 2001 NUM cana-1090 199 46 ) ) PUNCT cana-1090 199 47 , , PUNCT cana-1090 199 48 240 240 NUM cana-1090 199 49 248 248 NUM cana-1090 199 50 . . PUNCT cana-1090 200 1 [ [ X cana-1090 200 2 8 8 NUM cana-1090 200 3 ] ] X cana-1090 200 4 c.a c.a PROPN cana-1090 200 5 . . PROPN cana-1090 200 6 clark clark PROPN cana-1090 200 7 , , PUNCT cana-1090 200 8 m. m. PROPN cana-1090 200 9 r. r. PROPN cana-1090 200 10 s s PROPN cana-1090 200 11 kulenovic kulenovic PROPN cana-1090 200 12 and and CCONJ cana-1090 200 13 james james PROPN cana-1090 200 14 f. f. PROPN cana-1090 200 15 selgrade selgrade PROPN cana-1090 200 16 , , PUNCT cana-1090 200 17 on on ADP cana-1090 200 18 a a DET cana-1090 200 19 sytem sytem NOUN cana-1090 200 20 of of ADP cana-1090 200 21 rational rational ADJ cana-1090 200 22 difference difference NOUN cana-1090 200 23 equations equation NOUN cana-1090 200 24 , , PUNCT cana-1090 200 25 journal journal NOUN cana-1090 200 26 of of ADP cana-1090 200 27 difference difference NOUN cana-1090 200 28 equations equation NOUN cana-1090 200 29 and and CCONJ cana-1090 200 30 applications application NOUN cana-1090 200 31 , , PUNCT cana-1090 200 32 11(7 11(7 NUM cana-1090 200 33 ) ) PUNCT cana-1090 200 34 , , PUNCT cana-1090 200 35 ( ( PUNCT cana-1090 200 36 2005 2005 NUM cana-1090 200 37 ) ) PUNCT cana-1090 200 38 , , PUNCT cana-1090 200 39 565 565 NUM cana-1090 200 40 580 580 NUM cana-1090 200 41 . . PUNCT cana-1090 201 1 [ [ X cana-1090 201 2 9 9 NUM cana-1090 201 3 ] ] PUNCT cana-1090 201 4 elias elias PROPN cana-1090 201 5 camouzis camouzis PROPN cana-1090 201 6 and and CCONJ cana-1090 201 7 g. g. PROPN cana-1090 201 8 ladas ladas PROPN cana-1090 201 9 , , PUNCT cana-1090 201 10 dynamics dynamic NOUN cana-1090 201 11 of of ADP cana-1090 201 12 third third ADJ cana-1090 201 13 order order NOUN cana-1090 201 14 rational rational ADJ cana-1090 201 15 difference difference NOUN cana-1090 201 16 equations equation NOUN cana-1090 201 17 with with ADP cana-1090 201 18 open open ADJ cana-1090 201 19 problems problem NOUN cana-1090 201 20 , , PUNCT cana-1090 201 21 chapman chapman PROPN cana-1090 201 22 & & CCONJ cana-1090 201 23 hall hall PROPN cana-1090 201 24 / / SYM cana-1090 201 25 crc crc PROPN cana-1090 201 26 , , PUNCT cana-1090 201 27 2007 2007 NUM cana-1090 201 28 . . PUNCT cana-1090 202 1 [ [ X cana-1090 202 2 10 10 NUM cana-1090 202 3 ] ] X cana-1090 202 4 d d X cana-1090 202 5 s s X cana-1090 202 6 dilip dilip PROPN cana-1090 202 7 and and CCONJ cana-1090 202 8 smitha smitha PROPN cana-1090 202 9 mary mary PROPN cana-1090 202 10 mathew mathew PROPN cana-1090 202 11 , , PUNCT cana-1090 202 12 dynamics dynamic NOUN cana-1090 202 13 of of ADP cana-1090 202 14 a a DET cana-1090 202 15 second second ADJ cana-1090 202 16 order order NOUN cana-1090 202 17 nonlinear nonlinear ADJ cana-1090 202 18 difference difference NOUN cana-1090 202 19 system system NOUN cana-1090 202 20 with with ADP cana-1090 202 21 exponents exponent NOUN cana-1090 202 22 , , PUNCT cana-1090 202 23 journal journal NOUN cana-1090 202 24 of of ADP cana-1090 202 25 the the DET cana-1090 202 26 egyptian egyptian PROPN cana-1090 202 27 mathematical mathematical PROPN cana-1090 202 28 society society NOUN cana-1090 202 29 , , PUNCT cana-1090 202 30 ( ( PUNCT cana-1090 202 31 2021 2021 NUM cana-1090 202 32 ) ) PUNCT cana-1090 202 33 29:10 29:10 NUM cana-1090 202 34 . . PUNCT cana-1090 203 1 [ [ X cana-1090 203 2 11 11 NUM cana-1090 203 3 ] ] X cana-1090 203 4 d d X cana-1090 203 5 s s X cana-1090 203 6 dilip dilip PROPN cana-1090 203 7 and and CCONJ cana-1090 203 8 smitha smitha PROPN cana-1090 203 9 mary mary PROPN cana-1090 203 10 mathew mathew PROPN cana-1090 203 11 , , PUNCT cana-1090 203 12 stability stability NOUN cana-1090 203 13 analysis analysis NOUN cana-1090 203 14 of of ADP cana-1090 203 15 a a DET cana-1090 203 16 time time NOUN cana-1090 203 17 varying vary VERB cana-1090 203 18 population population NOUN cana-1090 203 19 model model NOUN cana-1090 203 20 without without ADP cana-1090 203 21 migration migration NOUN cana-1090 203 22 , , PUNCT cana-1090 203 23 journal journal NOUN cana-1090 203 24 of of ADP cana-1090 203 25 difference difference NOUN cana-1090 203 26 equations equation NOUN cana-1090 203 27 and and CCONJ cana-1090 203 28 applications application NOUN cana-1090 203 29 , , PUNCT cana-1090 203 30 27(11 27(11 NUM cana-1090 203 31 ) ) PUNCT cana-1090 203 32 , , PUNCT cana-1090 203 33 ( ( PUNCT cana-1090 203 34 2021 2021 NUM cana-1090 203 35 ) ) PUNCT cana-1090 203 36 , , PUNCT cana-1090 203 37 1525 1525 NUM cana-1090 203 38 - - SYM cana-1090 203 39 1536 1536 NUM cana-1090 203 40 . . PUNCT cana-1090 204 1 [ [ X cana-1090 204 2 12 12 NUM cana-1090 204 3 ] ] PUNCT cana-1090 204 4 d.s.dilip d.s.dilip NOUN cana-1090 204 5 and and CCONJ cana-1090 204 6 tony tony ADJ cana-1090 204 7 philip philip PROPN cana-1090 204 8 , , PUNCT cana-1090 204 9 stability stability NOUN cana-1090 204 10 and and CCONJ cana-1090 204 11 bifurcation bifurcation NOUN cana-1090 204 12 analysis analysis NOUN cana-1090 204 13 of of ADP cana-1090 204 14 two two NUM cana-1090 204 15 spatial spatial ADJ cana-1090 204 16 population population NOUN cana-1090 204 17 dynamics dynamic NOUN cana-1090 204 18 models model NOUN cana-1090 204 19 , , PUNCT cana-1090 204 20 international international ADJ cana-1090 204 21 journal journal NOUN cana-1090 204 22 of of ADP cana-1090 204 23 applied applied ADJ cana-1090 204 24 and and CCONJ cana-1090 204 25 computational computational ADJ cana-1090 204 26 mathematics mathematic NOUN cana-1090 204 27 , , PUNCT cana-1090 204 28 8 8 NUM cana-1090 204 29 , , PUNCT cana-1090 204 30 108(2022 108(2022 NUM cana-1090 204 31 ) ) PUNCT cana-1090 204 32 . . PUNCT cana-1090 205 1 [ [ X cana-1090 205 2 13 13 NUM cana-1090 205 3 ] ] X cana-1090 205 4 n. n. NOUN cana-1090 205 5 fotiades fotiade NOUN cana-1090 205 6 and and CCONJ cana-1090 205 7 g. g. PROPN cana-1090 205 8 papaschinopoulos papaschinopoulos PROPN cana-1090 205 9 , , PUNCT cana-1090 205 10 existence existence NOUN cana-1090 205 11 , , PUNCT cana-1090 205 12 uniqueness uniqueness NOUN cana-1090 205 13 and and CCONJ cana-1090 205 14 attractivity attractivity NOUN cana-1090 205 15 of of ADP cana-1090 205 16 prime prime ADJ cana-1090 205 17 period period NOUN cana-1090 205 18 two two NUM cana-1090 205 19 solution solution NOUN cana-1090 205 20 for for ADP cana-1090 205 21 a a DET cana-1090 205 22 difference difference NOUN cana-1090 205 23 equation equation NOUN cana-1090 205 24 of of ADP cana-1090 205 25 exponential exponential ADJ cana-1090 205 26 form form NOUN cana-1090 205 27 , , PUNCT cana-1090 205 28 applied apply VERB cana-1090 205 29 mathematics mathematic NOUN cana-1090 205 30 and and CCONJ cana-1090 205 31 computation computation NOUN cana-1090 205 32 , , PUNCT cana-1090 205 33 218 218 NUM cana-1090 205 34 ( ( PUNCT cana-1090 205 35 2012 2012 NUM cana-1090 205 36 ) ) PUNCT cana-1090 205 37 , , PUNCT cana-1090 205 38 11648 11648 NUM cana-1090 205 39 11653 11653 NUM cana-1090 205 40 . . PUNCT cana-1090 206 1 [ [ X cana-1090 206 2 14 14 NUM cana-1090 206 3 ] ] X cana-1090 206 4 m. m. NOUN cana-1090 206 5 kulenovic kulenovic PROPN cana-1090 206 6 and and CCONJ cana-1090 206 7 g. g. PROPN cana-1090 206 8 ladas ladas PROPN cana-1090 206 9 , , PUNCT cana-1090 206 10 dynamics dynamic NOUN cana-1090 206 11 of of ADP cana-1090 206 12 second second ADJ cana-1090 206 13 order order NOUN cana-1090 206 14 rational rational ADJ cana-1090 206 15 difference difference NOUN cana-1090 206 16 equations equation NOUN cana-1090 206 17 , , PUNCT cana-1090 206 18 chapman chapman PROPN cana-1090 206 19 & & CCONJ cana-1090 206 20 hall hall PROPN cana-1090 206 21 / / SYM cana-1090 206 22 crc crc PROPN cana-1090 206 23 , , PUNCT cana-1090 206 24 2002 2002 NUM cana-1090 206 25 . . PUNCT cana-1090 207 1 [ [ X cana-1090 207 2 15 15 NUM cana-1090 207 3 ] ] X cana-1090 207 4 g. g. PROPN cana-1090 207 5 papaschinopoulos papaschinopoulos PROPN cana-1090 207 6 , , PUNCT cana-1090 207 7 m. m. NOUN cana-1090 207 8 a. a. PROPN cana-1090 207 9 radin radin PROPN cana-1090 207 10 and and CCONJ cana-1090 207 11 c.j c.j PROPN cana-1090 207 12 . . PROPN cana-1090 207 13 schinas schinas PROPN cana-1090 207 14 , , PUNCT cana-1090 207 15 on on ADP cana-1090 207 16 the the DET cana-1090 207 17 system system NOUN cana-1090 207 18 of of ADP cana-1090 207 19 two two NUM cana-1090 207 20 difference difference NOUN cana-1090 207 21 equations equation NOUN cana-1090 207 22 of of ADP cana-1090 207 23 exponential exponential ADJ cana-1090 207 24 form form NOUN cana-1090 207 25 , , PUNCT cana-1090 207 26 mathematical mathematical ADJ cana-1090 207 27 and and CCONJ cana-1090 207 28 computer computer NOUN cana-1090 207 29 modelling modelling NOUN cana-1090 207 30 , , PUNCT cana-1090 207 31 54 54 NUM cana-1090 207 32 ( ( PUNCT cana-1090 207 33 2011 2011 NUM cana-1090 207 34 ) ) PUNCT cana-1090 207 35 , , PUNCT cana-1090 207 36 2969 2969 NUM cana-1090 207 37 2977 2977 NUM cana-1090 207 38 . . PUNCT cana-1090 208 1 [ [ X cana-1090 208 2 16 16 NUM cana-1090 208 3 ] ] X cana-1090 208 4 g. g. PROPN cana-1090 208 5 papaschinopoulos papaschinopoulos PROPN cana-1090 208 6 and and CCONJ cana-1090 208 7 c.j c.j PROPN cana-1090 208 8 . . PROPN cana-1090 208 9 schinas schinas PROPN cana-1090 208 10 , , PUNCT cana-1090 208 11 on on ADP cana-1090 208 12 the the DET cana-1090 208 13 dynamics dynamic NOUN cana-1090 208 14 of of ADP cana-1090 208 15 two two NUM cana-1090 208 16 exponential exponential ADJ cana-1090 208 17 type type NOUN cana-1090 208 18 systems system NOUN cana-1090 208 19 of of ADP cana-1090 208 20 difference difference NOUN cana-1090 208 21 equations equation NOUN cana-1090 208 22 , , PUNCT cana-1090 208 23 computers computer NOUN cana-1090 208 24 and and CCONJ cana-1090 208 25 mathematics mathematic NOUN cana-1090 208 26 with with ADP cana-1090 208 27 applications application NOUN cana-1090 208 28 , , PUNCT cana-1090 208 29 64 64 NUM cana-1090 208 30 ( ( PUNCT cana-1090 208 31 2012 2012 NUM cana-1090 208 32 ) ) PUNCT cana-1090 208 33 , , PUNCT cana-1090 208 34 2326 2326 NUM cana-1090 208 35 2334 2334 NUM cana-1090 208 36 . . PUNCT cana-1090 209 1 [ [ X cana-1090 209 2 17 17 NUM cana-1090 209 3 ] ] X cana-1090 209 4 g. g. PROPN cana-1090 209 5 papaschinopoulos papaschinopoulos PROPN cana-1090 209 6 , , PUNCT cana-1090 209 7 g. g. PROPN cana-1090 209 8 ellina ellina PROPN cana-1090 209 9 and and CCONJ cana-1090 209 10 k.b.papadopoulos k.b.papadopoulo NOUN cana-1090 209 11 , , PUNCT cana-1090 209 12 asymptotic asymptotic ADJ cana-1090 209 13 behavior behavior NOUN cana-1090 209 14 of of ADP cana-1090 209 15 the the DET cana-1090 209 16 positive positive ADJ cana-1090 209 17 solutions solution NOUN cana-1090 209 18 of of ADP cana-1090 209 19 an an DET cana-1090 209 20 exponential exponential ADJ cana-1090 209 21 type type NOUN cana-1090 209 22 system system NOUN cana-1090 209 23 of of ADP cana-1090 209 24 difference difference NOUN cana-1090 209 25 equations equation NOUN cana-1090 209 26 , , PUNCT cana-1090 209 27 applied apply VERB cana-1090 209 28 mathematics mathematic NOUN cana-1090 209 29 and and CCONJ cana-1090 209 30 computation computation NOUN cana-1090 209 31 , , PUNCT cana-1090 209 32 245 245 NUM cana-1090 209 33 ( ( PUNCT cana-1090 209 34 2014 2014 NUM cana-1090 209 35 ) ) PUNCT cana-1090 209 36 , , PUNCT cana-1090 209 37 181 181 NUM cana-1090 209 38 190 190 NUM cana-1090 209 39 . . PUNCT cana-1090 210 1 [ [ X cana-1090 210 2 18 18 NUM cana-1090 210 3 ] ] X cana-1090 210 4 g. g. PROPN cana-1090 210 5 papaschinopoulos papaschinopoulos PROPN cana-1090 210 6 , , PUNCT cana-1090 210 7 c.j c.j PROPN cana-1090 210 8 . . PROPN cana-1090 210 9 schinas schinas PROPN cana-1090 210 10 and and CCONJ cana-1090 210 11 g. g. PROPN cana-1090 210 12 ellina ellina PROPN cana-1090 210 13 , , PUNCT cana-1090 210 14 on on ADP cana-1090 210 15 the the DET cana-1090 210 16 dynamics dynamic NOUN cana-1090 210 17 of of ADP cana-1090 210 18 the the DET cana-1090 210 19 solutions solution NOUN cana-1090 210 20 of of ADP cana-1090 210 21 a a DET cana-1090 210 22 biological biological ADJ cana-1090 210 23 model model NOUN cana-1090 210 24 , , PUNCT cana-1090 210 25 journal journal NOUN cana-1090 210 26 of of ADP cana-1090 210 27 difference difference NOUN cana-1090 210 28 equations equation NOUN cana-1090 210 29 and and CCONJ cana-1090 210 30 applications application NOUN cana-1090 210 31 , , PUNCT cana-1090 210 32 20(5 20(5 NOUN cana-1090 210 33 6 6 NUM cana-1090 210 34 ) ) PUNCT cana-1090 210 35 , , PUNCT cana-1090 210 36 ( ( PUNCT cana-1090 210 37 2014 2014 NUM cana-1090 210 38 ) ) PUNCT cana-1090 210 39 , , PUNCT cana-1090 210 40 694 694 NUM cana-1090 210 41 705 705 NUM cana-1090 210 42 . . PUNCT cana-1090 211 1 [ [ X cana-1090 211 2 19 19 NUM cana-1090 211 3 ] ] X cana-1090 211 4 g. g. PROPN cana-1090 211 5 papaschinopoulos papaschinopoulos PROPN cana-1090 211 6 , , PUNCT cana-1090 211 7 n. n. PROPN cana-1090 211 8 fotiades fotiades PROPN cana-1090 211 9 and and CCONJ cana-1090 211 10 c.j c.j PROPN cana-1090 211 11 . . PROPN cana-1090 211 12 schinas schinas PROPN cana-1090 211 13 , , PUNCT cana-1090 211 14 on on ADP cana-1090 211 15 a a DET cana-1090 211 16 system system NOUN cana-1090 211 17 of of ADP cana-1090 211 18 difference difference NOUN cana-1090 211 19 equations equation NOUN cana-1090 211 20 including include VERB cana-1090 211 21 negative negative ADJ cana-1090 211 22 exponential exponential ADJ cana-1090 211 23 terms term NOUN cana-1090 211 24 , , PUNCT cana-1090 211 25 journal journal NOUN cana-1090 211 26 of of ADP cana-1090 211 27 difference difference NOUN cana-1090 211 28 equations equation NOUN cana-1090 211 29 and and CCONJ cana-1090 211 30 applications application NOUN cana-1090 211 31 , , PUNCT cana-1090 211 32 20(5 20(5 NOUN cana-1090 211 33 6 6 NUM cana-1090 211 34 ) ) PUNCT cana-1090 211 35 , , PUNCT cana-1090 211 36 ( ( PUNCT cana-1090 211 37 2014 2014 NUM cana-1090 211 38 ) ) PUNCT cana-1090 211 39 , , PUNCT cana-1090 211 40 717 717 NUM cana-1090 211 41 732 732 NUM cana-1090 211 42 . . PUNCT cana-1090 212 1 [ [ X cana-1090 212 2 20 20 NUM cana-1090 212 3 ] ] X cana-1090 212 4 n. n. NOUN cana-1090 212 5 psarros psarros PROPN cana-1090 212 6 and and CCONJ cana-1090 212 7 g. g. PROPN cana-1090 212 8 papaschinopoulos papaschinopoulos PROPN cana-1090 212 9 , , PUNCT cana-1090 212 10 long long ADJ cana-1090 212 11 - - PUNCT cana-1090 212 12 term term NOUN cana-1090 212 13 behavior behavior NOUN cana-1090 212 14 of of ADP cana-1090 212 15 positive positive ADJ cana-1090 212 16 solutions solution NOUN cana-1090 212 17 of of ADP cana-1090 212 18 an an DET cana-1090 212 19 exponentially exponentially ADV cana-1090 212 20 self self NOUN cana-1090 212 21 - - PUNCT cana-1090 212 22 regulating regulate VERB cana-1090 212 23 system system NOUN cana-1090 212 24 of of ADP cana-1090 212 25 difference difference NOUN cana-1090 212 26 equations equation NOUN cana-1090 212 27 , , PUNCT cana-1090 212 28 international international ADJ cana-1090 212 29 journal journal NOUN cana-1090 212 30 of of ADP cana-1090 212 31 biomathematics biomathematic NOUN cana-1090 212 32 , , PUNCT cana-1090 212 33 10(3 10(3 NUM cana-1090 212 34 ) ) PUNCT cana-1090 212 35 , , PUNCT cana-1090 212 36 ( ( PUNCT cana-1090 212 37 2017 2017 NUM cana-1090 212 38 ) ) PUNCT cana-1090 212 39 , , PUNCT cana-1090 212 40 1750045 1750045 NUM cana-1090 212 41 . . PUNCT cana-1090 213 1 [ [ X cana-1090 213 2 21 21 NUM cana-1090 213 3 ] ] X cana-1090 213 4 muhammad muhammad PROPN cana-1090 213 5 naeem naeem PROPN cana-1090 213 6 qureshi qureshi PROPN cana-1090 213 7 , , PUNCT cana-1090 213 8 a. a. NOUN cana-1090 213 9 qadeer qadeer PROPN cana-1090 213 10 khan khan PROPN cana-1090 213 11 and and CCONJ cana-1090 213 12 qamar qamar PROPN cana-1090 213 13 din din PROPN cana-1090 213 14 , , PUNCT cana-1090 213 15 asymptotic asymptotic ADJ cana-1090 213 16 behavior behavior NOUN cana-1090 213 17 of of ADP cana-1090 213 18 a a DET cana-1090 213 19 nicholson nicholson PROPN cana-1090 213 20 - - PUNCT cana-1090 213 21 bailey bailey PROPN cana-1090 213 22 model model PROPN cana-1090 213 23 , , PUNCT cana-1090 213 24 advances advance NOUN cana-1090 213 25 in in ADP cana-1090 213 26 difference difference NOUN cana-1090 213 27 equations equation NOUN cana-1090 213 28 , , PUNCT cana-1090 213 29 ( ( PUNCT cana-1090 213 30 2014 2014 NUM cana-1090 213 31 ) ) PUNCT cana-1090 213 32 , , PUNCT cana-1090 213 33 2014:62 2014:62 NUM cana-1090 213 34 . . PUNCT cana-1090 214 1 [ [ X cana-1090 214 2 22 22 NUM cana-1090 214 3 ] ] X cana-1090 214 4 samra samra NOUN cana-1090 214 5 moranjkic moranjkic PROPN cana-1090 214 6 and and CCONJ cana-1090 214 7 zehra zehra PROPN cana-1090 214 8 nurkanovic nurkanovic PROPN cana-1090 214 9 , , PUNCT cana-1090 214 10 basins basin NOUN cana-1090 214 11 of of ADP cana-1090 214 12 attraction attraction NOUN cana-1090 214 13 of of ADP cana-1090 214 14 certain certain ADJ cana-1090 214 15 rational rational ADJ cana-1090 214 16 anti anti ADJ cana-1090 214 17 - - ADJ cana-1090 214 18 competitive competitive ADJ cana-1090 214 19 system system NOUN cana-1090 214 20 of of ADP cana-1090 214 21 difference difference NOUN cana-1090 214 22 equations equation NOUN cana-1090 214 23 in in ADP cana-1090 214 24 the the DET cana-1090 214 25 plane plane NOUN cana-1090 214 26 , , PUNCT cana-1090 214 27 advances advance NOUN cana-1090 214 28 in in ADP cana-1090 214 29 difference difference NOUN cana-1090 214 30 equations equation NOUN cana-1090 214 31 , , PUNCT cana-1090 214 32 ( ( PUNCT cana-1090 214 33 2012 2012 NUM cana-1090 214 34 ) ) PUNCT cana-1090 214 35 , , PUNCT cana-1090 214 36 2012:153 2012:153 X cana-1090 214 37 . . PUNCT cana-1090 215 1 [ [ X cana-1090 215 2 23 23 NUM cana-1090 215 3 ] ] X cana-1090 215 4 hui hui PROPN cana-1090 215 5 feng feng PROPN cana-1090 215 6 , , PUNCT cana-1090 215 7 huili huili PROPN cana-1090 215 8 ma ma PROPN cana-1090 215 9 and and CCONJ cana-1090 215 10 wandi wandi VERB cana-1090 215 11 ding ding NOUN cana-1090 215 12 , , PUNCT cana-1090 215 13 global global ADJ cana-1090 215 14 asymptotic asymptotic ADJ cana-1090 215 15 behavior behavior NOUN cana-1090 215 16 of of ADP cana-1090 215 17 positive positive ADJ cana-1090 215 18 solutions solution NOUN cana-1090 215 19 for for ADP cana-1090 215 20 exponential exponential ADJ cana-1090 215 21 form form NOUN cana-1090 215 22 difference difference NOUN cana-1090 215 23 equations equation NOUN cana-1090 215 24 with with ADP cana-1090 215 25 three three NUM cana-1090 215 26 parameters parameter NOUN cana-1090 215 27 , , PUNCT cana-1090 215 28 journal journal NOUN cana-1090 215 29 of of ADP cana-1090 215 30 applied apply VERB cana-1090 215 31 analysis analysis NOUN cana-1090 215 32 and and CCONJ cana-1090 215 33 computation computation NOUN cana-1090 215 34 , , PUNCT cana-1090 215 35 6(3 6(3 NUM cana-1090 215 36 ) ) PUNCT cana-1090 215 37 , , PUNCT cana-1090 215 38 ( ( PUNCT cana-1090 215 39 2016 2016 NUM cana-1090 215 40 ) ) PUNCT cana-1090 215 41 , , PUNCT cana-1090 215 42 600 600 NUM cana-1090 215 43 606 606 NUM cana-1090 215 44 . . PUNCT cana-1090 216 1 [ [ X cana-1090 216 2 24 24 NUM cana-1090 216 3 ] ] X cana-1090 216 4 q q X cana-1090 216 5 din din PROPN cana-1090 216 6 , , PUNCT cana-1090 216 7 mn mn PROPN cana-1090 216 8 qureshi qureshi PROPN cana-1090 216 9 and and CCONJ cana-1090 216 10 a a DET cana-1090 216 11 qadeer qadeer NOUN cana-1090 216 12 khan khan PROPN cana-1090 216 13 , , PUNCT cana-1090 216 14 dynamics dynamic NOUN cana-1090 216 15 of of ADP cana-1090 216 16 a a DET cana-1090 216 17 fourth fourth ADJ cana-1090 216 18 - - PUNCT cana-1090 216 19 order order NOUN cana-1090 216 20 system system NOUN cana-1090 216 21 of of ADP cana-1090 216 22 rational rational ADJ cana-1090 216 23 difference difference NOUN cana-1090 216 24 equations equation NOUN cana-1090 216 25 , , PUNCT cana-1090 216 26 advances advance NOUN cana-1090 216 27 in in ADP cana-1090 216 28 difference difference NOUN cana-1090 216 29 equations equation NOUN cana-1090 216 30 , , PUNCT cana-1090 216 31 ( ( PUNCT cana-1090 216 32 2012 2012 NUM cana-1090 216 33 ) ) PUNCT cana-1090 216 34 , , PUNCT cana-1090 216 35 2012:215 2012:215 NOUN cana-1090 216 36 . . PUNCT cana-1090 217 1 [ [ X cana-1090 217 2 25 25 NUM cana-1090 217 3 ] ] PUNCT cana-1090 217 4 kocic kocic PROPN cana-1090 217 5 vl vl PROPN cana-1090 217 6 and and CCONJ cana-1090 217 7 ladas ladas PROPN cana-1090 217 8 g g PROPN cana-1090 217 9 , , PUNCT cana-1090 217 10 global global ADJ cana-1090 217 11 behavior behavior NOUN cana-1090 217 12 of of ADP cana-1090 217 13 nonlinear nonlinear ADJ cana-1090 217 14 difference difference NOUN cana-1090 217 15 equations equation NOUN cana-1090 217 16 of of ADP cana-1090 217 17 higher high ADJ cana-1090 217 18 order order NOUN cana-1090 217 19 with with ADP cana-1090 217 20 applications application NOUN cana-1090 217 21 , , PUNCT cana-1090 217 22 kluwer kluwer NOUN cana-1090 217 23 academic academic ADJ cana-1090 217 24 publishers publisher NOUN cana-1090 217 25 : : PUNCT cana-1090 217 26 dordrecht dordrecht PROPN cana-1090 217 27 / / SYM cana-1090 217 28 boston boston PROPN cana-1090 217 29 / / SYM cana-1090 217 30 london london PROPN cana-1090 217 31 , , PUNCT cana-1090 217 32 1993 1993 NUM cana-1090 217 33 . . PUNCT cana-1090 218 1 communications communication NOUN cana-1090 218 2 on on ADP cana-1090 218 3 applied apply VERB cana-1090 218 4 nonlinear nonlinear ADJ cana-1090 218 5 analysis analysis NOUN cana-1090 218 6 issn issn NOUN cana-1090 218 7 : : PUNCT cana-1090 218 8 1074 1074 NUM cana-1090 218 9 - - PUNCT cana-1090 218 10 133x 133x NUM cana-1090 218 11 vol vol NOUN cana-1090 218 12 31 31 NUM cana-1090 218 13 no no NOUN cana-1090 218 14 . . PUNCT cana-1090 219 1 5s 5s NUM cana-1090 219 2 ( ( PUNCT cana-1090 219 3 2024 2024 NUM cana-1090 219 4 ) ) PUNCT cana-1090 219 5 561 561 NUM cana-1090 219 6 https://internationalpubls.com https://internationalpubls.com X cana-1090 220 1 [ [ X cana-1090 220 2 26 26 NUM cana-1090 220 3 ] ] X cana-1090 220 4 e.a.grove e.a.grove NOUN cana-1090 220 5 and and CCONJ cana-1090 220 6 ladas ladas PROPN cana-1090 220 7 g g PROPN cana-1090 220 8 , , PUNCT cana-1090 220 9 periodicities periodicity NOUN cana-1090 220 10 in in ADP cana-1090 220 11 nonlinear nonlinear ADJ cana-1090 220 12 difference difference NOUN cana-1090 220 13 equations equation NOUN cana-1090 220 14 , , PUNCT cana-1090 220 15 chapman chapman PROPN cana-1090 220 16 & & CCONJ cana-1090 220 17 hall hall PROPN cana-1090 220 18 / / SYM cana-1090 220 19 crc crc PROPN cana-1090 220 20 , , PUNCT cana-1090 220 21 2005 2005 NUM cana-1090 220 22 . . PUNCT cana-1090 221 1 [ [ X cana-1090 221 2 27 27 NUM cana-1090 221 3 ] ] X cana-1090 221 4 qamar qamar PROPN cana-1090 221 5 din din PROPN cana-1090 221 6 , , PUNCT cana-1090 221 7 complexity complexity NOUN cana-1090 221 8 and and CCONJ cana-1090 221 9 chaos chaos NOUN cana-1090 221 10 control control NOUN cana-1090 221 11 in in ADP cana-1090 221 12 a a DET cana-1090 221 13 discrete discrete ADJ cana-1090 221 14 - - PUNCT cana-1090 221 15 time time NOUN cana-1090 221 16 prey prey NOUN cana-1090 221 17 - - PUNCT cana-1090 221 18 predator predator NOUN cana-1090 221 19 model model NOUN cana-1090 221 20 , , PUNCT cana-1090 221 21 communications communication NOUN cana-1090 221 22 in in ADP cana-1090 221 23 nonlinear nonlinear ADJ cana-1090 221 24 science science NOUN cana-1090 221 25 and and CCONJ cana-1090 221 26 numerical numerical PROPN cana-1090 221 27 simulation simulation PROPN cana-1090 221 28 , , PUNCT cana-1090 221 29 49 49 NUM cana-1090 221 30 ( ( PUNCT cana-1090 221 31 2017 2017 NUM cana-1090 221 32 ) ) PUNCT cana-1090 221 33 , , PUNCT cana-1090 221 34 113 113 NUM cana-1090 221 35 134 134 NUM cana-1090 221 36 . . PUNCT cana-1090 222 1 [ [ X cana-1090 222 2 28 28 NUM cana-1090 222 3 ] ] X cana-1090 222 4 guichen guichen NOUN cana-1090 222 5 lu lu PROPN cana-1090 222 6 and and CCONJ cana-1090 222 7 zhengyi zhengyi PROPN cana-1090 222 8 lu lu PROPN cana-1090 222 9 , , PUNCT cana-1090 222 10 non non ADJ cana-1090 222 11 - - NOUN cana-1090 222 12 permanence permanence NOUN cana-1090 222 13 for for ADP cana-1090 222 14 three three NUM cana-1090 222 15 - - PUNCT cana-1090 222 16 species species NOUN cana-1090 222 17 lotka lotka PROPN cana-1090 222 18 - - PUNCT cana-1090 222 19 volterra volterra PROPN cana-1090 222 20 cooperative cooperative ADJ cana-1090 222 21 difference difference NOUN cana-1090 222 22 systems system NOUN cana-1090 222 23 , , PUNCT cana-1090 222 24 advances advance NOUN cana-1090 222 25 in in ADP cana-1090 222 26 difference difference NOUN cana-1090 222 27 equations equation NOUN cana-1090 222 28 , , PUNCT cana-1090 222 29 ( ( PUNCT cana-1090 222 30 2017 2017 NUM cana-1090 222 31 ) ) PUNCT cana-1090 222 32 , , PUNCT cana-1090 222 33 2017:152 2017:152 NUM cana-1090 222 34 . . PUNCT cana-1090 223 1 [ [ X cana-1090 223 2 29 29 NUM cana-1090 223 3 ] ] PUNCT cana-1090 223 4 sibi sibi X cana-1090 223 5 c. c. PROPN cana-1090 223 6 babu babu PROPN cana-1090 223 7 , , PUNCT cana-1090 223 8 d.s.dilip d.s.dilip NOUN cana-1090 223 9 and and CCONJ cana-1090 223 10 smitha smitha PROPN cana-1090 223 11 mary mary PROPN cana-1090 223 12 mathew mathew PROPN cana-1090 223 13 , , PUNCT cana-1090 223 14 behavior behavior NOUN cana-1090 223 15 of of ADP cana-1090 223 16 solutions solution NOUN cana-1090 223 17 of of ADP cana-1090 223 18 a a DET cana-1090 223 19 discrete discrete ADJ cana-1090 223 20 population population NOUN cana-1090 223 21 model model NOUN cana-1090 223 22 with with ADP cana-1090 223 23 mutualistic mutualistic ADJ cana-1090 223 24 interaction interaction NOUN cana-1090 223 25 , , PUNCT cana-1090 223 26 computational computational ADJ cana-1090 223 27 and and CCONJ cana-1090 223 28 mathematical mathematical ADJ cana-1090 223 29 biophysics biophysic NOUN cana-1090 223 30 , , PUNCT cana-1090 223 31 12(1 12(1 NUM cana-1090 223 32 ) ) PUNCT cana-1090 223 33 , , PUNCT cana-1090 223 34 ( ( PUNCT cana-1090 223 35 2024 2024 NUM cana-1090 223 36 ) ) PUNCT cana-1090 223 37 , , PUNCT cana-1090 223 38 20230121 20230121 NUM cana-1090 223 39 . . PUNCT cana-1090 224 1 [ [ X cana-1090 224 2 30 30 NUM cana-1090 224 3 ] ] X cana-1090 224 4 smitha smitha PROPN cana-1090 224 5 mary mary PROPN cana-1090 224 6 mathew mathew PROPN cana-1090 224 7 and and CCONJ cana-1090 224 8 d.s.dilip d.s.dilip NOUN cana-1090 224 9 , , PUNCT cana-1090 224 10 dynamics dynamic NOUN cana-1090 224 11 of of ADP cana-1090 224 12 a a DET cana-1090 224 13 second second ADJ cana-1090 224 14 order order NOUN cana-1090 224 15 three three NUM cana-1090 224 16 species specie NOUN cana-1090 224 17 nonlinear nonlinear ADJ cana-1090 224 18 difference difference NOUN cana-1090 224 19 system system NOUN cana-1090 224 20 with with ADP cana-1090 224 21 exponents exponent NOUN cana-1090 224 22 , , PUNCT cana-1090 224 23 proyecciones proyeccione NOUN cana-1090 224 24 ( ( PUNCT cana-1090 224 25 antofagasta antofagasta PROPN cana-1090 224 26 , , PUNCT cana-1090 224 27 on on ADP cana-1090 224 28 line line NOUN cana-1090 224 29 ) ) PUNCT cana-1090 224 30 , , PUNCT cana-1090 224 31 41(4 41(4 NUM cana-1090 224 32 ) ) PUNCT cana-1090 224 33 , , PUNCT cana-1090 224 34 ( ( PUNCT cana-1090 224 35 2022 2022 NUM cana-1090 224 36 ) ) PUNCT cana-1090 224 37 , , PUNCT cana-1090 224 38 983 983 NUM cana-1090 224 39 - - SYM cana-1090 224 40 997 997 NUM cana-1090 224 41 . . PUNCT cana-1090 225 1 [ [ X cana-1090 225 2 31 31 NUM cana-1090 225 3 ] ] X cana-1090 225 4 smitha smitha PROPN cana-1090 225 5 mary mary PROPN cana-1090 225 6 mathew mathew PROPN cana-1090 225 7 and and CCONJ cana-1090 225 8 d.s.dilip d.s.dilip NOUN cana-1090 225 9 , , PUNCT cana-1090 225 10 dynamics dynamic NOUN cana-1090 225 11 of of ADP cana-1090 225 12 interspecific interspecific NOUN cana-1090 225 13 π‘˜ π‘˜ PROPN cana-1090 225 14 species specie NOUN cana-1090 225 15 competition competition NOUN cana-1090 225 16 model model NOUN cana-1090 225 17 , , PUNCT cana-1090 225 18 journal journal NOUN cana-1090 225 19 of of ADP cana-1090 225 20 interdisciplinary interdisciplinary ADJ cana-1090 225 21 mathematics mathematic NOUN cana-1090 225 22 , , PUNCT cana-1090 225 23 25(3 25(3 NUM cana-1090 225 24 ) ) PUNCT cana-1090 225 25 , , PUNCT cana-1090 225 26 ( ( PUNCT cana-1090 225 27 2022 2022 NUM cana-1090 225 28 ) ) PUNCT cana-1090 225 29 , , PUNCT cana-1090 225 30 629 629 NUM cana-1090 225 31 - - SYM cana-1090 225 32 638 638 NUM cana-1090 225 33 . . PUNCT cana-1090 226 1 [ [ X cana-1090 226 2 32 32 NUM cana-1090 226 3 ] ] X cana-1090 226 4 d. d. PROPN cana-1090 226 5 tilman tilman PROPN cana-1090 226 6 and and CCONJ cana-1090 226 7 d. d. PROPN cana-1090 226 8 wedin wedin PROPN cana-1090 226 9 , , PUNCT cana-1090 226 10 oscillations oscillation NOUN cana-1090 226 11 and and CCONJ cana-1090 226 12 chaos chaos NOUN cana-1090 226 13 in in ADP cana-1090 226 14 the the DET cana-1090 226 15 dynamics dynamic NOUN cana-1090 226 16 of of ADP cana-1090 226 17 a a DET cana-1090 226 18 perennial perennial ADJ cana-1090 226 19 grass grass NOUN cana-1090 226 20 , , PUNCT cana-1090 226 21 letters letter NOUN cana-1090 226 22 to to ADP cana-1090 226 23 nature nature NOUN cana-1090 226 24 , , PUNCT cana-1090 226 25 353 353 NUM cana-1090 226 26 ( ( PUNCT cana-1090 226 27 1991 1991 NUM cana-1090 226 28 ) ) PUNCT cana-1090 226 29 , , PUNCT cana-1090 226 30 653 653 NUM cana-1090 226 31 655 655 NUM cana-1090 226 32 . . PUNCT cana-1090 227 1 [ [ X cana-1090 227 2 33 33 NUM cana-1090 227 3 ] ] PUNCT cana-1090 227 4 yu yu PROPN cana-1090 227 5 x. x. PROPN cana-1090 227 6 , , PUNCT cana-1090 227 7 zhu zhu PROPN cana-1090 227 8 z. z. PROPN cana-1090 227 9 and and CCONJ cana-1090 227 10 li li PROPN cana-1090 227 11 z z PROPN cana-1090 227 12 , , PUNCT cana-1090 227 13 stability stability NOUN cana-1090 227 14 and and CCONJ cana-1090 227 15 bifurcation bifurcation NOUN cana-1090 227 16 analysis analysis NOUN cana-1090 227 17 of of ADP cana-1090 227 18 two two NUM cana-1090 227 19 - - PUNCT cana-1090 227 20 species specie NOUN cana-1090 227 21 competitive competitive ADJ cana-1090 227 22 model model NOUN cana-1090 227 23 with with ADP cana-1090 227 24 michaelis michaeli NOUN cana-1090 227 25 – – PUNCT cana-1090 227 26 menten menten ADJ cana-1090 227 27 type type NOUN cana-1090 227 28 harvesting harvesting NOUN cana-1090 227 29 in in ADP cana-1090 227 30 the the DET cana-1090 227 31 first first ADJ cana-1090 227 32 species specie NOUN cana-1090 227 33 , , PUNCT cana-1090 227 34 adv adv PROPN cana-1090 227 35 differ differ VERB cana-1090 227 36 equ equ PROPN cana-1090 227 37 2020 2020 NUM cana-1090 227 38 , , PUNCT cana-1090 227 39 397 397 NUM cana-1090 227 40 ( ( PUNCT cana-1090 227 41 2020 2020 NUM cana-1090 227 42 ) ) PUNCT cana-1090 227 43 . . PUNCT cana-1090 228 1 https://doi.org/10.1186/s13662-020-02817-4 https://doi.org/10.1186/s13662-020-02817-4 PROPN cana-1090 228 2 . . PUNCT