id sid tid token lemma pos ejpam-4172 1 1 european european PROPN ejpam-4172 1 2 journal journal PROPN ejpam-4172 1 3 of of ADP ejpam-4172 1 4 pure pure ADJ ejpam-4172 1 5 and and CCONJ ejpam-4172 1 6 applied apply VERB ejpam-4172 1 7 mathematics mathematic NOUN ejpam-4172 1 8 vol vol NOUN ejpam-4172 1 9 . . PROPN ejpam-4172 2 1 15 15 NUM ejpam-4172 2 2 , , PUNCT ejpam-4172 2 3 no no INTJ ejpam-4172 2 4 . . NOUN ejpam-4172 2 5 1 1 NUM ejpam-4172 2 6 , , PUNCT ejpam-4172 2 7 2022 2022 NUM ejpam-4172 2 8 , , PUNCT ejpam-4172 2 9 82 82 NUM ejpam-4172 2 10 - - SYM ejpam-4172 2 11 99 99 NUM ejpam-4172 2 12 issn issn PROPN ejpam-4172 2 13 1307 1307 NUM ejpam-4172 2 14 - - SYM ejpam-4172 2 15 5543 5543 NUM ejpam-4172 2 16 – – PUNCT ejpam-4172 2 17 ejpam.com ejpam.com X ejpam-4172 2 18 published publish VERB ejpam-4172 2 19 by by ADP ejpam-4172 2 20 new new PROPN ejpam-4172 2 21 york york PROPN ejpam-4172 2 22 business business PROPN ejpam-4172 2 23 global global ADJ ejpam-4172 2 24 application application NOUN ejpam-4172 2 25 of of ADP ejpam-4172 2 26 a a DET ejpam-4172 2 27 heuristic heuristic ADJ ejpam-4172 2 28 method method NOUN ejpam-4172 2 29 to to PART ejpam-4172 2 30 solve solve VERB ejpam-4172 2 31 nonlinear nonlinear ADJ ejpam-4172 2 32 oscillators oscillator NOUN ejpam-4172 2 33 with with ADP ejpam-4172 2 34 irrational irrational ADJ ejpam-4172 2 35 forces force NOUN ejpam-4172 2 36 o. o. PROPN ejpam-4172 2 37 gonzález gonzález PROPN ejpam-4172 2 38 - - PUNCT ejpam-4172 2 39 gaxiola1,∗ gaxiola1,∗ PROPN ejpam-4172 2 40 , , PUNCT ejpam-4172 2 41 j. j. PROPN ejpam-4172 2 42 ruiz ruiz PROPN ejpam-4172 2 43 de de PROPN ejpam-4172 2 44 chávez2 chávez2 PROPN ejpam-4172 2 45 1 1 NUM ejpam-4172 2 46 departamento departamento NOUN ejpam-4172 2 47 de de PROPN ejpam-4172 2 48 matemáticas matemáticas PROPN ejpam-4172 2 49 aplicadas aplicadas PROPN ejpam-4172 2 50 y y PROPN ejpam-4172 2 51 sistemas sistemas PROPN ejpam-4172 2 52 , , PUNCT ejpam-4172 2 53 universidad universidad PROPN ejpam-4172 2 54 autónoma autónoma ADP ejpam-4172 2 55 metropolitanacuajimalpa metropolitanacuajimalpa ADJ ejpam-4172 2 56 . . PUNCT ejpam-4172 3 1 vasco vasco PROPN ejpam-4172 3 2 de de PROPN ejpam-4172 3 3 quiroga quiroga PROPN ejpam-4172 3 4 4871 4871 NUM ejpam-4172 3 5 , , PUNCT ejpam-4172 3 6 santa santa PROPN ejpam-4172 3 7 fe fe PROPN ejpam-4172 3 8 , , PUNCT ejpam-4172 3 9 cuajimalpa cuajimalpa NOUN ejpam-4172 3 10 , , PUNCT ejpam-4172 3 11 05300 05300 NUM ejpam-4172 3 12 , , PUNCT ejpam-4172 3 13 mexico mexico PROPN ejpam-4172 3 14 city city PROPN ejpam-4172 3 15 , , PUNCT ejpam-4172 3 16 mexico mexico PROPN ejpam-4172 3 17 2 2 NUM ejpam-4172 3 18 departamento departamento NOUN ejpam-4172 3 19 de de PROPN ejpam-4172 3 20 matemáticas matemáticas PROPN ejpam-4172 3 21 , , PUNCT ejpam-4172 3 22 universidad universidad PROPN ejpam-4172 3 23 autónoma autónoma PROPN ejpam-4172 3 24 metropolitana metropolitana PROPN ejpam-4172 3 25 - - PUNCT ejpam-4172 3 26 iztapalapa iztapalapa PROPN ejpam-4172 3 27 . . PUNCT ejpam-4172 4 1 san san PROPN ejpam-4172 4 2 rafael rafael PROPN ejpam-4172 4 3 atlixco atlixco PROPN ejpam-4172 4 4 186 186 NUM ejpam-4172 4 5 , , PUNCT ejpam-4172 4 6 col col PROPN ejpam-4172 4 7 . . PROPN ejpam-4172 4 8 vicentina vicentina PROPN ejpam-4172 4 9 , , PUNCT ejpam-4172 4 10 iztapalapa iztapalapa PROPN ejpam-4172 4 11 , , PUNCT ejpam-4172 4 12 09340 09340 NUM ejpam-4172 4 13 , , PUNCT ejpam-4172 4 14 mexico mexico PROPN ejpam-4172 4 15 city city PROPN ejpam-4172 4 16 , , PUNCT ejpam-4172 4 17 mexico mexico PROPN ejpam-4172 4 18 abstract abstract NOUN ejpam-4172 4 19 . . PUNCT ejpam-4172 5 1 this this DET ejpam-4172 5 2 paper paper NOUN ejpam-4172 5 3 applies apply VERB ejpam-4172 5 4 another another DET ejpam-4172 5 5 amplitude amplitude NOUN ejpam-4172 5 6 - - PUNCT ejpam-4172 5 7 frequency frequency NOUN ejpam-4172 5 8 relationship relationship NOUN ejpam-4172 5 9 as as ADP ejpam-4172 5 10 of of ADP ejpam-4172 5 11 late late ADV ejpam-4172 5 12 established establish VERB ejpam-4172 5 13 by by ADP ejpam-4172 5 14 yazdi yazdi PROPN ejpam-4172 5 15 and and CCONJ ejpam-4172 5 16 tehrani tehrani PROPN ejpam-4172 5 17 ( ( PUNCT ejpam-4172 5 18 alexandria alexandria PROPN ejpam-4172 5 19 engineering engineering PROPN ejpam-4172 5 20 journal journal PROPN ejpam-4172 5 21 54 54 NUM ejpam-4172 5 22 99 99 NUM ejpam-4172 5 23 - - SYM ejpam-4172 5 24 103 103 NUM ejpam-4172 5 25 , , PUNCT ejpam-4172 5 26 2015 2015 NUM ejpam-4172 5 27 ) ) PUNCT ejpam-4172 5 28 to to PART ejpam-4172 5 29 analyze analyze VERB ejpam-4172 5 30 periodical periodical ADJ ejpam-4172 5 31 solutions solution NOUN ejpam-4172 5 32 of of ADP ejpam-4172 5 33 nonlinear nonlinear ADJ ejpam-4172 5 34 oscillators oscillator NOUN ejpam-4172 5 35 ; ; PUNCT ejpam-4172 5 36 such such ADJ ejpam-4172 5 37 oscillators oscillator NOUN ejpam-4172 5 38 are be AUX ejpam-4172 5 39 considered consider VERB ejpam-4172 5 40 emphatically emphatically ADV ejpam-4172 5 41 nonlinear nonlinear ADJ ejpam-4172 5 42 on on ADP ejpam-4172 5 43 the the DET ejpam-4172 5 44 grounds ground NOUN ejpam-4172 5 45 that that SCONJ ejpam-4172 5 46 they they PRON ejpam-4172 5 47 contain contain VERB ejpam-4172 5 48 an an DET ejpam-4172 5 49 irrational irrational ADJ ejpam-4172 5 50 force force NOUN ejpam-4172 5 51 term term NOUN ejpam-4172 5 52 . . PUNCT ejpam-4172 6 1 estimation estimation NOUN ejpam-4172 6 2 strategy strategy NOUN ejpam-4172 6 3 is be AUX ejpam-4172 6 4 straightforward straightforward ADJ ejpam-4172 6 5 yet yet ADV ejpam-4172 6 6 additionally additionally ADV ejpam-4172 6 7 helpful helpful ADJ ejpam-4172 6 8 and and CCONJ ejpam-4172 6 9 its its PRON ejpam-4172 6 10 formulation formulation NOUN ejpam-4172 6 11 depends depend VERB ejpam-4172 6 12 on on ADP ejpam-4172 6 13 combining combine VERB ejpam-4172 6 14 the the DET ejpam-4172 6 15 energy energy NOUN ejpam-4172 6 16 balance balance NOUN ejpam-4172 6 17 technique technique NOUN ejpam-4172 6 18 with with ADP ejpam-4172 6 19 an an DET ejpam-4172 6 20 uncommon uncommon ADJ ejpam-4172 6 21 collocation collocation NOUN ejpam-4172 6 22 point point NOUN ejpam-4172 6 23 . . PUNCT ejpam-4172 7 1 at at ADP ejpam-4172 7 2 long long ADV ejpam-4172 7 3 last last ADV ejpam-4172 7 4 , , PUNCT ejpam-4172 7 5 we we PRON ejpam-4172 7 6 uncover uncover VERB ejpam-4172 7 7 the the DET ejpam-4172 7 8 convenience convenience NOUN ejpam-4172 7 9 and and CCONJ ejpam-4172 7 10 proficiency proficiency NOUN ejpam-4172 7 11 of of ADP ejpam-4172 7 12 the the DET ejpam-4172 7 13 proposed propose VERB ejpam-4172 7 14 technique technique NOUN ejpam-4172 7 15 resolving resolve VERB ejpam-4172 7 16 three three NUM ejpam-4172 7 17 instances instance NOUN ejpam-4172 7 18 of of ADP ejpam-4172 7 19 conservative conservative ADJ ejpam-4172 7 20 nonlinear nonlinear ADJ ejpam-4172 7 21 oscillators oscillator NOUN ejpam-4172 7 22 in in ADP ejpam-4172 7 23 which which PRON ejpam-4172 7 24 the the DET ejpam-4172 7 25 maximum maximum ADJ ejpam-4172 7 26 relative relative ADJ ejpam-4172 7 27 error error NOUN ejpam-4172 7 28 acquired acquire VERB ejpam-4172 7 29 is be AUX ejpam-4172 7 30 2.5 2.5 NUM ejpam-4172 7 31 % % NOUN ejpam-4172 7 32 . . PUNCT ejpam-4172 8 1 exposed expose VERB ejpam-4172 8 2 models model NOUN ejpam-4172 8 3 show show VERB ejpam-4172 8 4 that that SCONJ ejpam-4172 8 5 the the DET ejpam-4172 8 6 strategy strategy NOUN ejpam-4172 8 7 has have VERB ejpam-4172 8 8 a a DET ejpam-4172 8 9 high high ADJ ejpam-4172 8 10 precision precision NOUN ejpam-4172 8 11 to to PART ejpam-4172 8 12 solve solve VERB ejpam-4172 8 13 mechanical mechanical ADJ ejpam-4172 8 14 issues issue NOUN ejpam-4172 8 15 of of ADP ejpam-4172 8 16 both both CCONJ ejpam-4172 8 17 small small ADJ ejpam-4172 8 18 and and CCONJ ejpam-4172 8 19 huge huge ADJ ejpam-4172 8 20 estimations estimation NOUN ejpam-4172 8 21 of of ADP ejpam-4172 8 22 the the DET ejpam-4172 8 23 oscillation oscillation NOUN ejpam-4172 8 24 amplitude amplitude NOUN ejpam-4172 8 25 . . PUNCT ejpam-4172 9 1 finally finally ADV ejpam-4172 9 2 , , PUNCT ejpam-4172 9 3 the the DET ejpam-4172 9 4 methodology methodology NOUN ejpam-4172 9 5 used use VERB ejpam-4172 9 6 can can AUX ejpam-4172 9 7 be be AUX ejpam-4172 9 8 very very ADV ejpam-4172 9 9 useful useful ADJ ejpam-4172 9 10 for for ADP ejpam-4172 9 11 the the DET ejpam-4172 9 12 study study NOUN ejpam-4172 9 13 of of ADP ejpam-4172 9 14 nonlinear nonlinear ADJ ejpam-4172 9 15 oscillators oscillator NOUN ejpam-4172 9 16 in in ADP ejpam-4172 9 17 basic basic ADJ ejpam-4172 9 18 undergraduate undergraduate ADJ ejpam-4172 9 19 physics physics NOUN ejpam-4172 9 20 and and CCONJ ejpam-4172 9 21 mechanics mechanic NOUN ejpam-4172 9 22 courses course NOUN ejpam-4172 9 23 . . PUNCT ejpam-4172 10 1 2020 2020 NUM ejpam-4172 10 2 mathematics mathematic NOUN ejpam-4172 10 3 subject subject NOUN ejpam-4172 10 4 classifications classification NOUN ejpam-4172 10 5 : : PUNCT ejpam-4172 10 6 34a34 34a34 NUM ejpam-4172 10 7 , , PUNCT ejpam-4172 10 8 97m10 97m10 NUM ejpam-4172 10 9 , , PUNCT ejpam-4172 10 10 97m50 97m50 NUM ejpam-4172 10 11 key key ADJ ejpam-4172 10 12 words word NOUN ejpam-4172 10 13 and and CCONJ ejpam-4172 10 14 phrases phrase NOUN ejpam-4172 10 15 : : PUNCT ejpam-4172 10 16 nonlinear nonlinear ADJ ejpam-4172 10 17 oscillators oscillator NOUN ejpam-4172 10 18 , , PUNCT ejpam-4172 10 19 periodic periodic ADJ ejpam-4172 10 20 solution solution NOUN ejpam-4172 10 21 , , PUNCT ejpam-4172 10 22 approximate approximate ADJ ejpam-4172 10 23 frequency frequency NOUN ejpam-4172 10 24 , , PUNCT ejpam-4172 10 25 conservative conservative ADJ ejpam-4172 10 26 oscillator oscillator NOUN ejpam-4172 10 27 , , PUNCT ejpam-4172 10 28 duffing duffing NOUN ejpam-4172 10 29 - - PUNCT ejpam-4172 10 30 relativistic relativistic ADJ ejpam-4172 10 31 oscillator oscillator NOUN ejpam-4172 10 32 , , PUNCT ejpam-4172 10 33 jacobi jacobi PROPN ejpam-4172 10 34 polynomials polynomial VERB ejpam-4172 10 35 1 1 NUM ejpam-4172 10 36 . . PUNCT ejpam-4172 11 1 introduction introduction NOUN ejpam-4172 11 2 most most ADJ ejpam-4172 11 3 of of ADP ejpam-4172 11 4 engineering engineering NOUN ejpam-4172 11 5 problems problem NOUN ejpam-4172 11 6 , , PUNCT ejpam-4172 11 7 especially especially ADV ejpam-4172 11 8 some some DET ejpam-4172 11 9 oscillation oscillation NOUN ejpam-4172 11 10 equations equation NOUN ejpam-4172 11 11 are be AUX ejpam-4172 11 12 nonlinear nonlinear ADJ ejpam-4172 11 13 , , PUNCT ejpam-4172 11 14 and and CCONJ ejpam-4172 11 15 in in ADP ejpam-4172 11 16 most most ADJ ejpam-4172 11 17 cases case NOUN ejpam-4172 11 18 it it PRON ejpam-4172 11 19 is be AUX ejpam-4172 11 20 difficult difficult ADJ ejpam-4172 11 21 to to PART ejpam-4172 11 22 solve solve VERB ejpam-4172 11 23 such such ADJ ejpam-4172 11 24 equations equation NOUN ejpam-4172 11 25 , , PUNCT ejpam-4172 11 26 especially especially ADV ejpam-4172 11 27 analytically analytically ADV ejpam-4172 11 28 . . PUNCT ejpam-4172 12 1 recently recently ADV ejpam-4172 12 2 , , PUNCT ejpam-4172 12 3 it it PRON ejpam-4172 12 4 is be AUX ejpam-4172 12 5 exceptionally exceptionally ADV ejpam-4172 12 6 valuable valuable ADJ ejpam-4172 12 7 to to PART ejpam-4172 12 8 have have VERB ejpam-4172 12 9 creative creative ADJ ejpam-4172 12 10 and and CCONJ ejpam-4172 12 11 effective effective ADJ ejpam-4172 12 12 numerical numerical ADJ ejpam-4172 12 13 techniques technique NOUN ejpam-4172 12 14 to to PART ejpam-4172 12 15 understand understand VERB ejpam-4172 12 16 nonlinear nonlinear ADJ ejpam-4172 12 17 differential differential ADJ ejpam-4172 12 18 equations equation NOUN ejpam-4172 12 19 , , PUNCT ejpam-4172 12 20 since since SCONJ ejpam-4172 12 21 huge huge ADJ ejpam-4172 12 22 numbers number NOUN ejpam-4172 12 23 of of ADP ejpam-4172 12 24 the the DET ejpam-4172 12 25 models model NOUN ejpam-4172 12 26 for for ADP ejpam-4172 12 27 depicting depict VERB ejpam-4172 12 28 phenomena phenomenon NOUN ejpam-4172 12 29 in in ADP ejpam-4172 12 30 both both DET ejpam-4172 12 31 applied apply VERB ejpam-4172 12 32 mathematics mathematic NOUN ejpam-4172 12 33 and and CCONJ ejpam-4172 12 34 engineering engineering NOUN ejpam-4172 12 35 end end VERB ejpam-4172 12 36 up up ADP ejpam-4172 12 37 being be AUX ejpam-4172 12 38 given give VERB ejpam-4172 12 39 through through ADP ejpam-4172 12 40 nonlinear nonlinear ADJ ejpam-4172 12 41 equations equation NOUN ejpam-4172 12 42 . . PUNCT ejpam-4172 13 1 in in ADP ejpam-4172 13 2 ongoing ongoing ADJ ejpam-4172 13 3 literature literature NOUN ejpam-4172 13 4 we we PRON ejpam-4172 13 5 can can AUX ejpam-4172 13 6 locate locate VERB ejpam-4172 13 7 that that SCONJ ejpam-4172 13 8 a a DET ejpam-4172 13 9 few few ADJ ejpam-4172 13 10 analytical analytical ADJ ejpam-4172 13 11 and and CCONJ ejpam-4172 13 12 estimated estimate VERB ejpam-4172 13 13 strategies strategy NOUN ejpam-4172 13 14 have have AUX ejpam-4172 13 15 been be AUX ejpam-4172 13 16 created create VERB ejpam-4172 13 17 to to PART ejpam-4172 13 18 solve solve VERB ejpam-4172 13 19 nonlinear nonlinear ADJ ejpam-4172 13 20 oscillators oscillator NOUN ejpam-4172 13 21 throughout throughout ADP ejpam-4172 13 22 the the DET ejpam-4172 13 23 most most ADV ejpam-4172 13 24 recent recent ADJ ejpam-4172 13 25 decades decade NOUN ejpam-4172 13 26 , , PUNCT ejpam-4172 13 27 for for ADP ejpam-4172 13 28 example example NOUN ejpam-4172 13 29 : : PUNCT ejpam-4172 13 30 the the DET ejpam-4172 13 31 harmonic harmonic ADJ ejpam-4172 13 32 balance balance NOUN ejpam-4172 13 33 method method NOUN ejpam-4172 13 34 [ [ X ejpam-4172 13 35 1 1 NUM ejpam-4172 13 36 , , PUNCT ejpam-4172 13 37 2 2 NUM ejpam-4172 13 38 ] ] PUNCT ejpam-4172 13 39 , , PUNCT ejpam-4172 13 40 which which PRON ejpam-4172 13 41 in in ADP ejpam-4172 13 42 practice practice NOUN ejpam-4172 13 43 is be AUX ejpam-4172 13 44 used use VERB ejpam-4172 13 45 to to PART ejpam-4172 13 46 replace replace VERB ejpam-4172 13 47 the the DET ejpam-4172 13 48 ∗corresponding ∗corresponding NOUN ejpam-4172 13 49 author author NOUN ejpam-4172 13 50 . . PUNCT ejpam-4172 14 1 doi doi NOUN ejpam-4172 14 2 : : PUNCT ejpam-4172 14 3 https://doi.org/10.29020/nybg.ejpam.v15i1.4172 https://doi.org/10.29020/nybg.ejpam.v15i1.4172 NUM ejpam-4172 14 4 email email NOUN ejpam-4172 14 5 addresses address NOUN ejpam-4172 14 6 : : PUNCT ejpam-4172 14 7 ogonzalez@cua.uam.mx ogonzalez@cua.uam.mx INTJ ejpam-4172 14 8 ( ( PUNCT ejpam-4172 14 9 o. o. PROPN ejpam-4172 14 10 gonzález gonzález PROPN ejpam-4172 14 11 - - PUNCT ejpam-4172 14 12 gaxiola gaxiola PROPN ejpam-4172 14 13 ) ) PUNCT ejpam-4172 14 14 jrch@xanum.uam.mx jrch@xanum.uam.mx NOUN ejpam-4172 14 15 ( ( PUNCT ejpam-4172 14 16 j. j. PROPN ejpam-4172 14 17 ruiz ruiz PROPN ejpam-4172 14 18 de de PROPN ejpam-4172 14 19 chávez chávez PROPN ejpam-4172 14 20 ) ) PUNCT ejpam-4172 14 21 http://www.ejpam.com http://www.ejpam.com X ejpam-4172 15 1 82 82 NUM ejpam-4172 16 1 © © ADP ejpam-4172 16 2 2022 2022 NUM ejpam-4172 16 3 ejpam ejpam VERB ejpam-4172 16 4 all all DET ejpam-4172 16 5 rights right NOUN ejpam-4172 16 6 reserved reserve VERB ejpam-4172 16 7 . . PUNCT ejpam-4172 17 1 o. o. PROPN ejpam-4172 17 2 gonzález gonzález PROPN ejpam-4172 17 3 - - PUNCT ejpam-4172 17 4 gaxiola gaxiola PROPN ejpam-4172 17 5 , , PUNCT ejpam-4172 17 6 j. j. PROPN ejpam-4172 17 7 ruiz ruiz PROPN ejpam-4172 17 8 de de PROPN ejpam-4172 17 9 chávez chávez PROPN ejpam-4172 17 10 / / SYM ejpam-4172 17 11 eur eur PROPN ejpam-4172 17 12 . . PUNCT ejpam-4172 18 1 j. j. PROPN ejpam-4172 18 2 pure pure PROPN ejpam-4172 18 3 appl appl PROPN ejpam-4172 18 4 . . PROPN ejpam-4172 18 5 math math PROPN ejpam-4172 18 6 , , PUNCT ejpam-4172 18 7 15 15 NUM ejpam-4172 18 8 ( ( PUNCT ejpam-4172 18 9 1 1 NUM ejpam-4172 18 10 ) ) PUNCT ejpam-4172 18 11 ( ( PUNCT ejpam-4172 18 12 2022 2022 NUM ejpam-4172 18 13 ) ) PUNCT ejpam-4172 18 14 , , PUNCT ejpam-4172 18 15 82 82 NUM ejpam-4172 18 16 - - SYM ejpam-4172 18 17 99 99 NUM ejpam-4172 18 18 83 83 NUM ejpam-4172 18 19 non non ADJ ejpam-4172 18 20 - - ADJ ejpam-4172 18 21 linear linear ADJ ejpam-4172 18 22 forces force NOUN ejpam-4172 18 23 in in ADP ejpam-4172 18 24 the the DET ejpam-4172 18 25 oscillating oscillate VERB ejpam-4172 18 26 systems system NOUN ejpam-4172 18 27 by by ADP ejpam-4172 18 28 specially specially ADV ejpam-4172 18 29 - - PUNCT ejpam-4172 18 30 constructed construct VERB ejpam-4172 18 31 linear linear NOUN ejpam-4172 18 32 functions function NOUN ejpam-4172 18 33 . . PUNCT ejpam-4172 19 1 the the DET ejpam-4172 19 2 he he PRON ejpam-4172 19 3 ’s ’s PART ejpam-4172 19 4 variational variational ADJ ejpam-4172 19 5 approach approach NOUN ejpam-4172 19 6 [ [ X ejpam-4172 19 7 3 3 X ejpam-4172 19 8 ] ] PUNCT ejpam-4172 19 9 which which PRON ejpam-4172 19 10 is be AUX ejpam-4172 19 11 widely widely ADV ejpam-4172 19 12 used use VERB ejpam-4172 19 13 to to PART ejpam-4172 19 14 obtain obtain VERB ejpam-4172 19 15 an an DET ejpam-4172 19 16 approximate approximate ADJ ejpam-4172 19 17 analytical analytical ADJ ejpam-4172 19 18 solutions solution NOUN ejpam-4172 19 19 for for ADP ejpam-4172 19 20 nonlinear nonlinear ADJ ejpam-4172 19 21 problems problem NOUN ejpam-4172 19 22 . . PUNCT ejpam-4172 20 1 the the DET ejpam-4172 20 2 energy energy NOUN ejpam-4172 20 3 balance balance NOUN ejpam-4172 20 4 technique technique NOUN ejpam-4172 20 5 [ [ X ejpam-4172 20 6 4 4 NUM ejpam-4172 20 7 , , PUNCT ejpam-4172 20 8 5 5 NUM ejpam-4172 20 9 , , PUNCT ejpam-4172 20 10 6 6 NUM ejpam-4172 20 11 ] ] PUNCT ejpam-4172 20 12 , , PUNCT ejpam-4172 20 13 which which PRON ejpam-4172 20 14 can can AUX ejpam-4172 20 15 be be AUX ejpam-4172 20 16 seen see VERB ejpam-4172 20 17 as as ADP ejpam-4172 20 18 a a DET ejpam-4172 20 19 ritz ritz PROPN ejpam-4172 20 20 method method NOUN ejpam-4172 20 21 and and CCONJ ejpam-4172 20 22 leads lead VERB ejpam-4172 20 23 to to ADP ejpam-4172 20 24 a a DET ejpam-4172 20 25 very very ADV ejpam-4172 20 26 rapid rapid ADJ ejpam-4172 20 27 convergence convergence NOUN ejpam-4172 20 28 of of ADP ejpam-4172 20 29 the the DET ejpam-4172 20 30 solution solution NOUN ejpam-4172 20 31 , , PUNCT ejpam-4172 20 32 and and CCONJ ejpam-4172 20 33 can can AUX ejpam-4172 20 34 be be AUX ejpam-4172 20 35 easily easily ADV ejpam-4172 20 36 extended extend VERB ejpam-4172 20 37 to to ADP ejpam-4172 20 38 other other ADJ ejpam-4172 20 39 nonlinear nonlinear ADJ ejpam-4172 20 40 oscillations oscillation NOUN ejpam-4172 20 41 . . PUNCT ejpam-4172 21 1 the the DET ejpam-4172 21 2 hamiltonian hamiltonian ADJ ejpam-4172 21 3 approach approach NOUN ejpam-4172 21 4 [ [ X ejpam-4172 21 5 7 7 NUM ejpam-4172 21 6 , , PUNCT ejpam-4172 21 7 8 8 NUM ejpam-4172 21 8 ] ] PUNCT ejpam-4172 21 9 , , PUNCT ejpam-4172 21 10 which which PRON ejpam-4172 21 11 is be AUX ejpam-4172 21 12 very very ADV ejpam-4172 21 13 simple simple ADJ ejpam-4172 21 14 but but CCONJ ejpam-4172 21 15 strongly strongly ADV ejpam-4172 21 16 depends depend VERB ejpam-4172 21 17 upon upon SCONJ ejpam-4172 21 18 the the DET ejpam-4172 21 19 chosen choose VERB ejpam-4172 21 20 location location NOUN ejpam-4172 21 21 point point NOUN ejpam-4172 21 22 . . PUNCT ejpam-4172 22 1 the the DET ejpam-4172 22 2 use use NOUN ejpam-4172 22 3 of of ADP ejpam-4172 22 4 special special ADJ ejpam-4172 22 5 functions function NOUN ejpam-4172 22 6 [ [ X ejpam-4172 22 7 9 9 NUM ejpam-4172 22 8 , , PUNCT ejpam-4172 22 9 10 10 NUM ejpam-4172 22 10 ] ] PUNCT ejpam-4172 22 11 . . PUNCT ejpam-4172 23 1 the the DET ejpam-4172 23 2 max max PROPN ejpam-4172 23 3 - - PUNCT ejpam-4172 23 4 min min NOUN ejpam-4172 23 5 approach approach NOUN ejpam-4172 23 6 [ [ X ejpam-4172 23 7 11 11 NUM ejpam-4172 23 8 , , PUNCT ejpam-4172 23 9 12 12 NUM ejpam-4172 23 10 ] ] PUNCT ejpam-4172 23 11 , , PUNCT ejpam-4172 23 12 which which PRON ejpam-4172 23 13 has have VERB ejpam-4172 23 14 millennia millennia NOUN ejpam-4172 23 15 history history NOUN ejpam-4172 23 16 . . PUNCT ejpam-4172 24 1 the the DET ejpam-4172 24 2 adomian adomian NOUN ejpam-4172 24 3 decomposition decomposition NOUN ejpam-4172 24 4 technique technique NOUN ejpam-4172 24 5 [ [ X ejpam-4172 24 6 13 13 NUM ejpam-4172 24 7 ] ] PUNCT ejpam-4172 24 8 , , PUNCT ejpam-4172 24 9 consisting consist VERB ejpam-4172 24 10 of of ADP ejpam-4172 24 11 decomposing decompose VERB ejpam-4172 24 12 the the DET ejpam-4172 24 13 solution solution NOUN ejpam-4172 24 14 into into ADP ejpam-4172 24 15 rapidly rapidly ADV ejpam-4172 24 16 convergent convergent ADJ ejpam-4172 24 17 series series NOUN ejpam-4172 24 18 . . PUNCT ejpam-4172 25 1 the the DET ejpam-4172 25 2 variational variational ADJ ejpam-4172 25 3 iteration iteration NOUN ejpam-4172 25 4 method method NOUN ejpam-4172 25 5 [ [ X ejpam-4172 25 6 14 14 NUM ejpam-4172 25 7 , , PUNCT ejpam-4172 25 8 15 15 NUM ejpam-4172 25 9 , , PUNCT ejpam-4172 25 10 16 16 NUM ejpam-4172 25 11 ] ] PUNCT ejpam-4172 25 12 , , PUNCT ejpam-4172 25 13 based base VERB ejpam-4172 25 14 on on ADP ejpam-4172 25 15 the the DET ejpam-4172 25 16 concepts concept NOUN ejpam-4172 25 17 of of ADP ejpam-4172 25 18 force force NOUN ejpam-4172 25 19 and and CCONJ ejpam-4172 25 20 momentum momentum NOUN ejpam-4172 25 21 , , PUNCT ejpam-4172 25 22 and and CCONJ ejpam-4172 25 23 homotopy homotopy VERB ejpam-4172 25 24 perturbation perturbation NOUN ejpam-4172 25 25 [ [ X ejpam-4172 25 26 17 17 NUM ejpam-4172 25 27 , , PUNCT ejpam-4172 25 28 18 18 NUM ejpam-4172 25 29 ] ] PUNCT ejpam-4172 25 30 , , PUNCT ejpam-4172 25 31 which which PRON ejpam-4172 25 32 is be AUX ejpam-4172 25 33 an an DET ejpam-4172 25 34 effective effective ADJ ejpam-4172 25 35 technique technique NOUN ejpam-4172 25 36 and and CCONJ ejpam-4172 25 37 convenient convenient ADJ ejpam-4172 25 38 one one NUM ejpam-4172 25 39 for for ADP ejpam-4172 25 40 both both CCONJ ejpam-4172 25 41 weakly weakly ADJ ejpam-4172 25 42 and and CCONJ ejpam-4172 25 43 strongly strongly ADV ejpam-4172 25 44 nonlinear nonlinear ADJ ejpam-4172 25 45 equations equation NOUN ejpam-4172 25 46 . . PUNCT ejpam-4172 26 1 an an DET ejpam-4172 26 2 investigation investigation NOUN ejpam-4172 26 3 that that PRON ejpam-4172 26 4 can can AUX ejpam-4172 26 5 be be AUX ejpam-4172 26 6 an an DET ejpam-4172 26 7 exceptionally exceptionally ADV ejpam-4172 26 8 valuable valuable ADJ ejpam-4172 26 9 tool tool NOUN ejpam-4172 26 10 , , PUNCT ejpam-4172 26 11 where where SCONJ ejpam-4172 26 12 a a DET ejpam-4172 26 13 large large ADJ ejpam-4172 26 14 numbers number NOUN ejpam-4172 26 15 of of ADP ejpam-4172 26 16 these these DET ejpam-4172 26 17 techniques technique NOUN ejpam-4172 26 18 can can AUX ejpam-4172 26 19 be be AUX ejpam-4172 26 20 found find VERB ejpam-4172 26 21 in in ADP ejpam-4172 26 22 detail detail NOUN ejpam-4172 26 23 to to PART ejpam-4172 26 24 take take VERB ejpam-4172 26 25 care care NOUN ejpam-4172 26 26 of of ADP ejpam-4172 26 27 nonlinear nonlinear ADJ ejpam-4172 26 28 issues issue NOUN ejpam-4172 26 29 of of ADP ejpam-4172 26 30 oscillatory oscillatory ADJ ejpam-4172 26 31 type type NOUN ejpam-4172 26 32 can can AUX ejpam-4172 26 33 be be AUX ejpam-4172 26 34 seen see VERB ejpam-4172 26 35 in in ADP ejpam-4172 26 36 [ [ X ejpam-4172 26 37 19 19 NUM ejpam-4172 26 38 , , PUNCT ejpam-4172 26 39 20 20 NUM ejpam-4172 26 40 ] ] PUNCT ejpam-4172 26 41 . . PUNCT ejpam-4172 27 1 as as ADP ejpam-4172 27 2 of of ADP ejpam-4172 27 3 late late ADJ ejpam-4172 27 4 , , PUNCT ejpam-4172 27 5 in in ADP ejpam-4172 27 6 [ [ PUNCT ejpam-4172 27 7 21 21 NUM ejpam-4172 27 8 ] ] X ejpam-4172 27 9 yazdi yazdi NOUN ejpam-4172 27 10 and and CCONJ ejpam-4172 27 11 tehrani tehrani PROPN ejpam-4172 27 12 improved improve VERB ejpam-4172 27 13 improved improved ADJ ejpam-4172 27 14 exactness exactness NOUN ejpam-4172 27 15 of of ADP ejpam-4172 27 16 the the DET ejpam-4172 27 17 energy energy NOUN ejpam-4172 27 18 balance balance NOUN ejpam-4172 27 19 through through ADP ejpam-4172 27 20 jacobi jacobi PROPN ejpam-4172 27 21 collocation collocation NOUN ejpam-4172 27 22 technique technique NOUN ejpam-4172 27 23 to to ADP ejpam-4172 27 24 moderate moderate ADJ ejpam-4172 27 25 nonlinear nonlinear ADJ ejpam-4172 27 26 oscillators oscillator NOUN ejpam-4172 27 27 . . PUNCT ejpam-4172 28 1 in in ADP ejpam-4172 28 2 addition addition NOUN ejpam-4172 28 3 , , PUNCT ejpam-4172 28 4 we we PRON ejpam-4172 28 5 can can AUX ejpam-4172 28 6 see see VERB ejpam-4172 28 7 techniques technique NOUN ejpam-4172 28 8 for for ADP ejpam-4172 28 9 solving solve VERB ejpam-4172 28 10 oscillatory oscillatory ADJ ejpam-4172 28 11 problems problem NOUN ejpam-4172 28 12 for for ADP ejpam-4172 28 13 teaching teach VERB ejpam-4172 28 14 purposes purpose NOUN ejpam-4172 28 15 in in ADP ejpam-4172 28 16 [ [ X ejpam-4172 28 17 22 22 NUM ejpam-4172 28 18 , , PUNCT ejpam-4172 28 19 23 23 NUM ejpam-4172 28 20 ] ] PUNCT ejpam-4172 28 21 . . PUNCT ejpam-4172 29 1 in in ADP ejpam-4172 29 2 this this DET ejpam-4172 29 3 paper paper NOUN ejpam-4172 29 4 , , PUNCT ejpam-4172 29 5 we we PRON ejpam-4172 29 6 will will AUX ejpam-4172 29 7 uncover uncover VERB ejpam-4172 29 8 the the DET ejpam-4172 29 9 new new ADJ ejpam-4172 29 10 strategy strategy NOUN ejpam-4172 29 11 proposed propose VERB ejpam-4172 29 12 in in ADP ejpam-4172 29 13 [ [ X ejpam-4172 29 14 21 21 NUM ejpam-4172 29 15 ] ] PUNCT ejpam-4172 29 16 and and CCONJ ejpam-4172 29 17 we we PRON ejpam-4172 29 18 will will AUX ejpam-4172 29 19 likewise likewise ADV ejpam-4172 29 20 apply apply VERB ejpam-4172 29 21 it it PRON ejpam-4172 29 22 to to PART ejpam-4172 29 23 discover discover VERB ejpam-4172 29 24 approximate approximate ADJ ejpam-4172 29 25 solutions solution NOUN ejpam-4172 29 26 of of ADP ejpam-4172 29 27 certain certain ADJ ejpam-4172 29 28 nonlinear nonlinear ADJ ejpam-4172 29 29 differential differential ADJ ejpam-4172 29 30 equations equation NOUN ejpam-4172 29 31 that that PRON ejpam-4172 29 32 model model VERB ejpam-4172 29 33 emphatically emphatically ADV ejpam-4172 29 34 nonlinear nonlinear ADJ ejpam-4172 29 35 oscillators oscillator NOUN ejpam-4172 29 36 with with ADP ejpam-4172 29 37 a a DET ejpam-4172 29 38 nonsensical nonsensical ADJ ejpam-4172 29 39 force force NOUN ejpam-4172 29 40 and and CCONJ ejpam-4172 29 41 have have AUX ejpam-4172 29 42 made make VERB ejpam-4172 29 43 a a DET ejpam-4172 29 44 comparison comparison NOUN ejpam-4172 29 45 with with ADP ejpam-4172 29 46 the the DET ejpam-4172 29 47 specific specific ADJ ejpam-4172 29 48 solution solution NOUN ejpam-4172 29 49 . . PUNCT ejpam-4172 30 1 the the DET ejpam-4172 30 2 present present ADJ ejpam-4172 30 3 method method NOUN ejpam-4172 30 4 has have VERB ejpam-4172 30 5 the the DET ejpam-4172 30 6 versatility versatility NOUN ejpam-4172 30 7 and and CCONJ ejpam-4172 30 8 advantages advantage NOUN ejpam-4172 30 9 of of ADP ejpam-4172 30 10 numerical numerical ADJ ejpam-4172 30 11 methods method NOUN ejpam-4172 30 12 for for ADP ejpam-4172 30 13 being be AUX ejpam-4172 30 14 applied apply VERB ejpam-4172 30 15 directly directly ADV ejpam-4172 30 16 to to ADP ejpam-4172 30 17 highly highly ADV ejpam-4172 30 18 nonlinear nonlinear ADJ ejpam-4172 30 19 problems problem NOUN ejpam-4172 30 20 and and CCONJ ejpam-4172 30 21 also also ADV ejpam-4172 30 22 have have VERB ejpam-4172 30 23 the the DET ejpam-4172 30 24 elegance elegance NOUN ejpam-4172 30 25 and and CCONJ ejpam-4172 30 26 other other ADJ ejpam-4172 30 27 benefits benefit NOUN ejpam-4172 30 28 of of ADP ejpam-4172 30 29 analytical analytical ADJ ejpam-4172 30 30 techniques technique NOUN ejpam-4172 30 31 . . PUNCT ejpam-4172 31 1 2 2 X ejpam-4172 31 2 . . X ejpam-4172 31 3 solution solution NOUN ejpam-4172 31 4 method method VERB ejpam-4172 31 5 the the DET ejpam-4172 31 6 technique technique NOUN ejpam-4172 31 7 that that PRON ejpam-4172 31 8 we we PRON ejpam-4172 31 9 will will AUX ejpam-4172 31 10 uncover uncover VERB ejpam-4172 31 11 straightaway straightaway NOUN ejpam-4172 31 12 and and CCONJ ejpam-4172 31 13 which which PRON ejpam-4172 31 14 the the DET ejpam-4172 31 15 perusers peruser NOUN ejpam-4172 31 16 can can AUX ejpam-4172 31 17 discover discover VERB ejpam-4172 31 18 in in ADP ejpam-4172 31 19 [ [ X ejpam-4172 31 20 21 21 NUM ejpam-4172 31 21 ] ] PUNCT ejpam-4172 31 22 with with ADP ejpam-4172 31 23 more more ADV ejpam-4172 31 24 prominent prominent ADJ ejpam-4172 31 25 detail detail NOUN ejpam-4172 31 26 , , PUNCT ejpam-4172 31 27 ends end VERB ejpam-4172 31 28 up up ADP ejpam-4172 31 29 being be AUX ejpam-4172 31 30 the the DET ejpam-4172 31 31 combination combination NOUN ejpam-4172 31 32 of of ADP ejpam-4172 31 33 the the DET ejpam-4172 31 34 energy energy NOUN ejpam-4172 31 35 balance balance NOUN ejpam-4172 31 36 strategy strategy NOUN ejpam-4172 31 37 with with ADP ejpam-4172 31 38 points point NOUN ejpam-4172 31 39 that that PRON ejpam-4172 31 40 have have VERB ejpam-4172 31 41 an an DET ejpam-4172 31 42 extremely extremely ADV ejpam-4172 31 43 extraordinary extraordinary ADJ ejpam-4172 31 44 collocation collocation NOUN ejpam-4172 31 45 point point NOUN ejpam-4172 31 46 . . PUNCT ejpam-4172 32 1 it it PRON ejpam-4172 32 2 is be AUX ejpam-4172 32 3 worth worth ADJ ejpam-4172 32 4 mentioning mention VERB ejpam-4172 32 5 that that SCONJ ejpam-4172 32 6 many many ADJ ejpam-4172 32 7 of of ADP ejpam-4172 32 8 the the DET ejpam-4172 32 9 modern modern ADJ ejpam-4172 32 10 techniques technique NOUN ejpam-4172 32 11 for for ADP ejpam-4172 32 12 solving solve VERB ejpam-4172 32 13 oscillatory oscillatory ADJ ejpam-4172 32 14 problems problem NOUN ejpam-4172 32 15 have have AUX ejpam-4172 32 16 been be AUX ejpam-4172 32 17 established establish VERB ejpam-4172 32 18 by by ADP ejpam-4172 32 19 j. j. PROPN ejpam-4172 32 20 h. h. PROPN ejpam-4172 32 21 he he PRON ejpam-4172 32 22 . . PUNCT ejpam-4172 33 1 if if SCONJ ejpam-4172 33 2 the the DET ejpam-4172 33 3 reader reader NOUN ejpam-4172 33 4 wishes wish VERB ejpam-4172 33 5 to to PART ejpam-4172 33 6 go go VERB ejpam-4172 33 7 deeper deeply ADV ejpam-4172 33 8 into into ADP ejpam-4172 33 9 the the DET ejpam-4172 33 10 study study NOUN ejpam-4172 33 11 , , PUNCT ejpam-4172 33 12 origin origin NOUN ejpam-4172 33 13 and and CCONJ ejpam-4172 33 14 fundamentals fundamental NOUN ejpam-4172 33 15 of of ADP ejpam-4172 33 16 collocation collocation NOUN ejpam-4172 33 17 methods method NOUN ejpam-4172 33 18 , , PUNCT ejpam-4172 33 19 energy energy NOUN ejpam-4172 33 20 balance balance NOUN ejpam-4172 33 21 methods method NOUN ejpam-4172 33 22 , , PUNCT ejpam-4172 33 23 variational variational ADJ ejpam-4172 33 24 methods method NOUN ejpam-4172 33 25 and and CCONJ ejpam-4172 33 26 hamiltonian hamiltonian ADJ ejpam-4172 33 27 techniques technique NOUN ejpam-4172 33 28 we we PRON ejpam-4172 33 29 recommend recommend VERB ejpam-4172 33 30 references reference NOUN ejpam-4172 33 31 [ [ X ejpam-4172 33 32 24 24 NUM ejpam-4172 33 33 , , PUNCT ejpam-4172 33 34 25 25 NUM ejpam-4172 33 35 , , PUNCT ejpam-4172 33 36 26 26 NUM ejpam-4172 33 37 , , PUNCT ejpam-4172 33 38 27 27 NUM ejpam-4172 33 39 , , PUNCT ejpam-4172 33 40 28 28 NUM ejpam-4172 33 41 , , PUNCT ejpam-4172 33 42 29 29 NUM ejpam-4172 33 43 , , PUNCT ejpam-4172 33 44 30 30 NUM ejpam-4172 33 45 , , PUNCT ejpam-4172 33 46 31 31 NUM ejpam-4172 33 47 ] ] PUNCT ejpam-4172 33 48 . . PUNCT ejpam-4172 34 1 2.1 2.1 NUM ejpam-4172 34 2 . . PUNCT ejpam-4172 35 1 the the DET ejpam-4172 35 2 jacobi jacobi PROPN ejpam-4172 35 3 polynomials polynomial NOUN ejpam-4172 35 4 and and CCONJ ejpam-4172 35 5 its its PRON ejpam-4172 35 6 basic basic ADJ ejpam-4172 35 7 properties property NOUN ejpam-4172 35 8 the the DET ejpam-4172 35 9 jacobi jacobi PROPN ejpam-4172 35 10 polynomials polynomial NOUN ejpam-4172 35 11 are be AUX ejpam-4172 35 12 generally generally ADV ejpam-4172 35 13 used use VERB ejpam-4172 35 14 to to PART ejpam-4172 35 15 tackle tackle VERB ejpam-4172 35 16 numerous numerous ADJ ejpam-4172 35 17 issues issue NOUN ejpam-4172 35 18 in in ADP ejpam-4172 35 19 mathematics mathematic NOUN ejpam-4172 35 20 , , PUNCT ejpam-4172 35 21 physics physics NOUN ejpam-4172 35 22 , , PUNCT ejpam-4172 35 23 chemistry chemistry NOUN ejpam-4172 35 24 , , PUNCT ejpam-4172 35 25 biology biology NOUN ejpam-4172 35 26 , , PUNCT ejpam-4172 35 27 engineering engineering NOUN ejpam-4172 35 28 and and CCONJ ejpam-4172 35 29 applied applied ADJ ejpam-4172 35 30 sciences science NOUN ejpam-4172 35 31 . . PUNCT ejpam-4172 36 1 the the DET ejpam-4172 36 2 jacobi jacobi PROPN ejpam-4172 36 3 polynomials polynomial VERB ejpam-4172 36 4 with with ADP ejpam-4172 36 5 general general ADJ ejpam-4172 36 6 parameters parameter NOUN ejpam-4172 36 7 ( ( PUNCT ejpam-4172 36 8 α α X ejpam-4172 36 9 , , PUNCT ejpam-4172 36 10 β β X ejpam-4172 36 11 > > X ejpam-4172 36 12 −1 −1 NOUN ejpam-4172 36 13 ) ) PUNCT ejpam-4172 36 14 are be AUX ejpam-4172 36 15 a a DET ejpam-4172 36 16 sequence sequence NOUN ejpam-4172 36 17 of of ADP ejpam-4172 36 18 polynomials polynomial NOUN ejpam-4172 36 19 p p NOUN ejpam-4172 36 20 ( ( PUNCT ejpam-4172 36 21 α α NOUN ejpam-4172 36 22 , , PUNCT ejpam-4172 36 23 β β NOUN ejpam-4172 36 24 ) ) PUNCT ejpam-4172 36 25 n n PROPN ejpam-4172 36 26 ( ( PUNCT ejpam-4172 36 27 x x NOUN ejpam-4172 36 28 ) ) PUNCT ejpam-4172 36 29 , , PUNCT ejpam-4172 36 30 ( ( PUNCT ejpam-4172 36 31 n n X ejpam-4172 36 32 = = SYM ejpam-4172 36 33 0 0 NUM ejpam-4172 36 34 , , PUNCT ejpam-4172 36 35 1 1 NUM ejpam-4172 36 36 , , PUNCT ejpam-4172 36 37 2 2 NUM ejpam-4172 36 38 , , PUNCT ejpam-4172 36 39 . . PUNCT ejpam-4172 36 40 . . PUNCT ejpam-4172 37 1 . . PUNCT ejpam-4172 37 2 ) ) PUNCT ejpam-4172 38 1 , , PUNCT ejpam-4172 38 2 every every DET ejpam-4172 38 3 one one NUM ejpam-4172 38 4 of of ADP ejpam-4172 38 5 degree degree NOUN ejpam-4172 38 6 n. n. PROPN ejpam-4172 38 7 o. o. PROPN ejpam-4172 39 1 gonzález gonzález PROPN ejpam-4172 39 2 - - PUNCT ejpam-4172 39 3 gaxiola gaxiola PROPN ejpam-4172 39 4 , , PUNCT ejpam-4172 39 5 j. j. PROPN ejpam-4172 39 6 ruiz ruiz PROPN ejpam-4172 39 7 de de PROPN ejpam-4172 39 8 chávez chávez PROPN ejpam-4172 39 9 / / SYM ejpam-4172 39 10 eur eur PROPN ejpam-4172 39 11 . . PUNCT ejpam-4172 40 1 j. j. PROPN ejpam-4172 40 2 pure pure PROPN ejpam-4172 40 3 appl appl PROPN ejpam-4172 40 4 . . PROPN ejpam-4172 40 5 math math PROPN ejpam-4172 40 6 , , PUNCT ejpam-4172 40 7 15 15 NUM ejpam-4172 40 8 ( ( PUNCT ejpam-4172 40 9 1 1 NUM ejpam-4172 40 10 ) ) PUNCT ejpam-4172 40 11 ( ( PUNCT ejpam-4172 40 12 2022 2022 NUM ejpam-4172 40 13 ) ) PUNCT ejpam-4172 40 14 , , PUNCT ejpam-4172 40 15 82 82 NUM ejpam-4172 40 16 - - SYM ejpam-4172 40 17 99 99 NUM ejpam-4172 40 18 84 84 NUM ejpam-4172 40 19 the the DET ejpam-4172 40 20 jacobi jacobi PROPN ejpam-4172 40 21 polynomials polynomial VERB ejpam-4172 40 22 p p X ejpam-4172 40 23 ( ( PUNCT ejpam-4172 40 24 α α X ejpam-4172 40 25 , , PUNCT ejpam-4172 40 26 β β NOUN ejpam-4172 40 27 ) ) PUNCT ejpam-4172 40 28 n n PROPN ejpam-4172 40 29 ( ( PUNCT ejpam-4172 40 30 x x X ejpam-4172 40 31 ) ) PUNCT ejpam-4172 40 32 are be AUX ejpam-4172 40 33 given give VERB ejpam-4172 40 34 explicitly explicitly ADV ejpam-4172 40 35 by by ADP ejpam-4172 40 36 p p PROPN ejpam-4172 40 37 ( ( PUNCT ejpam-4172 40 38 α α X ejpam-4172 40 39 , , PUNCT ejpam-4172 40 40 β β NOUN ejpam-4172 40 41 ) ) PUNCT ejpam-4172 40 42 n n PROPN ejpam-4172 40 43 ( ( PUNCT ejpam-4172 40 44 x x X ejpam-4172 40 45 ) ) PUNCT ejpam-4172 40 46 = = SYM ejpam-4172 40 47 1 1 NUM ejpam-4172 40 48 2n 2n NUM ejpam-4172 40 49 n∑ n∑ X ejpam-4172 40 50 k=0 k=0 PROPN ejpam-4172 40 51 ( ( PUNCT ejpam-4172 40 52 n+ n+ NUM ejpam-4172 40 53 α α NUM ejpam-4172 40 54 n− n− NOUN ejpam-4172 40 55 k k NOUN ejpam-4172 40 56 ) ) PUNCT ejpam-4172 40 57 ( ( PUNCT ejpam-4172 40 58 n+ n+ X ejpam-4172 40 59 β β X ejpam-4172 40 60 k k X ejpam-4172 40 61 ) ) PUNCT ejpam-4172 40 62 ( ( PUNCT ejpam-4172 40 63 x− x− PROPN ejpam-4172 40 64 1)k(x+ 1)k(x+ PROPN ejpam-4172 40 65 1)n−k 1)n−k NUM ejpam-4172 40 66 , , PUNCT ejpam-4172 40 67 ( ( PUNCT ejpam-4172 40 68 1 1 NUM ejpam-4172 40 69 ) ) PUNCT ejpam-4172 40 70 or or CCONJ ejpam-4172 40 71 , , PUNCT ejpam-4172 40 72 equivalently equivalently ADV ejpam-4172 40 73 , , PUNCT ejpam-4172 40 74 by by ADP ejpam-4172 40 75 the the DET ejpam-4172 40 76 well well ADV ejpam-4172 40 77 - - PUNCT ejpam-4172 40 78 known know VERB ejpam-4172 40 79 rodrigues rodrigue NOUN ejpam-4172 40 80 formula formula NOUN ejpam-4172 40 81 ( ( PUNCT ejpam-4172 40 82 x− x− PROPN ejpam-4172 40 83 1)α(x+ 1)α(x+ PROPN ejpam-4172 40 84 1)βp 1)βp PROPN ejpam-4172 40 85 ( ( PUNCT ejpam-4172 40 86 α α X ejpam-4172 40 87 , , PUNCT ejpam-4172 40 88 β β NOUN ejpam-4172 40 89 ) ) PUNCT ejpam-4172 40 90 n n PROPN ejpam-4172 40 91 ( ( PUNCT ejpam-4172 40 92 x x X ejpam-4172 40 93 ) ) PUNCT ejpam-4172 40 94 = = SYM ejpam-4172 40 95 ( ( PUNCT ejpam-4172 40 96 −1)n −1)n PROPN ejpam-4172 40 97 2nn 2nn PROPN ejpam-4172 40 98 ! ! PUNCT ejpam-4172 41 1 dn dn PROPN ejpam-4172 41 2 dxn dxn PROPN ejpam-4172 41 3 [ [ PUNCT ejpam-4172 41 4 ( ( PUNCT ejpam-4172 41 5 1− 1− NUM ejpam-4172 41 6 x)n+α(1 x)n+α(1 PUNCT ejpam-4172 41 7 + + CCONJ ejpam-4172 41 8 x)n+β x)n+β PRON ejpam-4172 41 9 ] ] PUNCT ejpam-4172 41 10 , , PUNCT ejpam-4172 41 11 x x PUNCT ejpam-4172 41 12 ∈ ∈ PROPN ejpam-4172 42 1 [ [ X ejpam-4172 42 2 −1 −1 NOUN ejpam-4172 42 3 , , PUNCT ejpam-4172 42 4 1 1 NUM ejpam-4172 42 5 ] ] PUNCT ejpam-4172 42 6 . . PUNCT ejpam-4172 43 1 ( ( PUNCT ejpam-4172 43 2 2 2 X ejpam-4172 43 3 ) ) PUNCT ejpam-4172 43 4 let let VERB ejpam-4172 43 5 w(α w(α NOUN ejpam-4172 43 6 , , PUNCT ejpam-4172 43 7 β)(x β)(x NOUN ejpam-4172 43 8 ) ) PUNCT ejpam-4172 44 1 = = PUNCT ejpam-4172 44 2 ( ( PUNCT ejpam-4172 44 3 x− x− PROPN ejpam-4172 44 4 1)α(x+1)β 1)α(x+1)β NUM ejpam-4172 44 5 be be AUX ejpam-4172 44 6 the the DET ejpam-4172 44 7 jacobi jacobi PROPN ejpam-4172 44 8 weight weight NOUN ejpam-4172 44 9 function function NOUN ejpam-4172 44 10 . . PUNCT ejpam-4172 45 1 for for ADP ejpam-4172 45 2 α α NOUN ejpam-4172 45 3 , , PUNCT ejpam-4172 45 4 β β X ejpam-4172 45 5 > > X ejpam-4172 45 6 −1 −1 NOUN ejpam-4172 45 7 , , PUNCT ejpam-4172 45 8 the the DET ejpam-4172 45 9 jacobi jacobi PROPN ejpam-4172 45 10 polynomials polynomial NOUN ejpam-4172 45 11 satisfy satisfy VERB ejpam-4172 45 12 the the DET ejpam-4172 45 13 following follow VERB ejpam-4172 45 14 orthogonality orthogonality NOUN ejpam-4172 45 15 relation:∫ relation:∫ PROPN ejpam-4172 45 16 1 1 NUM ejpam-4172 45 17 −1 −1 NOUN ejpam-4172 45 18 p p NOUN ejpam-4172 45 19 ( ( PUNCT ejpam-4172 45 20 α α X ejpam-4172 45 21 , , PUNCT ejpam-4172 45 22 β β NOUN ejpam-4172 45 23 ) ) PUNCT ejpam-4172 45 24 n n PROPN ejpam-4172 45 25 ( ( PUNCT ejpam-4172 45 26 x)p x)p X ejpam-4172 45 27 ( ( PUNCT ejpam-4172 45 28 α α X ejpam-4172 45 29 , , PUNCT ejpam-4172 45 30 β β NOUN ejpam-4172 45 31 ) ) PUNCT ejpam-4172 45 32 m m VERB ejpam-4172 45 33 ( ( PUNCT ejpam-4172 45 34 x)w(α x)w(α ADJ ejpam-4172 45 35 , , PUNCT ejpam-4172 45 36 β)(x)dx β)(x)dx NOUN ejpam-4172 45 37 = = SYM ejpam-4172 45 38 cα cα NOUN ejpam-4172 45 39 , , PUNCT ejpam-4172 45 40 β β X ejpam-4172 45 41 n n PRON ejpam-4172 45 42 δn δn NOUN ejpam-4172 45 43 , , PUNCT ejpam-4172 45 44 m m VERB ejpam-4172 45 45 ( ( PUNCT ejpam-4172 45 46 3 3 NUM ejpam-4172 45 47 ) ) PUNCT ejpam-4172 45 48 where where SCONJ ejpam-4172 45 49 δn δn NOUN ejpam-4172 45 50 , , PUNCT ejpam-4172 45 51 m m VERB ejpam-4172 45 52 is be AUX ejpam-4172 45 53 the the DET ejpam-4172 45 54 kronecker kronecker NOUN ejpam-4172 45 55 function function NOUN ejpam-4172 45 56 , , PUNCT ejpam-4172 45 57 and and CCONJ ejpam-4172 45 58 cα cα ADP ejpam-4172 45 59 , , PUNCT ejpam-4172 45 60 β β X ejpam-4172 45 61 n n NOUN ejpam-4172 45 62 = = SYM ejpam-4172 45 63 2α+β+1γ(n+ 2α+β+1γ(n+ NUM ejpam-4172 45 64 α+ α+ PUNCT ejpam-4172 45 65 1)γ(n+ 1)γ(n+ PROPN ejpam-4172 45 66 β β X ejpam-4172 45 67 + + NOUN ejpam-4172 45 68 1 1 X ejpam-4172 45 69 ) ) PUNCT ejpam-4172 45 70 n!(2n+ n!(2n+ PRON ejpam-4172 46 1 α+ α+ X ejpam-4172 46 2 β β X ejpam-4172 46 3 + + CCONJ ejpam-4172 46 4 1)γ(n+ 1)γ(n+ NUM ejpam-4172 46 5 α+ α+ PUNCT ejpam-4172 46 6 β β NOUN ejpam-4172 46 7 + + NOUN ejpam-4172 46 8 1 1 NUM ejpam-4172 46 9 ) ) PUNCT ejpam-4172 46 10 . . PUNCT ejpam-4172 47 1 ( ( PUNCT ejpam-4172 47 2 4 4 X ejpam-4172 47 3 ) ) PUNCT ejpam-4172 47 4 the the DET ejpam-4172 47 5 jacobi jacobi PROPN ejpam-4172 47 6 polynomial polynomial PROPN ejpam-4172 47 7 is be AUX ejpam-4172 47 8 the the DET ejpam-4172 47 9 solution solution NOUN ejpam-4172 47 10 the the DET ejpam-4172 47 11 following follow VERB ejpam-4172 47 12 second second ADJ ejpam-4172 47 13 - - PUNCT ejpam-4172 47 14 order order NOUN ejpam-4172 47 15 , , PUNCT ejpam-4172 47 16 homogeneous homogeneous ADJ ejpam-4172 47 17 differential differential NOUN ejpam-4172 47 18 equation equation NOUN ejpam-4172 47 19 : : PUNCT ejpam-4172 47 20 ( ( PUNCT ejpam-4172 47 21 1− 1− NUM ejpam-4172 47 22 x2)y′′ x2)y′′ NOUN ejpam-4172 48 1 + + CCONJ ejpam-4172 49 1 [ [ X ejpam-4172 49 2 β β X ejpam-4172 49 3 − − NOUN ejpam-4172 49 4 α− α− ADP ejpam-4172 49 5 x(α+ x(α+ X ejpam-4172 49 6 β β PROPN ejpam-4172 49 7 + + PROPN ejpam-4172 49 8 2)]y′ 2)]y′ PROPN ejpam-4172 49 9 + + CCONJ ejpam-4172 49 10 n(n+ n(n+ X ejpam-4172 49 11 α+ α+ X ejpam-4172 49 12 β β NOUN ejpam-4172 49 13 + + NOUN ejpam-4172 49 14 1)y 1)y NUM ejpam-4172 49 15 = = SYM ejpam-4172 49 16 0 0 NUM ejpam-4172 49 17 . . PUNCT ejpam-4172 50 1 ( ( PUNCT ejpam-4172 50 2 5 5 X ejpam-4172 50 3 ) ) PUNCT ejpam-4172 50 4 the the DET ejpam-4172 50 5 following follow VERB ejpam-4172 50 6 recurrence recurrence NOUN ejpam-4172 50 7 relation relation NOUN ejpam-4172 50 8 generates generate VERB ejpam-4172 50 9 the the DET ejpam-4172 50 10 jacobi jacobi PROPN ejpam-4172 50 11 polynomials polynomial NOUN ejpam-4172 50 12 [ [ X ejpam-4172 50 13 32 32 NUM ejpam-4172 50 14 ] ] X ejpam-4172 50 15 p p X ejpam-4172 50 16 ( ( PUNCT ejpam-4172 50 17 α α X ejpam-4172 50 18 , , PUNCT ejpam-4172 50 19 β β NOUN ejpam-4172 50 20 ) ) PUNCT ejpam-4172 50 21 n+1 n+1 PROPN ejpam-4172 50 22 ( ( PUNCT ejpam-4172 50 23 x x NOUN ejpam-4172 50 24 ) ) PUNCT ejpam-4172 50 25 = = SYM ejpam-4172 50 26 ( ( PUNCT ejpam-4172 50 27 a(α a(α NOUN ejpam-4172 50 28 , , PUNCT ejpam-4172 50 29 β)n β)n SYM ejpam-4172 50 30 x− x− PROPN ejpam-4172 50 31 b(α b(α PROPN ejpam-4172 50 32 , , PUNCT ejpam-4172 50 33 β)n β)n PUNCT ejpam-4172 50 34 ) ) PUNCT ejpam-4172 50 35 p p NOUN ejpam-4172 50 36 ( ( PUNCT ejpam-4172 50 37 α α X ejpam-4172 50 38 , , PUNCT ejpam-4172 50 39 β β NOUN ejpam-4172 50 40 ) ) PUNCT ejpam-4172 50 41 n n PROPN ejpam-4172 50 42 ( ( PUNCT ejpam-4172 50 43 x)− x)− PROPN ejpam-4172 50 44 c(α c(α PROPN ejpam-4172 50 45 , , PUNCT ejpam-4172 50 46 β)n β)n ADP ejpam-4172 50 47 p p X ejpam-4172 50 48 ( ( PUNCT ejpam-4172 50 49 α α X ejpam-4172 50 50 , , PUNCT ejpam-4172 50 51 β β NOUN ejpam-4172 50 52 ) ) PUNCT ejpam-4172 50 53 n−1 n−1 PROPN ejpam-4172 50 54 ( ( PUNCT ejpam-4172 50 55 x x NOUN ejpam-4172 50 56 ) ) PUNCT ejpam-4172 50 57 , , PUNCT ejpam-4172 50 58 n n X ejpam-4172 50 59 ≥ ≥ NOUN ejpam-4172 50 60 1 1 NUM ejpam-4172 50 61 ( ( PUNCT ejpam-4172 50 62 6 6 NUM ejpam-4172 50 63 ) ) PUNCT ejpam-4172 50 64 p p NOUN ejpam-4172 50 65 ( ( PUNCT ejpam-4172 50 66 α α X ejpam-4172 50 67 , , PUNCT ejpam-4172 50 68 β β NOUN ejpam-4172 50 69 ) ) PUNCT ejpam-4172 50 70 0 0 NUM ejpam-4172 51 1 ( ( PUNCT ejpam-4172 51 2 x x X ejpam-4172 51 3 ) ) PUNCT ejpam-4172 51 4 = = SYM ejpam-4172 51 5 1 1 NUM ejpam-4172 51 6 , , PUNCT ejpam-4172 51 7 p p X ejpam-4172 51 8 ( ( PUNCT ejpam-4172 51 9 α α NOUN ejpam-4172 51 10 , , PUNCT ejpam-4172 51 11 β β NOUN ejpam-4172 51 12 ) ) PUNCT ejpam-4172 51 13 1 1 NUM ejpam-4172 51 14 ( ( PUNCT ejpam-4172 51 15 x x NOUN ejpam-4172 51 16 ) ) PUNCT ejpam-4172 51 17 = = SYM ejpam-4172 51 18 1 1 NUM ejpam-4172 51 19 2 2 NUM ejpam-4172 51 20 ( ( PUNCT ejpam-4172 51 21 ( ( PUNCT ejpam-4172 51 22 α+ α+ X ejpam-4172 51 23 β β X ejpam-4172 51 24 + + NOUN ejpam-4172 51 25 2)x+ 2)x+ NUM ejpam-4172 51 26 α− α− ADP ejpam-4172 51 27 β β X ejpam-4172 51 28 ) ) PUNCT ejpam-4172 51 29 , , PUNCT ejpam-4172 51 30 ( ( PUNCT ejpam-4172 51 31 7 7 X ejpam-4172 51 32 ) ) PUNCT ejpam-4172 51 33 where where SCONJ ejpam-4172 51 34 a(α a(α NOUN ejpam-4172 51 35 , , PUNCT ejpam-4172 51 36 β)n β)n X ejpam-4172 51 37 = = SYM ejpam-4172 51 38 ( ( PUNCT ejpam-4172 51 39 2n+ 2n+ NUM ejpam-4172 51 40 α+ α+ X ejpam-4172 52 1 β β PROPN ejpam-4172 52 2 + + ADJ ejpam-4172 52 3 1)(2n+ 1)(2n+ NUM ejpam-4172 52 4 α+ α+ X ejpam-4172 52 5 β β X ejpam-4172 52 6 + + NOUN ejpam-4172 52 7 2 2 NUM ejpam-4172 52 8 ) ) PUNCT ejpam-4172 52 9 2(n+ 2(n+ NOUN ejpam-4172 53 1 1)(n+ 1)(n+ NUM ejpam-4172 53 2 α+ α+ X ejpam-4172 53 3 β β X ejpam-4172 53 4 + + NOUN ejpam-4172 53 5 1 1 NUM ejpam-4172 53 6 ) ) PUNCT ejpam-4172 53 7 , , PUNCT ejpam-4172 53 8 ( ( PUNCT ejpam-4172 53 9 8) 8) NUM ejpam-4172 53 10 b(α b(α NOUN ejpam-4172 53 11 , , PUNCT ejpam-4172 53 12 β)n β)n PUNCT ejpam-4172 53 13 = = SYM ejpam-4172 53 14 ( ( PUNCT ejpam-4172 53 15 β2 β2 NOUN ejpam-4172 53 16 − − PROPN ejpam-4172 53 17 α2)(2n+ α2)(2n+ NUM ejpam-4172 53 18 α+ α+ X ejpam-4172 53 19 β β X ejpam-4172 53 20 + + NOUN ejpam-4172 53 21 1 1 NUM ejpam-4172 53 22 ) ) PUNCT ejpam-4172 53 23 2(n+ 2(n+ NOUN ejpam-4172 54 1 1)(n+ 1)(n+ NUM ejpam-4172 54 2 α+ α+ X ejpam-4172 54 3 β β X ejpam-4172 55 1 + + SYM ejpam-4172 55 2 1)(2n+ 1)(2n+ NUM ejpam-4172 55 3 α+ α+ X ejpam-4172 55 4 β β NOUN ejpam-4172 55 5 ) ) PUNCT ejpam-4172 55 6 , , PUNCT ejpam-4172 55 7 ( ( PUNCT ejpam-4172 55 8 9 9 X ejpam-4172 55 9 ) ) PUNCT ejpam-4172 55 10 c(α c(α NOUN ejpam-4172 55 11 , , PUNCT ejpam-4172 55 12 β)n β)n X ejpam-4172 55 13 = = SYM ejpam-4172 55 14 ( ( PUNCT ejpam-4172 55 15 n+ n+ NUM ejpam-4172 55 16 α)(n+ α)(n+ NOUN ejpam-4172 55 17 β)(2n+ β)(2n+ PROPN ejpam-4172 55 18 α+ α+ X ejpam-4172 55 19 β β X ejpam-4172 55 20 + + NOUN ejpam-4172 55 21 2 2 NUM ejpam-4172 55 22 ) ) PUNCT ejpam-4172 55 23 ( ( PUNCT ejpam-4172 55 24 n+ n+ NUM ejpam-4172 55 25 1)(n+ 1)(n+ NUM ejpam-4172 55 26 α+ α+ X ejpam-4172 55 27 β β X ejpam-4172 56 1 + + SYM ejpam-4172 56 2 1)(2n+ 1)(2n+ NUM ejpam-4172 56 3 α+ α+ X ejpam-4172 56 4 β β NOUN ejpam-4172 56 5 ) ) PUNCT ejpam-4172 56 6 . . PUNCT ejpam-4172 57 1 ( ( PUNCT ejpam-4172 57 2 10 10 NUM ejpam-4172 57 3 ) ) PUNCT ejpam-4172 57 4 o. o. PROPN ejpam-4172 57 5 gonzález gonzález PROPN ejpam-4172 57 6 - - PUNCT ejpam-4172 57 7 gaxiola gaxiola PROPN ejpam-4172 57 8 , , PUNCT ejpam-4172 57 9 j. j. PROPN ejpam-4172 57 10 ruiz ruiz PROPN ejpam-4172 57 11 de de PROPN ejpam-4172 57 12 chávez chávez PROPN ejpam-4172 57 13 / / SYM ejpam-4172 57 14 eur eur PROPN ejpam-4172 57 15 . . PUNCT ejpam-4172 58 1 j. j. PROPN ejpam-4172 58 2 pure pure PROPN ejpam-4172 58 3 appl appl PROPN ejpam-4172 58 4 . . PROPN ejpam-4172 58 5 math math PROPN ejpam-4172 58 6 , , PUNCT ejpam-4172 58 7 15 15 NUM ejpam-4172 58 8 ( ( PUNCT ejpam-4172 58 9 1 1 NUM ejpam-4172 58 10 ) ) PUNCT ejpam-4172 58 11 ( ( PUNCT ejpam-4172 58 12 2022 2022 NUM ejpam-4172 58 13 ) ) PUNCT ejpam-4172 58 14 , , PUNCT ejpam-4172 58 15 82 82 NUM ejpam-4172 58 16 - - SYM ejpam-4172 58 17 99 99 NUM ejpam-4172 58 18 85 85 NUM ejpam-4172 58 19 2.2 2.2 NUM ejpam-4172 58 20 . . PUNCT ejpam-4172 59 1 basic basic ADJ ejpam-4172 59 2 idea idea NOUN ejpam-4172 59 3 of of ADP ejpam-4172 59 4 energy energy NOUN ejpam-4172 59 5 balance balance NOUN ejpam-4172 59 6 method method NOUN ejpam-4172 59 7 with with ADP ejpam-4172 59 8 a a DET ejpam-4172 59 9 special special ADJ ejpam-4172 59 10 collocation collocation NOUN ejpam-4172 59 11 point point NOUN ejpam-4172 59 12 consider consider VERB ejpam-4172 59 13 a a DET ejpam-4172 59 14 one one NUM ejpam-4172 59 15 - - PUNCT ejpam-4172 59 16 dimensional dimensional ADJ ejpam-4172 59 17 , , PUNCT ejpam-4172 59 18 free free ADJ ejpam-4172 59 19 nonlinear nonlinear ADJ ejpam-4172 59 20 oscillator oscillator NOUN ejpam-4172 59 21 ( ( PUNCT ejpam-4172 59 22 undamped undampe VERB ejpam-4172 59 23 and and CCONJ ejpam-4172 59 24 unforced unforced ADJ ejpam-4172 59 25 ) ) PUNCT ejpam-4172 59 26 governed govern VERB ejpam-4172 59 27 by by ADP ejpam-4172 59 28 ü+ ü+ PROPN ejpam-4172 59 29 f(u f(u PROPN ejpam-4172 59 30 ) ) PUNCT ejpam-4172 59 31 = = SYM ejpam-4172 60 1 0 0 NUM ejpam-4172 60 2 , , PUNCT ejpam-4172 60 3 ( ( PUNCT ejpam-4172 60 4 11 11 NUM ejpam-4172 60 5 ) ) PUNCT ejpam-4172 60 6 with with ADP ejpam-4172 60 7 the the DET ejpam-4172 60 8 initial initial ADJ ejpam-4172 60 9 conditions condition NOUN ejpam-4172 60 10 u(0 u(0 NOUN ejpam-4172 60 11 ) ) PUNCT ejpam-4172 60 12 = = SYM ejpam-4172 61 1 a a PRON ejpam-4172 61 2 , , PUNCT ejpam-4172 61 3 u̇(0 u̇(0 NOUN ejpam-4172 61 4 ) ) PUNCT ejpam-4172 61 5 = = SYM ejpam-4172 61 6 0 0 NUM ejpam-4172 61 7 , , PUNCT ejpam-4172 61 8 ( ( PUNCT ejpam-4172 61 9 12 12 NUM ejpam-4172 61 10 ) ) PUNCT ejpam-4172 61 11 where where SCONJ ejpam-4172 61 12 a a DET ejpam-4172 61 13 dot dot NOUN ejpam-4172 61 14 denotes denote NOUN ejpam-4172 61 15 differentiation differentiation NOUN ejpam-4172 61 16 with with ADP ejpam-4172 61 17 respect respect NOUN ejpam-4172 61 18 to to ADP ejpam-4172 61 19 the the DET ejpam-4172 61 20 time time NOUN ejpam-4172 61 21 t t PROPN ejpam-4172 61 22 , , PUNCT ejpam-4172 61 23 u u PROPN ejpam-4172 61 24 is be AUX ejpam-4172 61 25 the the DET ejpam-4172 61 26 dimensionless dimensionless ADJ ejpam-4172 61 27 displacement displacement NOUN ejpam-4172 61 28 , , PUNCT ejpam-4172 61 29 a a PRON ejpam-4172 61 30 will will AUX ejpam-4172 61 31 be be AUX ejpam-4172 61 32 a a DET ejpam-4172 61 33 given give VERB ejpam-4172 61 34 constant constant ADJ ejpam-4172 61 35 , , PUNCT ejpam-4172 61 36 and and CCONJ ejpam-4172 61 37 the the DET ejpam-4172 61 38 nonlinear nonlinear ADJ ejpam-4172 61 39 restoring restore VERB ejpam-4172 61 40 force force NOUN ejpam-4172 61 41 f(u f(u PROPN ejpam-4172 61 42 ) ) PUNCT ejpam-4172 61 43 is be AUX ejpam-4172 61 44 an an DET ejpam-4172 61 45 odd odd ADJ ejpam-4172 61 46 function function NOUN ejpam-4172 61 47 of of ADP ejpam-4172 61 48 u u NOUN ejpam-4172 61 49 , , PUNCT ejpam-4172 61 50 i.e. i.e. X ejpam-4172 61 51 f(−u f(−u NOUN ejpam-4172 61 52 ) ) PUNCT ejpam-4172 61 53 = = SYM ejpam-4172 61 54 −f(u −f(u NOUN ejpam-4172 61 55 ) ) PUNCT ejpam-4172 61 56 for for ADP ejpam-4172 61 57 u u PROPN ejpam-4172 61 58 ∈ ∈ PROPN ejpam-4172 61 59 [ [ X ejpam-4172 61 60 −a −a X ejpam-4172 61 61 , , PUNCT ejpam-4172 61 62 a a PRON ejpam-4172 61 63 ] ] X ejpam-4172 61 64 . . PUNCT ejpam-4172 62 1 clearly clearly ADV ejpam-4172 62 2 u u X ejpam-4172 62 3 = = NOUN ejpam-4172 62 4 0 0 NUM ejpam-4172 62 5 is be AUX ejpam-4172 62 6 the the DET ejpam-4172 62 7 equilibrium equilibrium NOUN ejpam-4172 62 8 position position NOUN ejpam-4172 62 9 . . PUNCT ejpam-4172 63 1 the the DET ejpam-4172 63 2 system system NOUN ejpam-4172 63 3 oscillates oscillate NOUN ejpam-4172 63 4 between between ADP ejpam-4172 63 5 the the DET ejpam-4172 63 6 symmetric symmetric ADJ ejpam-4172 63 7 limits limit NOUN ejpam-4172 63 8 −a −a ADJ ejpam-4172 63 9 and and CCONJ ejpam-4172 63 10 a. a. NOUN ejpam-4172 63 11 in in ADP ejpam-4172 63 12 the the DET ejpam-4172 63 13 event event NOUN ejpam-4172 63 14 that that PRON ejpam-4172 63 15 f(u f(u PROPN ejpam-4172 63 16 ) ) PUNCT ejpam-4172 63 17 is be AUX ejpam-4172 63 18 a a DET ejpam-4172 63 19 nonlinear nonlinear ADJ ejpam-4172 63 20 function function NOUN ejpam-4172 63 21 , , PUNCT ejpam-4172 63 22 both both DET ejpam-4172 63 23 period period NOUN ejpam-4172 63 24 t t NOUN ejpam-4172 63 25 and and CCONJ ejpam-4172 63 26 frequency frequency PROPN ejpam-4172 63 27 ω ω NOUN ejpam-4172 63 28 = = PUNCT ejpam-4172 63 29 2π 2π NOUN ejpam-4172 63 30 / / SYM ejpam-4172 63 31 t t NOUN ejpam-4172 63 32 of of ADP ejpam-4172 63 33 the the DET ejpam-4172 63 34 relating relate VERB ejpam-4172 63 35 oscillation oscillation NOUN ejpam-4172 63 36 are be AUX ejpam-4172 63 37 reliant reliant ADJ ejpam-4172 63 38 upon upon SCONJ ejpam-4172 63 39 the the DET ejpam-4172 63 40 amplitude amplitude NOUN ejpam-4172 63 41 of of ADP ejpam-4172 63 42 oscillation oscillation NOUN ejpam-4172 63 43 a. a. NOUN ejpam-4172 63 44 the the DET ejpam-4172 63 45 connection connection NOUN ejpam-4172 63 46 between between ADP ejpam-4172 63 47 the the DET ejpam-4172 63 48 frequency frequency NOUN ejpam-4172 63 49 and and CCONJ ejpam-4172 63 50 amplitude amplitude NOUN ejpam-4172 63 51 is be AUX ejpam-4172 63 52 the the DET ejpam-4172 63 53 main main ADJ ejpam-4172 63 54 property property NOUN ejpam-4172 63 55 of of ADP ejpam-4172 63 56 a a DET ejpam-4172 63 57 nonlinear nonlinear ADJ ejpam-4172 63 58 oscillator oscillator NOUN ejpam-4172 63 59 ; ; PUNCT ejpam-4172 63 60 see see VERB ejpam-4172 63 61 [ [ X ejpam-4172 63 62 33 33 NUM ejpam-4172 63 63 , , PUNCT ejpam-4172 63 64 34 34 NUM ejpam-4172 63 65 , , PUNCT ejpam-4172 63 66 35 35 NUM ejpam-4172 63 67 , , PUNCT ejpam-4172 63 68 36 36 NUM ejpam-4172 63 69 , , PUNCT ejpam-4172 63 70 37 37 NUM ejpam-4172 63 71 , , PUNCT ejpam-4172 63 72 38 38 NUM ejpam-4172 63 73 ] ] PUNCT ejpam-4172 63 74 and and CCONJ ejpam-4172 63 75 references reference NOUN ejpam-4172 63 76 therein therein ADV ejpam-4172 63 77 . . PUNCT ejpam-4172 64 1 the the DET ejpam-4172 64 2 variational variational ADJ ejpam-4172 64 3 principle principle NOUN ejpam-4172 64 4 applied apply VERB ejpam-4172 64 5 to to ADP ejpam-4172 64 6 eq eq PROPN ejpam-4172 64 7 . . PUNCT ejpam-4172 65 1 ( ( PUNCT ejpam-4172 65 2 11 11 NUM ejpam-4172 65 3 ) ) PUNCT ejpam-4172 65 4 can can AUX ejpam-4172 65 5 be be AUX ejpam-4172 65 6 expressed express VERB ejpam-4172 65 7 as as ADP ejpam-4172 65 8 v v NOUN ejpam-4172 65 9 ( ( PUNCT ejpam-4172 65 10 u u NOUN ejpam-4172 65 11 ) ) PUNCT ejpam-4172 65 12 = = SYM ejpam-4172 66 1 ∫ ∫ PROPN ejpam-4172 66 2 t t PROPN ejpam-4172 66 3 0 0 NUM ejpam-4172 66 4 { { PUNCT ejpam-4172 66 5 − − PROPN ejpam-4172 66 6 1 1 NUM ejpam-4172 66 7 2 2 NUM ejpam-4172 66 8 u̇2 u̇2 PROPN ejpam-4172 66 9 + + NUM ejpam-4172 66 10 f f X ejpam-4172 66 11 ( ( PUNCT ejpam-4172 66 12 u u NOUN ejpam-4172 66 13 ) ) PUNCT ejpam-4172 66 14 } } PUNCT ejpam-4172 66 15 dt̂ dt̂ PROPN ejpam-4172 66 16 , , PUNCT ejpam-4172 66 17 ( ( PUNCT ejpam-4172 66 18 13 13 NUM ejpam-4172 66 19 ) ) PUNCT ejpam-4172 66 20 in in ADP ejpam-4172 66 21 which which PRON ejpam-4172 66 22 , , PUNCT ejpam-4172 66 23 f f PROPN ejpam-4172 66 24 ( ( PUNCT ejpam-4172 66 25 u u NOUN ejpam-4172 66 26 ) ) PUNCT ejpam-4172 66 27 = = SYM ejpam-4172 66 28 ∫ ∫ PROPN ejpam-4172 66 29 f(u f(u PROPN ejpam-4172 66 30 ) ) PUNCT ejpam-4172 66 31 du du PROPN ejpam-4172 66 32 . . PUNCT ejpam-4172 67 1 hence hence ADV ejpam-4172 67 2 , , PUNCT ejpam-4172 67 3 its its PRON ejpam-4172 67 4 hamiltonian hamiltonian ADJ ejpam-4172 67 5 function function NOUN ejpam-4172 67 6 becomes become VERB ejpam-4172 67 7 h h NOUN ejpam-4172 67 8 = = NOUN ejpam-4172 67 9 1 1 NUM ejpam-4172 67 10 2 2 NUM ejpam-4172 67 11 u̇2 u̇2 PROPN ejpam-4172 67 12 + + NUM ejpam-4172 67 13 f f X ejpam-4172 67 14 ( ( PUNCT ejpam-4172 67 15 u u NOUN ejpam-4172 67 16 ) ) PUNCT ejpam-4172 67 17 = = SYM ejpam-4172 67 18 f f X ejpam-4172 67 19 ( ( PUNCT ejpam-4172 67 20 a a NOUN ejpam-4172 67 21 ) ) PUNCT ejpam-4172 67 22 , , PUNCT ejpam-4172 67 23 ( ( PUNCT ejpam-4172 67 24 14 14 NUM ejpam-4172 67 25 ) ) PUNCT ejpam-4172 67 26 or or CCONJ ejpam-4172 67 27 equivalently equivalently ADV ejpam-4172 67 28 r(t r(t VERB ejpam-4172 67 29 ) ) PUNCT ejpam-4172 67 30 = = SYM ejpam-4172 67 31 1 1 NUM ejpam-4172 67 32 2 2 NUM ejpam-4172 67 33 u̇2 u̇2 PROPN ejpam-4172 67 34 + + NUM ejpam-4172 67 35 f f X ejpam-4172 67 36 ( ( PUNCT ejpam-4172 67 37 u)− u)− PROPN ejpam-4172 67 38 f f PROPN ejpam-4172 67 39 ( ( PUNCT ejpam-4172 67 40 a a X ejpam-4172 67 41 ) ) PUNCT ejpam-4172 67 42 = = SYM ejpam-4172 67 43 0 0 X ejpam-4172 67 44 . . PUNCT ejpam-4172 68 1 ( ( PUNCT ejpam-4172 68 2 15 15 NUM ejpam-4172 68 3 ) ) PUNCT ejpam-4172 68 4 according accord VERB ejpam-4172 68 5 to to ADP ejpam-4172 68 6 the the DET ejpam-4172 68 7 initial initial ADJ ejpam-4172 68 8 conditions condition NOUN ejpam-4172 68 9 ( ( PUNCT ejpam-4172 68 10 12 12 NUM ejpam-4172 68 11 ) ) PUNCT ejpam-4172 68 12 , , PUNCT ejpam-4172 68 13 the the DET ejpam-4172 68 14 trial trial NOUN ejpam-4172 68 15 function function NOUN ejpam-4172 68 16 for for ADP ejpam-4172 68 17 eq eq PROPN ejpam-4172 68 18 . . PUNCT ejpam-4172 69 1 ( ( PUNCT ejpam-4172 69 2 11 11 NUM ejpam-4172 69 3 ) ) PUNCT ejpam-4172 69 4 is be AUX ejpam-4172 69 5 introduced introduce VERB ejpam-4172 69 6 to to PART ejpam-4172 69 7 determine determine VERB ejpam-4172 69 8 the the DET ejpam-4172 69 9 angular angular ADJ ejpam-4172 69 10 frequency frequency NOUN ejpam-4172 69 11 : : PUNCT ejpam-4172 69 12 u(t u(t NOUN ejpam-4172 69 13 ) ) PUNCT ejpam-4172 69 14 = = PUNCT ejpam-4172 70 1 a a DET ejpam-4172 70 2 cos(ωt cos(ωt NOUN ejpam-4172 70 3 ) ) PUNCT ejpam-4172 70 4 . . PUNCT ejpam-4172 71 1 ( ( PUNCT ejpam-4172 71 2 16 16 X ejpam-4172 71 3 ) ) PUNCT ejpam-4172 71 4 substituting substitute VERB ejpam-4172 71 5 ( ( PUNCT ejpam-4172 71 6 16 16 NUM ejpam-4172 71 7 ) ) PUNCT ejpam-4172 71 8 into into ADP ejpam-4172 71 9 ( ( PUNCT ejpam-4172 71 10 15 15 NUM ejpam-4172 71 11 ) ) PUNCT ejpam-4172 71 12 , , PUNCT ejpam-4172 71 13 the the DET ejpam-4172 71 14 corresponding corresponding ADJ ejpam-4172 71 15 residual residual ADJ ejpam-4172 71 16 turns turn VERB ejpam-4172 71 17 out out ADP ejpam-4172 71 18 to to PART ejpam-4172 71 19 be be AUX ejpam-4172 71 20 [ [ PUNCT ejpam-4172 71 21 39 39 NUM ejpam-4172 71 22 ] ] PUNCT ejpam-4172 71 23 : : PUNCT ejpam-4172 71 24 r(t r(t NOUN ejpam-4172 71 25 ) ) PUNCT ejpam-4172 71 26 = = SYM ejpam-4172 71 27 1 1 NUM ejpam-4172 71 28 2 2 NUM ejpam-4172 71 29 a2ω2 a2ω2 NOUN ejpam-4172 71 30 sin2(ωt sin2(ωt NOUN ejpam-4172 71 31 ) ) PUNCT ejpam-4172 72 1 + + NUM ejpam-4172 72 2 f f X ejpam-4172 72 3 ( ( PUNCT ejpam-4172 72 4 a a DET ejpam-4172 72 5 cos(ωt))− cos(ωt))− NUM ejpam-4172 72 6 f f NOUN ejpam-4172 72 7 ( ( PUNCT ejpam-4172 72 8 a a NOUN ejpam-4172 72 9 ) ) PUNCT ejpam-4172 72 10 . . PUNCT ejpam-4172 73 1 ( ( PUNCT ejpam-4172 73 2 17 17 NUM ejpam-4172 73 3 ) ) PUNCT ejpam-4172 73 4 according accord VERB ejpam-4172 73 5 to to ADP ejpam-4172 73 6 the the DET ejpam-4172 73 7 classical classical ADJ ejpam-4172 73 8 energy energy NOUN ejpam-4172 73 9 balance balance NOUN ejpam-4172 73 10 method method NOUN ejpam-4172 73 11 [ [ X ejpam-4172 73 12 39 39 NUM ejpam-4172 73 13 , , PUNCT ejpam-4172 73 14 40 40 NUM ejpam-4172 73 15 , , PUNCT ejpam-4172 73 16 41 41 NUM ejpam-4172 73 17 ] ] PUNCT ejpam-4172 73 18 , , PUNCT ejpam-4172 73 19 we we PRON ejpam-4172 73 20 have have VERB ejpam-4172 73 21 ωebm ωebm NOUN ejpam-4172 73 22 ( ( PUNCT ejpam-4172 73 23 a a X ejpam-4172 73 24 ) ) PUNCT ejpam-4172 73 25 = = SYM ejpam-4172 74 1 √ √ NOUN ejpam-4172 74 2 2(f 2(f NUM ejpam-4172 74 3 ( ( PUNCT ejpam-4172 74 4 a)− a)− PROPN ejpam-4172 74 5 f f PROPN ejpam-4172 74 6 ( ( PUNCT ejpam-4172 74 7 a a DET ejpam-4172 74 8 cos(ωt cos(ωt NOUN ejpam-4172 74 9 ) ) PUNCT ejpam-4172 74 10 ) ) PUNCT ejpam-4172 74 11 ) ) PUNCT ejpam-4172 75 1 a a DET ejpam-4172 75 2 sin(ωt sin(ωt NOUN ejpam-4172 75 3 ) ) PUNCT ejpam-4172 75 4 , , PUNCT ejpam-4172 75 5 ( ( PUNCT ejpam-4172 75 6 18 18 NUM ejpam-4172 75 7 ) ) PUNCT ejpam-4172 75 8 with with ADP ejpam-4172 75 9 the the DET ejpam-4172 75 10 collocation collocation NOUN ejpam-4172 75 11 point point NOUN ejpam-4172 75 12 in in ADP ejpam-4172 75 13 ωt ωt NOUN ejpam-4172 75 14 = = SYM ejpam-4172 75 15 π π PROPN ejpam-4172 75 16 4 4 X ejpam-4172 75 17 . . PUNCT ejpam-4172 76 1 o. o. PROPN ejpam-4172 76 2 gonzález gonzález PROPN ejpam-4172 76 3 - - PUNCT ejpam-4172 76 4 gaxiola gaxiola PROPN ejpam-4172 76 5 , , PUNCT ejpam-4172 76 6 j. j. PROPN ejpam-4172 76 7 ruiz ruiz PROPN ejpam-4172 76 8 de de PROPN ejpam-4172 76 9 chávez chávez PROPN ejpam-4172 76 10 / / SYM ejpam-4172 76 11 eur eur PROPN ejpam-4172 76 12 . . PUNCT ejpam-4172 77 1 j. j. PROPN ejpam-4172 77 2 pure pure PROPN ejpam-4172 77 3 appl appl PROPN ejpam-4172 77 4 . . PROPN ejpam-4172 77 5 math math PROPN ejpam-4172 77 6 , , PUNCT ejpam-4172 77 7 15 15 NUM ejpam-4172 77 8 ( ( PUNCT ejpam-4172 77 9 1 1 NUM ejpam-4172 77 10 ) ) PUNCT ejpam-4172 77 11 ( ( PUNCT ejpam-4172 77 12 2022 2022 NUM ejpam-4172 77 13 ) ) PUNCT ejpam-4172 77 14 , , PUNCT ejpam-4172 77 15 82 82 NUM ejpam-4172 77 16 - - SYM ejpam-4172 77 17 99 99 NUM ejpam-4172 77 18 86 86 NUM ejpam-4172 77 19 as as ADP ejpam-4172 77 20 of of ADP ejpam-4172 77 21 late late ADJ ejpam-4172 77 22 , , PUNCT ejpam-4172 77 23 the the DET ejpam-4172 77 24 regular regular ADJ ejpam-4172 77 25 energy energy NOUN ejpam-4172 77 26 balance balance NOUN ejpam-4172 77 27 method method NOUN ejpam-4172 77 28 has have AUX ejpam-4172 77 29 been be AUX ejpam-4172 77 30 changed change VERB ejpam-4172 77 31 by by ADP ejpam-4172 77 32 m. m. PROPN ejpam-4172 77 33 k. k. PROPN ejpam-4172 77 34 yazdi yazdi PROPN ejpam-4172 77 35 and and CCONJ ejpam-4172 77 36 p. p. PROPN ejpam-4172 77 37 h. h. PROPN ejpam-4172 77 38 tehrani tehrani PROPN ejpam-4172 77 39 in in ADP ejpam-4172 77 40 [ [ X ejpam-4172 77 41 21 21 NUM ejpam-4172 77 42 ] ] PUNCT ejpam-4172 77 43 , , PUNCT ejpam-4172 77 44 utilizing utilize VERB ejpam-4172 77 45 an an DET ejpam-4172 77 46 extraordinary extraordinary ADJ ejpam-4172 77 47 collocation collocation NOUN ejpam-4172 77 48 point point NOUN ejpam-4172 77 49 . . PUNCT ejpam-4172 78 1 as as ADP ejpam-4172 78 2 per per ADP ejpam-4172 78 3 the the DET ejpam-4172 78 4 investigation investigation NOUN ejpam-4172 78 5 done do VERB ejpam-4172 78 6 in in ADP ejpam-4172 78 7 [ [ X ejpam-4172 78 8 21 21 NUM ejpam-4172 78 9 ] ] PUNCT ejpam-4172 78 10 , , PUNCT ejpam-4172 78 11 the the DET ejpam-4172 78 12 approximate approximate ADJ ejpam-4172 78 13 frequency frequency NOUN ejpam-4172 78 14 as as ADP ejpam-4172 78 15 a a DET ejpam-4172 78 16 function function NOUN ejpam-4172 78 17 of of ADP ejpam-4172 78 18 a a PRON ejpam-4172 78 19 can can AUX ejpam-4172 78 20 be be AUX ejpam-4172 78 21 gotten get VERB ejpam-4172 78 22 as as SCONJ ejpam-4172 78 23 follows follow VERB ejpam-4172 78 24 : : PUNCT ejpam-4172 78 25 ωapp(a ωapp(a NUM ejpam-4172 78 26 ) ) PUNCT ejpam-4172 78 27 = = PUNCT ejpam-4172 79 1 √ √ NOUN ejpam-4172 79 2 2(f 2(f NUM ejpam-4172 79 3 ( ( PUNCT ejpam-4172 79 4 a)− a)− PROPN ejpam-4172 79 5 f f PROPN ejpam-4172 79 6 ( ( PUNCT ejpam-4172 79 7 a a DET ejpam-4172 79 8 cos(x0π cos(x0π PROPN ejpam-4172 79 9 ) ) PUNCT ejpam-4172 79 10 ) ) PUNCT ejpam-4172 79 11 ) ) PUNCT ejpam-4172 79 12 a a DET ejpam-4172 79 13 sin(x0π sin(x0π PROPN ejpam-4172 79 14 ) ) PUNCT ejpam-4172 79 15 , , PUNCT ejpam-4172 79 16 ( ( PUNCT ejpam-4172 79 17 19 19 NUM ejpam-4172 79 18 ) ) PUNCT ejpam-4172 79 19 where where SCONJ ejpam-4172 79 20 x0 x0 PROPN ejpam-4172 79 21 is be AUX ejpam-4172 79 22 the the DET ejpam-4172 79 23 root root NOUN ejpam-4172 79 24 of of ADP ejpam-4172 79 25 some some DET ejpam-4172 79 26 one one NUM ejpam-4172 79 27 - - PUNCT ejpam-4172 79 28 degree degree NOUN ejpam-4172 79 29 jacobi jacobi NOUN ejpam-4172 79 30 polynomial polynomial NOUN ejpam-4172 79 31 , , PUNCT ejpam-4172 79 32 this this PRON ejpam-4172 79 33 is be AUX ejpam-4172 79 34 , , PUNCT ejpam-4172 79 35 p p X ejpam-4172 79 36 ( ( PUNCT ejpam-4172 79 37 α α X ejpam-4172 79 38 , , PUNCT ejpam-4172 79 39 β β NOUN ejpam-4172 79 40 ) ) PUNCT ejpam-4172 79 41 1 1 NUM ejpam-4172 79 42 ( ( PUNCT ejpam-4172 79 43 x0 x0 PROPN ejpam-4172 79 44 ) ) PUNCT ejpam-4172 80 1 = = SYM ejpam-4172 80 2 0 0 X ejpam-4172 80 3 . . PUNCT ejpam-4172 81 1 both both DET ejpam-4172 81 2 jacobi jacobi PROPN ejpam-4172 81 3 polynomial polynomial ADJ ejpam-4172 81 4 parameters parameter NOUN ejpam-4172 81 5 ( ( PUNCT ejpam-4172 81 6 α;β α;β X ejpam-4172 81 7 > > X ejpam-4172 81 8 −1 −1 NOUN ejpam-4172 81 9 ) ) PUNCT ejpam-4172 81 10 can can AUX ejpam-4172 81 11 be be AUX ejpam-4172 81 12 utilized utilize VERB ejpam-4172 81 13 to to PART ejpam-4172 81 14 acquire acquire VERB ejpam-4172 81 15 an an DET ejpam-4172 81 16 optimal optimal ADJ ejpam-4172 81 17 estimation estimation NOUN ejpam-4172 81 18 of of ADP ejpam-4172 81 19 x0 x0 PROPN ejpam-4172 81 20 . . PUNCT ejpam-4172 82 1 the the DET ejpam-4172 82 2 precision precision NOUN ejpam-4172 82 3 relies rely VERB ejpam-4172 82 4 extraordinarily extraordinarily ADV ejpam-4172 82 5 on on ADP ejpam-4172 82 6 the the DET ejpam-4172 82 7 collocation collocation NOUN ejpam-4172 82 8 point point NOUN ejpam-4172 82 9 . . PUNCT ejpam-4172 83 1 from from ADP ejpam-4172 83 2 ( ( PUNCT ejpam-4172 83 3 19 19 NUM ejpam-4172 83 4 ) ) PUNCT ejpam-4172 83 5 we we PRON ejpam-4172 83 6 get get VERB ejpam-4172 83 7 the the DET ejpam-4172 83 8 following follow VERB ejpam-4172 83 9 approximate approximate ADJ ejpam-4172 83 10 periodic periodic ADJ ejpam-4172 83 11 solution solution NOUN ejpam-4172 83 12 for for ADP ejpam-4172 83 13 ( ( PUNCT ejpam-4172 83 14 11 11 NUM ejpam-4172 83 15 ) ) PUNCT ejpam-4172 83 16 uapp(t uapp(t NOUN ejpam-4172 83 17 ) ) PUNCT ejpam-4172 83 18 = = PUNCT ejpam-4172 83 19 a a DET ejpam-4172 83 20 cos cos PROPN ejpam-4172 83 21 ( ( PUNCT ejpam-4172 83 22 ωapp ωapp PROPN ejpam-4172 83 23 · · PUNCT ejpam-4172 83 24 t t PROPN ejpam-4172 83 25 ) ) PUNCT ejpam-4172 83 26 . . PUNCT ejpam-4172 84 1 ( ( PUNCT ejpam-4172 84 2 20 20 NUM ejpam-4172 84 3 ) ) PUNCT ejpam-4172 84 4 the the DET ejpam-4172 84 5 technique technique NOUN ejpam-4172 84 6 to to PART ejpam-4172 84 7 find find VERB ejpam-4172 84 8 x0 x0 PROPN ejpam-4172 84 9 is be AUX ejpam-4172 84 10 an an DET ejpam-4172 84 11 open open ADJ ejpam-4172 84 12 problem problem NOUN ejpam-4172 84 13 , , PUNCT ejpam-4172 84 14 however however ADV ejpam-4172 84 15 we we PRON ejpam-4172 84 16 have have AUX ejpam-4172 84 17 chosen choose VERB ejpam-4172 84 18 it it PRON ejpam-4172 84 19 in in ADP ejpam-4172 84 20 a a DET ejpam-4172 84 21 heuristic heuristic ADJ ejpam-4172 84 22 way way NOUN ejpam-4172 84 23 . . PUNCT ejpam-4172 85 1 3 3 X ejpam-4172 85 2 . . X ejpam-4172 85 3 numerical numerical ADJ ejpam-4172 85 4 examples example NOUN ejpam-4172 85 5 in in ADP ejpam-4172 85 6 this this DET ejpam-4172 85 7 section section NOUN ejpam-4172 85 8 , , PUNCT ejpam-4172 85 9 we we PRON ejpam-4172 85 10 provide provide VERB ejpam-4172 85 11 three three NUM ejpam-4172 85 12 examples example NOUN ejpam-4172 85 13 to to PART ejpam-4172 85 14 illustrate illustrate VERB ejpam-4172 85 15 the the DET ejpam-4172 85 16 use use NOUN ejpam-4172 85 17 , , PUNCT ejpam-4172 85 18 ease ease NOUN ejpam-4172 85 19 and and CCONJ ejpam-4172 85 20 effectiveness effectiveness NOUN ejpam-4172 85 21 of of ADP ejpam-4172 85 22 the the DET ejpam-4172 85 23 proposed propose VERB ejpam-4172 85 24 method method NOUN ejpam-4172 85 25 . . PUNCT ejpam-4172 86 1 in in ADP ejpam-4172 86 2 each each DET ejpam-4172 86 3 example example NOUN ejpam-4172 86 4 we we PRON ejpam-4172 86 5 compare compare VERB ejpam-4172 86 6 our our PRON ejpam-4172 86 7 results result NOUN ejpam-4172 86 8 with with ADP ejpam-4172 86 9 the the DET ejpam-4172 86 10 exact exact ADJ ejpam-4172 86 11 results result NOUN ejpam-4172 86 12 found find VERB ejpam-4172 86 13 in in ADP ejpam-4172 86 14 the the DET ejpam-4172 86 15 literature literature NOUN ejpam-4172 86 16 to to PART ejpam-4172 86 17 show show VERB ejpam-4172 86 18 that that SCONJ ejpam-4172 86 19 the the DET ejpam-4172 86 20 proposed propose VERB ejpam-4172 86 21 method method NOUN ejpam-4172 86 22 is be AUX ejpam-4172 86 23 accurate accurate ADJ ejpam-4172 86 24 and and CCONJ ejpam-4172 86 25 efficient efficient ADJ ejpam-4172 86 26 . . PUNCT ejpam-4172 87 1 it it PRON ejpam-4172 87 2 is be AUX ejpam-4172 87 3 not not PART ejpam-4172 87 4 the the DET ejpam-4172 87 5 objective objective NOUN ejpam-4172 87 6 of of ADP ejpam-4172 87 7 the the DET ejpam-4172 87 8 present present ADJ ejpam-4172 87 9 study study NOUN ejpam-4172 87 10 to to PART ejpam-4172 87 11 compare compare VERB ejpam-4172 87 12 the the DET ejpam-4172 87 13 method method NOUN ejpam-4172 87 14 with with ADP ejpam-4172 87 15 other other ADJ ejpam-4172 87 16 existing exist VERB ejpam-4172 87 17 methods method NOUN ejpam-4172 87 18 since since SCONJ ejpam-4172 87 19 we we PRON ejpam-4172 87 20 intend intend VERB ejpam-4172 87 21 to to PART ejpam-4172 87 22 expose expose VERB ejpam-4172 87 23 this this DET ejpam-4172 87 24 technique technique NOUN ejpam-4172 87 25 for for ADP ejpam-4172 87 26 didactic didactic ADJ ejpam-4172 87 27 purposes purpose NOUN ejpam-4172 87 28 only only ADV ejpam-4172 87 29 . . PUNCT ejpam-4172 88 1 example example NOUN ejpam-4172 89 1 1 1 NUM ejpam-4172 89 2 . . PUNCT ejpam-4172 90 1 the the DET ejpam-4172 90 2 governing govern VERB ejpam-4172 90 3 non non ADJ ejpam-4172 90 4 - - ADJ ejpam-4172 90 5 dimensional dimensional ADJ ejpam-4172 90 6 nonlinear nonlinear ADJ ejpam-4172 90 7 differential differential ADJ ejpam-4172 90 8 equation equation NOUN ejpam-4172 90 9 of of ADP ejpam-4172 90 10 motion motion NOUN ejpam-4172 90 11 for for ADP ejpam-4172 90 12 the the DET ejpam-4172 90 13 relativistic relativistic ADJ ejpam-4172 90 14 oscillator oscillator NOUN ejpam-4172 90 15 is be AUX ejpam-4172 90 16 [ [ X ejpam-4172 90 17 42 42 NUM ejpam-4172 90 18 ] ] X ejpam-4172 90 19 : : PUNCT ejpam-4172 90 20 ü+ ü+ PROPN ejpam-4172 90 21 u√ u√ ADJ ejpam-4172 90 22 1 1 NUM ejpam-4172 90 23 + + NUM ejpam-4172 90 24 u2 u2 NOUN ejpam-4172 90 25 = = SYM ejpam-4172 90 26 0 0 NUM ejpam-4172 90 27 , , PUNCT ejpam-4172 90 28 u(0 u(0 NOUN ejpam-4172 90 29 ) ) PUNCT ejpam-4172 90 30 = = SYM ejpam-4172 91 1 a a PRON ejpam-4172 91 2 , , PUNCT ejpam-4172 91 3 u̇(0 u̇(0 NOUN ejpam-4172 91 4 ) ) PUNCT ejpam-4172 91 5 = = SYM ejpam-4172 91 6 0 0 NUM ejpam-4172 91 7 , , PUNCT ejpam-4172 91 8 ( ( PUNCT ejpam-4172 91 9 21 21 NUM ejpam-4172 91 10 ) ) PUNCT ejpam-4172 91 11 where where SCONJ ejpam-4172 91 12 f(u f(u NOUN ejpam-4172 91 13 ) ) PUNCT ejpam-4172 91 14 = = PUNCT ejpam-4172 91 15 u√ u√ PROPN ejpam-4172 91 16 1+u2 1+u2 NUM ejpam-4172 91 17 . . PUNCT ejpam-4172 92 1 calculating calculate VERB ejpam-4172 92 2 we we PRON ejpam-4172 92 3 have have VERB ejpam-4172 92 4 f f PROPN ejpam-4172 92 5 ( ( PUNCT ejpam-4172 92 6 u u NOUN ejpam-4172 92 7 ) ) PUNCT ejpam-4172 92 8 = = SYM ejpam-4172 93 1 ∫ ∫ PROPN ejpam-4172 93 2 f(u)du f(u)du PROPN ejpam-4172 93 3 = = SYM ejpam-4172 93 4 ∫ ∫ PROPN ejpam-4172 93 5 u√ u√ PROPN ejpam-4172 93 6 1 1 NUM ejpam-4172 93 7 + + CCONJ ejpam-4172 93 8 u2 u2 PROPN ejpam-4172 93 9 du du PROPN ejpam-4172 93 10 = = NOUN ejpam-4172 93 11 √ √ NUM ejpam-4172 93 12 1 1 NUM ejpam-4172 93 13 + + CCONJ ejpam-4172 93 14 u2 u2 NOUN ejpam-4172 93 15 , , PUNCT ejpam-4172 93 16 ( ( PUNCT ejpam-4172 93 17 22 22 NUM ejpam-4172 93 18 ) ) PUNCT ejpam-4172 93 19 therefore therefore ADV ejpam-4172 93 20 , , PUNCT ejpam-4172 93 21 we we PRON ejpam-4172 93 22 obtain obtain VERB ejpam-4172 93 23 f f X ejpam-4172 93 24 ( ( PUNCT ejpam-4172 93 25 a a X ejpam-4172 93 26 ) ) PUNCT ejpam-4172 93 27 = = SYM ejpam-4172 94 1 √ √ ADP ejpam-4172 94 2 1 1 NUM ejpam-4172 94 3 + + NUM ejpam-4172 94 4 a2 a2 PROPN ejpam-4172 94 5 and and CCONJ ejpam-4172 94 6 f f PROPN ejpam-4172 94 7 ( ( PUNCT ejpam-4172 94 8 a a DET ejpam-4172 94 9 cos(πx0 cos(πx0 NOUN ejpam-4172 94 10 ) ) PUNCT ejpam-4172 94 11 ) ) PUNCT ejpam-4172 95 1 = = PUNCT ejpam-4172 96 1 √ √ ADP ejpam-4172 96 2 1 1 NUM ejpam-4172 96 3 + + NOUN ejpam-4172 96 4 a2 a2 NOUN ejpam-4172 96 5 cos2(πx0 cos2(πx0 NOUN ejpam-4172 96 6 ) ) PUNCT ejpam-4172 96 7 , , PUNCT ejpam-4172 96 8 ( ( PUNCT ejpam-4172 96 9 23 23 NUM ejpam-4172 96 10 ) ) PUNCT ejpam-4172 96 11 and and CCONJ ejpam-4172 96 12 the the DET ejpam-4172 96 13 exact exact ADJ ejpam-4172 96 14 frequency frequency NOUN ejpam-4172 96 15 is be AUX ejpam-4172 96 16 given give VERB ejpam-4172 96 17 by by ADP ejpam-4172 96 18 [ [ X ejpam-4172 96 19 41 41 NUM ejpam-4172 96 20 ] ] PUNCT ejpam-4172 96 21 : : PUNCT ejpam-4172 96 22 ωex(a ωex(a PROPN ejpam-4172 96 23 ) ) PUNCT ejpam-4172 96 24 = = PUNCT ejpam-4172 97 1 π π NOUN ejpam-4172 97 2 2 2 NUM ejpam-4172 97 3 ∫ ∫ NOUN ejpam-4172 97 4 a a DET ejpam-4172 97 5 0 0 NUM ejpam-4172 97 6 dy√ dy√ PROPN ejpam-4172 97 7 2 2 NUM ejpam-4172 97 8 ( ( PUNCT ejpam-4172 97 9 √ √ NUM ejpam-4172 97 10 1 1 NUM ejpam-4172 97 11 + + NOUN ejpam-4172 97 12 a2 a2 PROPN ejpam-4172 97 13 − − NOUN ejpam-4172 97 14 √ √ NOUN ejpam-4172 97 15 1 1 NUM ejpam-4172 97 16 + + NUM ejpam-4172 97 17 y2 y2 NOUN ejpam-4172 97 18 ) ) PUNCT ejpam-4172 97 19 . . PUNCT ejpam-4172 98 1 ( ( PUNCT ejpam-4172 98 2 24 24 NUM ejpam-4172 98 3 ) ) PUNCT ejpam-4172 98 4 so so SCONJ ejpam-4172 98 5 as as SCONJ ejpam-4172 98 6 to to PART ejpam-4172 98 7 apply apply VERB ejpam-4172 98 8 the the DET ejpam-4172 98 9 previously previously ADV ejpam-4172 98 10 mentioned mention VERB ejpam-4172 98 11 method method NOUN ejpam-4172 98 12 for for ADP ejpam-4172 98 13 finding find VERB ejpam-4172 98 14 the the DET ejpam-4172 98 15 approximate approximate ADJ ejpam-4172 98 16 frequency frequency NOUN ejpam-4172 98 17 and and CCONJ ejpam-4172 98 18 solution solution NOUN ejpam-4172 98 19 of of ADP ejpam-4172 98 20 the the DET ejpam-4172 98 21 oscillator oscillator NOUN ejpam-4172 98 22 ( ( PUNCT ejpam-4172 98 23 21 21 NUM ejpam-4172 98 24 ) ) PUNCT ejpam-4172 98 25 , , PUNCT ejpam-4172 98 26 we we PRON ejpam-4172 98 27 propose propose VERB ejpam-4172 98 28 to to PART ejpam-4172 98 29 consider consider VERB ejpam-4172 98 30 the the DET ejpam-4172 98 31 jacobi jacobi PROPN ejpam-4172 98 32 polynomial polynomial PROPN ejpam-4172 98 33 p p PROPN ejpam-4172 98 34 ( ( PUNCT ejpam-4172 98 35 864,1441 864,1441 NOUN ejpam-4172 98 36 ) ) PUNCT ejpam-4172 98 37 1 1 NUM ejpam-4172 98 38 ( ( PUNCT ejpam-4172 98 39 x x NOUN ejpam-4172 98 40 ) ) PUNCT ejpam-4172 98 41 , , PUNCT ejpam-4172 98 42 o. o. PROPN ejpam-4172 98 43 gonzález gonzález PROPN ejpam-4172 98 44 - - PUNCT ejpam-4172 98 45 gaxiola gaxiola PROPN ejpam-4172 98 46 , , PUNCT ejpam-4172 98 47 j. j. PROPN ejpam-4172 98 48 ruiz ruiz PROPN ejpam-4172 98 49 de de PROPN ejpam-4172 98 50 chávez chávez PROPN ejpam-4172 98 51 / / SYM ejpam-4172 98 52 eur eur PROPN ejpam-4172 98 53 . . PUNCT ejpam-4172 99 1 j. j. PROPN ejpam-4172 99 2 pure pure PROPN ejpam-4172 99 3 appl appl PROPN ejpam-4172 99 4 . . PROPN ejpam-4172 99 5 math math PROPN ejpam-4172 99 6 , , PUNCT ejpam-4172 99 7 15 15 NUM ejpam-4172 99 8 ( ( PUNCT ejpam-4172 99 9 1 1 NUM ejpam-4172 99 10 ) ) PUNCT ejpam-4172 99 11 ( ( PUNCT ejpam-4172 99 12 2022 2022 NUM ejpam-4172 99 13 ) ) PUNCT ejpam-4172 99 14 , , PUNCT ejpam-4172 99 15 82 82 NUM ejpam-4172 99 16 - - SYM ejpam-4172 99 17 99 99 NUM ejpam-4172 99 18 87 87 NUM ejpam-4172 99 19 whose whose DET ejpam-4172 99 20 root root NOUN ejpam-4172 99 21 is be AUX ejpam-4172 99 22 x0 x0 PROPN ejpam-4172 99 23 = = SYM ejpam-4172 99 24 0.250108 0.250108 NUM ejpam-4172 99 25 . . PUNCT ejpam-4172 100 1 choosing choose VERB ejpam-4172 100 2 the the DET ejpam-4172 100 3 collocation collocation NOUN ejpam-4172 100 4 point point NOUN ejpam-4172 100 5 ωt ωt ADP ejpam-4172 100 6 = = SYM ejpam-4172 100 7 0.250108π 0.250108π NOUN ejpam-4172 100 8 , , PUNCT ejpam-4172 100 9 the the DET ejpam-4172 100 10 analytical analytical ADJ ejpam-4172 100 11 approximation approximation NOUN ejpam-4172 100 12 to to ADP ejpam-4172 100 13 periodic periodic ADJ ejpam-4172 100 14 solution solution NOUN ejpam-4172 100 15 of of ADP ejpam-4172 100 16 the the DET ejpam-4172 100 17 nonlinear nonlinear ADJ ejpam-4172 100 18 oscillator oscillator NOUN ejpam-4172 100 19 is be AUX ejpam-4172 100 20 u(t u(t NOUN ejpam-4172 100 21 ) ) PUNCT ejpam-4172 100 22 = = PUNCT ejpam-4172 101 1 a a DET ejpam-4172 101 2 cos(ωapp(a)t cos(ωapp(a)t PROPN ejpam-4172 101 3 ) ) PUNCT ejpam-4172 101 4 , , PUNCT ejpam-4172 101 5 ( ( PUNCT ejpam-4172 101 6 25 25 NUM ejpam-4172 101 7 ) ) PUNCT ejpam-4172 101 8 where where SCONJ ejpam-4172 101 9 ωapp(a ωapp(a NUM ejpam-4172 101 10 ) ) PUNCT ejpam-4172 101 11 = = PUNCT ejpam-4172 102 1 √ √ ADV ejpam-4172 102 2 2 2 NUM ejpam-4172 102 3 √ √ NUM ejpam-4172 102 4 1 1 NUM ejpam-4172 102 5 + + SYM ejpam-4172 102 6 a2 a2 PROPN ejpam-4172 102 7 − − ADP ejpam-4172 102 8 2 2 NUM ejpam-4172 102 9 √ √ NOUN ejpam-4172 102 10 1 1 NUM ejpam-4172 102 11 + + SYM ejpam-4172 102 12 a2 a2 PROPN ejpam-4172 102 13 cos2(0.250108π cos2(0.250108π PROPN ejpam-4172 102 14 ) ) PUNCT ejpam-4172 102 15 a a DET ejpam-4172 102 16 sin(0.250108π sin(0.250108π NOUN ejpam-4172 102 17 ) ) PUNCT ejpam-4172 102 18 , , PUNCT ejpam-4172 102 19 ( ( PUNCT ejpam-4172 102 20 26 26 NUM ejpam-4172 102 21 ) ) PUNCT ejpam-4172 102 22 which which PRON ejpam-4172 102 23 has have VERB ejpam-4172 102 24 a a DET ejpam-4172 102 25 high high ADJ ejpam-4172 102 26 accuracy accuracy NOUN ejpam-4172 102 27 , , PUNCT ejpam-4172 102 28 see see VERB ejpam-4172 102 29 figs fig NOUN ejpam-4172 102 30 . . PUNCT ejpam-4172 103 1 1 1 NUM ejpam-4172 103 2 and and CCONJ ejpam-4172 103 3 2 2 NUM ejpam-4172 103 4 . . NOUN ejpam-4172 103 5 table table NOUN ejpam-4172 103 6 1 1 NUM ejpam-4172 103 7 shows show VERB ejpam-4172 103 8 a a DET ejpam-4172 103 9 comparison comparison NOUN ejpam-4172 103 10 between between ADP ejpam-4172 103 11 the the DET ejpam-4172 103 12 approximate approximate ADJ ejpam-4172 103 13 frequencies frequency NOUN ejpam-4172 103 14 ωapp(a ωapp(a NUM ejpam-4172 103 15 ) ) PUNCT ejpam-4172 103 16 and and CCONJ ejpam-4172 103 17 the the DET ejpam-4172 103 18 exact exact ADJ ejpam-4172 103 19 frequencies frequency NOUN ejpam-4172 103 20 ωex(a ωex(a NOUN ejpam-4172 103 21 ) ) PUNCT ejpam-4172 103 22 for for ADP ejpam-4172 103 23 different different ADJ ejpam-4172 103 24 values value NOUN ejpam-4172 103 25 of of ADP ejpam-4172 103 26 a a PRON ejpam-4172 103 27 , , PUNCT ejpam-4172 103 28 a a DET ejpam-4172 103 29 decent decent ADJ ejpam-4172 103 30 solution solution NOUN ejpam-4172 103 31 is be AUX ejpam-4172 103 32 acquired acquire VERB ejpam-4172 103 33 . . PUNCT ejpam-4172 104 1 a a DET ejpam-4172 104 2 ωapp(a ωapp(a PROPN ejpam-4172 104 3 ) ) PUNCT ejpam-4172 104 4 eq eq NOUN ejpam-4172 104 5 . . PUNCT ejpam-4172 105 1 ( ( PUNCT ejpam-4172 105 2 26 26 NUM ejpam-4172 105 3 ) ) PUNCT ejpam-4172 105 4 ωex(a ωex(a PROPN ejpam-4172 105 5 ) ) PUNCT ejpam-4172 105 6 eq eq NOUN ejpam-4172 105 7 . . PUNCT ejpam-4172 106 1 ( ( PUNCT ejpam-4172 106 2 24 24 NUM ejpam-4172 106 3 ) ) PUNCT ejpam-4172 106 4 relative relative ADJ ejpam-4172 106 5 error error NOUN ejpam-4172 106 6 ( ( PUNCT ejpam-4172 106 7 % % INTJ ejpam-4172 106 8 ) ) PUNCT ejpam-4172 106 9 0.01 0.01 NUM ejpam-4172 106 10 0.999981 0.999981 NUM ejpam-4172 106 11 0.999981 0.999981 NUM ejpam-4172 106 12 0.00 0.00 NUM ejpam-4172 106 13 % % NOUN ejpam-4172 106 14 0.1 0.1 NUM ejpam-4172 106 15 0.998135 0.998135 NUM ejpam-4172 106 16 0.998135 0.998135 NUM ejpam-4172 106 17 0.00 0.00 NUM ejpam-4172 106 18 % % NOUN ejpam-4172 106 19 0.5 0.5 NUM ejpam-4172 106 20 0.958123 0.958123 NUM ejpam-4172 106 21 0.958338 0.958338 NUM ejpam-4172 106 22 0.02 0.02 NUM ejpam-4172 106 23 % % NOUN ejpam-4172 106 24 1 1 NUM ejpam-4172 106 25 0.870586 0.870586 NUM ejpam-4172 106 26 0.872342 0.872342 NUM ejpam-4172 106 27 0.20 0.20 NUM ejpam-4172 106 28 % % NOUN ejpam-4172 106 29 5 5 NUM ejpam-4172 106 30 0.477493 0.477493 NUM ejpam-4172 106 31 0.486634 0.486634 NUM ejpam-4172 106 32 1.87 1.87 NUM ejpam-4172 106 33 % % NOUN ejpam-4172 106 34 10 10 NUM ejpam-4172 106 35 0.341110 0.341110 NUM ejpam-4172 106 36 0.349115 0.349115 NUM ejpam-4172 106 37 2.29 2.29 NUM ejpam-4172 106 38 % % NOUN ejpam-4172 106 39 50 50 NUM ejpam-4172 106 40 0.153064 0.153064 NUM ejpam-4172 106 41 0.157029 0.157029 NUM ejpam-4172 106 42 2.52 2.52 NUM ejpam-4172 106 43 % % NOUN ejpam-4172 106 44 100 100 NUM ejpam-4172 106 45 0.108244 0.108244 NUM ejpam-4172 106 46 0.111062 0.111062 NUM ejpam-4172 106 47 2.53 2.53 NUM ejpam-4172 106 48 % % NOUN ejpam-4172 106 49 1000 1000 NUM ejpam-4172 106 50 0.034231 0.034231 NUM ejpam-4172 106 51 0.035124 0.035124 NUM ejpam-4172 106 52 2.54 2.54 NUM ejpam-4172 106 53 % % NOUN ejpam-4172 106 54 10000 10000 NUM ejpam-4172 106 55 0.010824 0.010824 NUM ejpam-4172 106 56 0.011107 0.011107 NUM ejpam-4172 106 57 2.54 2.54 NUM ejpam-4172 106 58 % % NOUN ejpam-4172 106 59 100000 100000 NUM ejpam-4172 106 60 0.003423 0.003423 NUM ejpam-4172 106 61 0.003512 0.003512 NUM ejpam-4172 106 62 2.54 2.54 NUM ejpam-4172 106 63 % % NOUN ejpam-4172 106 64 table table NOUN ejpam-4172 106 65 1 1 NUM ejpam-4172 106 66 : : PUNCT ejpam-4172 106 67 comparison comparison NOUN ejpam-4172 106 68 between between ADP ejpam-4172 106 69 frequencies frequency NOUN ejpam-4172 106 70 ωapp(a ωapp(a NUM ejpam-4172 106 71 ) ) PUNCT ejpam-4172 106 72 and and CCONJ ejpam-4172 106 73 ωex(a ωex(a NOUN ejpam-4172 106 74 ) ) PUNCT ejpam-4172 106 75 for for ADP ejpam-4172 106 76 different different ADJ ejpam-4172 106 77 values value NOUN ejpam-4172 106 78 of of ADP ejpam-4172 106 79 a. a. PROPN ejpam-4172 106 80 o. o. PROPN ejpam-4172 106 81 gonzález gonzález PROPN ejpam-4172 106 82 - - PUNCT ejpam-4172 106 83 gaxiola gaxiola PROPN ejpam-4172 106 84 , , PUNCT ejpam-4172 106 85 j. j. PROPN ejpam-4172 106 86 ruiz ruiz PROPN ejpam-4172 106 87 de de PROPN ejpam-4172 106 88 chávez chávez PROPN ejpam-4172 106 89 / / SYM ejpam-4172 106 90 eur eur PROPN ejpam-4172 106 91 . . PUNCT ejpam-4172 107 1 j. j. PROPN ejpam-4172 107 2 pure pure PROPN ejpam-4172 107 3 appl appl PROPN ejpam-4172 107 4 . . PROPN ejpam-4172 107 5 math math PROPN ejpam-4172 107 6 , , PUNCT ejpam-4172 107 7 15 15 NUM ejpam-4172 107 8 ( ( PUNCT ejpam-4172 107 9 1 1 NUM ejpam-4172 107 10 ) ) PUNCT ejpam-4172 107 11 ( ( PUNCT ejpam-4172 107 12 2022 2022 NUM ejpam-4172 107 13 ) ) PUNCT ejpam-4172 107 14 , , PUNCT ejpam-4172 107 15 82 82 NUM ejpam-4172 107 16 - - SYM ejpam-4172 107 17 99 99 NUM ejpam-4172 107 18 88 88 NUM ejpam-4172 107 19 0 0 NUM ejpam-4172 107 20 2 2 NUM ejpam-4172 107 21 4 4 NUM ejpam-4172 107 22 6 6 NUM ejpam-4172 107 23 8 8 NUM ejpam-4172 107 24 10 10 NUM ejpam-4172 107 25 12 12 NUM ejpam-4172 107 26 -4 -4 INTJ ejpam-4172 107 27 -2 -2 NOUN ejpam-4172 107 28 0 0 NUM ejpam-4172 107 29 2 2 NUM ejpam-4172 107 30 4 4 NUM ejpam-4172 107 31 t t NOUN ejpam-4172 107 32 u(t u(t NOUN ejpam-4172 107 33 ) ) PUNCT ejpam-4172 107 34 figure figure NOUN ejpam-4172 107 35 1 1 NUM ejpam-4172 107 36 : : PUNCT ejpam-4172 107 37 example example NOUN ejpam-4172 108 1 1 1 NUM ejpam-4172 108 2 : : PUNCT ejpam-4172 108 3 comparison comparison NOUN ejpam-4172 108 4 of of ADP ejpam-4172 108 5 the the DET ejpam-4172 108 6 approximate approximate ADJ ejpam-4172 108 7 periodic periodic ADJ ejpam-4172 108 8 solution solution NOUN ejpam-4172 108 9 ( ( PUNCT ejpam-4172 108 10 dashed dash VERB ejpam-4172 108 11 ) ) PUNCT ejpam-4172 108 12 with with ADP ejpam-4172 108 13 exact exact ADJ ejpam-4172 108 14 solution solution NOUN ejpam-4172 108 15 ( ( PUNCT ejpam-4172 108 16 blue blue NOUN ejpam-4172 108 17 ) ) PUNCT ejpam-4172 108 18 for for ADP ejpam-4172 108 19 a a DET ejpam-4172 108 20 = = SYM ejpam-4172 108 21 5 5 NUM ejpam-4172 108 22 . . NOUN ejpam-4172 108 23 0 0 NUM ejpam-4172 108 24 10 10 NUM ejpam-4172 108 25 20 20 NUM ejpam-4172 108 26 30 30 NUM ejpam-4172 108 27 40 40 NUM ejpam-4172 108 28 -40 -40 NUM ejpam-4172 108 29 -20 -20 NUM ejpam-4172 108 30 0 0 NUM ejpam-4172 108 31 20 20 NUM ejpam-4172 108 32 40 40 NUM ejpam-4172 108 33 t t NOUN ejpam-4172 108 34 u(t u(t NOUN ejpam-4172 108 35 ) ) PUNCT ejpam-4172 108 36 figure figure NOUN ejpam-4172 108 37 2 2 NUM ejpam-4172 108 38 : : PUNCT ejpam-4172 108 39 example example NOUN ejpam-4172 108 40 1 1 NUM ejpam-4172 108 41 : : PUNCT ejpam-4172 108 42 comparison comparison NOUN ejpam-4172 108 43 of of ADP ejpam-4172 108 44 the the DET ejpam-4172 108 45 approximate approximate ADJ ejpam-4172 108 46 periodic periodic ADJ ejpam-4172 108 47 solution solution NOUN ejpam-4172 108 48 ( ( PUNCT ejpam-4172 108 49 dashed dash VERB ejpam-4172 108 50 ) ) PUNCT ejpam-4172 108 51 with with ADP ejpam-4172 108 52 exact exact ADJ ejpam-4172 108 53 solution solution NOUN ejpam-4172 108 54 ( ( PUNCT ejpam-4172 108 55 blue blue ADJ ejpam-4172 108 56 ) ) PUNCT ejpam-4172 108 57 for for ADP ejpam-4172 108 58 for for ADP ejpam-4172 108 59 a a DET ejpam-4172 108 60 = = SYM ejpam-4172 108 61 50 50 NUM ejpam-4172 108 62 . . PUNCT ejpam-4172 109 1 we we PRON ejpam-4172 109 2 consider consider VERB ejpam-4172 109 3 our our PRON ejpam-4172 109 4 approximation approximation NOUN ejpam-4172 109 5 to to PART ejpam-4172 109 6 be be AUX ejpam-4172 109 7 acceptable acceptable ADJ ejpam-4172 109 8 , , PUNCT ejpam-4172 109 9 since since SCONJ ejpam-4172 109 10 the the DET ejpam-4172 109 11 best good ADJ ejpam-4172 109 12 approximation approximation NOUN ejpam-4172 109 13 to to ADP ejpam-4172 109 14 this this DET ejpam-4172 109 15 problem problem NOUN ejpam-4172 109 16 had have AUX ejpam-4172 109 17 shown show VERB ejpam-4172 109 18 a a DET ejpam-4172 109 19 maximum maximum ADJ ejpam-4172 109 20 error error NOUN ejpam-4172 109 21 of of ADP ejpam-4172 109 22 1.6 1.6 NUM ejpam-4172 109 23 % % NOUN ejpam-4172 109 24 for for ADP ejpam-4172 109 25 all all DET ejpam-4172 109 26 the the DET ejpam-4172 109 27 range range NOUN ejpam-4172 109 28 of of ADP ejpam-4172 109 29 values value NOUN ejpam-4172 109 30 of of ADP ejpam-4172 109 31 amplitude amplitude NOUN ejpam-4172 109 32 of of ADP ejpam-4172 109 33 oscillation oscillation NOUN ejpam-4172 110 1 a a PRON ejpam-4172 111 1 [ [ X ejpam-4172 111 2 43 43 NUM ejpam-4172 111 3 ] ] PUNCT ejpam-4172 111 4 . . PUNCT ejpam-4172 112 1 example example NOUN ejpam-4172 112 2 2 2 NUM ejpam-4172 112 3 the the DET ejpam-4172 112 4 motion motion NOUN ejpam-4172 112 5 equation equation NOUN ejpam-4172 112 6 for for ADP ejpam-4172 112 7 a a DET ejpam-4172 112 8 mass mass NOUN ejpam-4172 112 9 attached attach VERB ejpam-4172 112 10 to to ADP ejpam-4172 112 11 a a DET ejpam-4172 112 12 stretched stretch VERB ejpam-4172 112 13 elastic elastic ADJ ejpam-4172 112 14 wire wire NOUN ejpam-4172 112 15 is be AUX ejpam-4172 112 16 governed govern VERB ejpam-4172 112 17 by by ADP ejpam-4172 112 18 the the DET ejpam-4172 112 19 o. o. PROPN ejpam-4172 112 20 gonzález gonzález PROPN ejpam-4172 112 21 - - PUNCT ejpam-4172 112 22 gaxiola gaxiola PROPN ejpam-4172 112 23 , , PUNCT ejpam-4172 112 24 j. j. PROPN ejpam-4172 112 25 ruiz ruiz PROPN ejpam-4172 112 26 de de PROPN ejpam-4172 112 27 chávez chávez PROPN ejpam-4172 112 28 / / SYM ejpam-4172 112 29 eur eur PROPN ejpam-4172 112 30 . . PUNCT ejpam-4172 113 1 j. j. PROPN ejpam-4172 113 2 pure pure PROPN ejpam-4172 113 3 appl appl PROPN ejpam-4172 113 4 . . PROPN ejpam-4172 113 5 math math PROPN ejpam-4172 113 6 , , PUNCT ejpam-4172 113 7 15 15 NUM ejpam-4172 113 8 ( ( PUNCT ejpam-4172 113 9 1 1 NUM ejpam-4172 113 10 ) ) PUNCT ejpam-4172 113 11 ( ( PUNCT ejpam-4172 113 12 2022 2022 NUM ejpam-4172 113 13 ) ) PUNCT ejpam-4172 113 14 , , PUNCT ejpam-4172 113 15 82 82 NUM ejpam-4172 113 16 - - SYM ejpam-4172 113 17 99 99 NUM ejpam-4172 113 18 89 89 NUM ejpam-4172 113 19 mathematical mathematical ADJ ejpam-4172 113 20 model model NOUN ejpam-4172 113 21 with with ADP ejpam-4172 113 22 initial initial ADJ ejpam-4172 113 23 conditions condition NOUN ejpam-4172 113 24 given give VERB ejpam-4172 113 25 by by ADP ejpam-4172 113 26 [ [ X ejpam-4172 113 27 1 1 NUM ejpam-4172 113 28 , , PUNCT ejpam-4172 113 29 8 8 NUM ejpam-4172 113 30 ] ] PUNCT ejpam-4172 113 31 ü+ ü+ PROPN ejpam-4172 114 1 u− u− PROPN ejpam-4172 114 2 γu√ γu√ ADJ ejpam-4172 114 3 1 1 NUM ejpam-4172 114 4 + + NUM ejpam-4172 114 5 u2 u2 NOUN ejpam-4172 114 6 = = SYM ejpam-4172 114 7 0 0 NUM ejpam-4172 114 8 , , PUNCT ejpam-4172 114 9 u(0 u(0 NOUN ejpam-4172 114 10 ) ) PUNCT ejpam-4172 114 11 = = SYM ejpam-4172 115 1 a a PRON ejpam-4172 115 2 , , PUNCT ejpam-4172 115 3 u̇(0 u̇(0 NOUN ejpam-4172 115 4 ) ) PUNCT ejpam-4172 115 5 = = SYM ejpam-4172 115 6 0 0 NUM ejpam-4172 115 7 , , PUNCT ejpam-4172 115 8 0 0 PUNCT ejpam-4172 115 9 < < X ejpam-4172 115 10 γ γ X ejpam-4172 115 11 < < X ejpam-4172 115 12 1 1 NUM ejpam-4172 115 13 . . PUNCT ejpam-4172 116 1 ( ( PUNCT ejpam-4172 116 2 27 27 NUM ejpam-4172 116 3 ) ) PUNCT ejpam-4172 116 4 if if SCONJ ejpam-4172 116 5 f(u f(u PROPN ejpam-4172 116 6 ) ) PUNCT ejpam-4172 117 1 = = PROPN ejpam-4172 117 2 u− u− PROPN ejpam-4172 117 3 γu√ γu√ ADJ ejpam-4172 117 4 1+u2 1+u2 NUM ejpam-4172 117 5 , , PUNCT ejpam-4172 117 6 calculating calculate VERB ejpam-4172 117 7 we we PRON ejpam-4172 117 8 have have VERB ejpam-4172 117 9 f f PROPN ejpam-4172 117 10 ( ( PUNCT ejpam-4172 117 11 u u NOUN ejpam-4172 117 12 ) ) PUNCT ejpam-4172 117 13 = = SYM ejpam-4172 118 1 ∫ ∫ PROPN ejpam-4172 118 2 f(u)du f(u)du PROPN ejpam-4172 118 3 = = PROPN ejpam-4172 118 4 u2 u2 PROPN ejpam-4172 118 5 2 2 NUM ejpam-4172 118 6 − − NOUN ejpam-4172 118 7 γ γ NOUN ejpam-4172 118 8 √ √ VERB ejpam-4172 118 9 1 1 NUM ejpam-4172 118 10 + + CCONJ ejpam-4172 118 11 u2 u2 NOUN ejpam-4172 118 12 . . PUNCT ejpam-4172 119 1 ( ( PUNCT ejpam-4172 119 2 28 28 NUM ejpam-4172 119 3 ) ) PUNCT ejpam-4172 119 4 considering consider VERB ejpam-4172 119 5 the the DET ejpam-4172 119 6 above above ADJ ejpam-4172 119 7 , , PUNCT ejpam-4172 119 8 we we PRON ejpam-4172 119 9 obtain obtain VERB ejpam-4172 119 10 f f X ejpam-4172 119 11 ( ( PUNCT ejpam-4172 119 12 a a NOUN ejpam-4172 119 13 ) ) PUNCT ejpam-4172 119 14 = = SYM ejpam-4172 119 15 a2 a2 PROPN ejpam-4172 119 16 2 2 NUM ejpam-4172 119 17 − − NOUN ejpam-4172 119 18 γ γ NOUN ejpam-4172 119 19 √ √ ADP ejpam-4172 119 20 1 1 NUM ejpam-4172 119 21 + + SYM ejpam-4172 119 22 a2 a2 PROPN ejpam-4172 119 23 , , PUNCT ejpam-4172 119 24 f f PROPN ejpam-4172 119 25 ( ( PUNCT ejpam-4172 119 26 a a DET ejpam-4172 119 27 cos(πx0 cos(πx0 NOUN ejpam-4172 119 28 ) ) PUNCT ejpam-4172 119 29 ) ) PUNCT ejpam-4172 120 1 = = SYM ejpam-4172 120 2 a2 a2 PROPN ejpam-4172 120 3 cos2(πx0 cos2(πx0 NOUN ejpam-4172 120 4 ) ) PUNCT ejpam-4172 120 5 2 2 NUM ejpam-4172 120 6 − − NOUN ejpam-4172 120 7 γ γ NOUN ejpam-4172 120 8 √ √ VERB ejpam-4172 120 9 1 1 NUM ejpam-4172 120 10 + + NOUN ejpam-4172 120 11 a2 a2 NOUN ejpam-4172 120 12 cos2(πx0 cos2(πx0 NOUN ejpam-4172 120 13 ) ) PUNCT ejpam-4172 120 14 . . PUNCT ejpam-4172 121 1 ( ( PUNCT ejpam-4172 121 2 29 29 NUM ejpam-4172 121 3 ) ) PUNCT ejpam-4172 121 4 the the DET ejpam-4172 121 5 exact exact ADJ ejpam-4172 121 6 vibration vibration NOUN ejpam-4172 121 7 frequency frequency NOUN ejpam-4172 121 8 ωex ωex NOUN ejpam-4172 121 9 can can AUX ejpam-4172 121 10 be be AUX ejpam-4172 121 11 derived derive VERB ejpam-4172 121 12 by by ADP ejpam-4172 121 13 direct direct ADJ ejpam-4172 121 14 integration integration NOUN ejpam-4172 121 15 of of ADP ejpam-4172 121 16 governing govern VERB ejpam-4172 121 17 eq eq ADP ejpam-4172 121 18 . . PUNCT ejpam-4172 122 1 ( ( PUNCT ejpam-4172 122 2 27 27 NUM ejpam-4172 122 3 ) ) PUNCT ejpam-4172 122 4 as as SCONJ ejpam-4172 122 5 follows follow VERB ejpam-4172 122 6 [ [ X ejpam-4172 122 7 41 41 NUM ejpam-4172 122 8 , , PUNCT ejpam-4172 122 9 44 44 NUM ejpam-4172 122 10 ] ] PUNCT ejpam-4172 122 11 : : PUNCT ejpam-4172 122 12 ωex(a ωex(a PROPN ejpam-4172 122 13 ) ) PUNCT ejpam-4172 122 14 = = PUNCT ejpam-4172 123 1 π π PROPN ejpam-4172 124 1 2 2 PROPN ejpam-4172 124 2 ∫ ∫ PROPN ejpam-4172 124 3 a a DET ejpam-4172 124 4 0 0 NUM ejpam-4172 124 5 dy√ dy√ PROPN ejpam-4172 124 6 ( ( PUNCT ejpam-4172 124 7 a2 a2 PROPN ejpam-4172 124 8 − − PROPN ejpam-4172 124 9 y2)− y2)− NUM ejpam-4172 124 10 2γ 2γ NOUN ejpam-4172 124 11 ( ( PUNCT ejpam-4172 124 12 √ √ ADP ejpam-4172 124 13 1 1 NUM ejpam-4172 124 14 + + NOUN ejpam-4172 124 15 a2 a2 PROPN ejpam-4172 124 16 − − NOUN ejpam-4172 124 17 √ √ NOUN ejpam-4172 124 18 1 1 NUM ejpam-4172 124 19 + + NUM ejpam-4172 124 20 y2 y2 NOUN ejpam-4172 124 21 ) ) PUNCT ejpam-4172 125 1 −1 −1 INTJ ejpam-4172 125 2 . . PUNCT ejpam-4172 126 1 ( ( PUNCT ejpam-4172 126 2 30 30 NUM ejpam-4172 126 3 ) ) PUNCT ejpam-4172 126 4 case case NOUN ejpam-4172 126 5 γ γ X ejpam-4172 126 6 = = PROPN ejpam-4172 126 7 0.25 0.25 NUM ejpam-4172 126 8 for for ADP ejpam-4172 126 9 this this DET ejpam-4172 126 10 situation situation NOUN ejpam-4172 126 11 , , PUNCT ejpam-4172 126 12 we we PRON ejpam-4172 126 13 propose propose VERB ejpam-4172 126 14 to to PART ejpam-4172 126 15 consider consider VERB ejpam-4172 126 16 the the DET ejpam-4172 126 17 jacobi jacobi PROPN ejpam-4172 126 18 polynomial polynomial PROPN ejpam-4172 126 19 p p PROPN ejpam-4172 126 20 ( ( PUNCT ejpam-4172 126 21 856,1425 856,1425 NUM ejpam-4172 126 22 ) ) PUNCT ejpam-4172 126 23 1 1 NUM ejpam-4172 126 24 ( ( PUNCT ejpam-4172 126 25 x x X ejpam-4172 126 26 ) ) PUNCT ejpam-4172 126 27 whose whose DET ejpam-4172 126 28 root root NOUN ejpam-4172 126 29 is be AUX ejpam-4172 126 30 x0 x0 PROPN ejpam-4172 126 31 = = SYM ejpam-4172 126 32 0.24923 0.24923 NUM ejpam-4172 126 33 . . PUNCT ejpam-4172 127 1 by by ADP ejpam-4172 127 2 choosing choose VERB ejpam-4172 127 3 the the DET ejpam-4172 127 4 collocation collocation NOUN ejpam-4172 127 5 point point NOUN ejpam-4172 127 6 ωt ωt PROPN ejpam-4172 127 7 = = SYM ejpam-4172 127 8 0.24923π 0.24923π PROPN ejpam-4172 127 9 , , PUNCT ejpam-4172 127 10 the the DET ejpam-4172 127 11 approximation approximation NOUN ejpam-4172 127 12 to to ADP ejpam-4172 127 13 periodic periodic ADJ ejpam-4172 127 14 solution solution NOUN ejpam-4172 127 15 of of ADP ejpam-4172 127 16 the the DET ejpam-4172 127 17 nonlinear nonlinear ADJ ejpam-4172 127 18 oscillator oscillator NOUN ejpam-4172 127 19 is be AUX ejpam-4172 127 20 u(t u(t NOUN ejpam-4172 127 21 ) ) PUNCT ejpam-4172 127 22 = = PUNCT ejpam-4172 128 1 a a DET ejpam-4172 128 2 cos(ωapp(a)t cos(ωapp(a)t PROPN ejpam-4172 128 3 ) ) PUNCT ejpam-4172 128 4 , , PUNCT ejpam-4172 128 5 ( ( PUNCT ejpam-4172 128 6 31 31 NUM ejpam-4172 128 7 ) ) PUNCT ejpam-4172 128 8 where where SCONJ ejpam-4172 128 9 ωapp(a ωapp(a NUM ejpam-4172 128 10 ) ) PUNCT ejpam-4172 128 11 = = SYM ejpam-4172 128 12 √ √ PROPN ejpam-4172 128 13 a2 a2 PROPN ejpam-4172 128 14 sin2(0.24923π sin2(0.24923π PROPN ejpam-4172 128 15 ) ) PUNCT ejpam-4172 128 16 + + CCONJ ejpam-4172 128 17 ( ( PUNCT ejpam-4172 128 18 0.5 0.5 NUM ejpam-4172 128 19 ) ) PUNCT ejpam-4172 128 20 [ [ PUNCT ejpam-4172 129 1 √ √ ADP ejpam-4172 129 2 1 1 NUM ejpam-4172 129 3 + + NOUN ejpam-4172 129 4 a2 a2 PROPN ejpam-4172 129 5 cos2(0.24923π)− cos2(0.24923π)− NOUN ejpam-4172 130 1 √ √ ADP ejpam-4172 130 2 1 1 NUM ejpam-4172 130 3 + + SYM ejpam-4172 130 4 a2 a2 PROPN ejpam-4172 130 5 ] ] X ejpam-4172 130 6 a a DET ejpam-4172 130 7 sin(0.24923π sin(0.24923π NOUN ejpam-4172 130 8 ) ) PUNCT ejpam-4172 130 9 . . PUNCT ejpam-4172 131 1 ( ( PUNCT ejpam-4172 131 2 32 32 NUM ejpam-4172 131 3 ) ) PUNCT ejpam-4172 131 4 o. o. PROPN ejpam-4172 131 5 gonzález gonzález PROPN ejpam-4172 131 6 - - PUNCT ejpam-4172 131 7 gaxiola gaxiola PROPN ejpam-4172 131 8 , , PUNCT ejpam-4172 131 9 j. j. PROPN ejpam-4172 131 10 ruiz ruiz PROPN ejpam-4172 131 11 de de PROPN ejpam-4172 131 12 chávez chávez PROPN ejpam-4172 131 13 / / SYM ejpam-4172 131 14 eur eur PROPN ejpam-4172 131 15 . . PUNCT ejpam-4172 132 1 j. j. PROPN ejpam-4172 132 2 pure pure PROPN ejpam-4172 132 3 appl appl PROPN ejpam-4172 132 4 . . PROPN ejpam-4172 132 5 math math PROPN ejpam-4172 132 6 , , PUNCT ejpam-4172 132 7 15 15 NUM ejpam-4172 132 8 ( ( PUNCT ejpam-4172 132 9 1 1 NUM ejpam-4172 132 10 ) ) PUNCT ejpam-4172 132 11 ( ( PUNCT ejpam-4172 132 12 2022 2022 NUM ejpam-4172 132 13 ) ) PUNCT ejpam-4172 132 14 , , PUNCT ejpam-4172 132 15 82 82 NUM ejpam-4172 132 16 - - SYM ejpam-4172 132 17 99 99 NUM ejpam-4172 132 18 90 90 NUM ejpam-4172 132 19 γ γ NOUN ejpam-4172 132 20 = = SYM ejpam-4172 132 21 0.25 0.25 NUM ejpam-4172 132 22 γ γ X ejpam-4172 132 23 = = PROPN ejpam-4172 132 24 0.75 0.75 NUM ejpam-4172 132 25 a a DET ejpam-4172 132 26 ωapp(a ωapp(a PROPN ejpam-4172 132 27 ) ) PUNCT ejpam-4172 132 28 eq eq NOUN ejpam-4172 132 29 . . PUNCT ejpam-4172 132 30 ( ( PUNCT ejpam-4172 132 31 32)ωex(a 32)ωex(a NUM ejpam-4172 132 32 ) ) PUNCT ejpam-4172 132 33 eq eq NOUN ejpam-4172 132 34 . . PUNCT ejpam-4172 133 1 ( ( PUNCT ejpam-4172 133 2 30)rel 30)rel PROPN ejpam-4172 133 3 . . PUNCT ejpam-4172 133 4 error error NOUN ejpam-4172 133 5 ( ( PUNCT ejpam-4172 133 6 % % INTJ ejpam-4172 133 7 ) ) PUNCT ejpam-4172 133 8 ωapp(a ωapp(a NUM ejpam-4172 133 9 ) ) PUNCT ejpam-4172 133 10 eq eq NOUN ejpam-4172 133 11 . . PUNCT ejpam-4172 134 1 ( ( PUNCT ejpam-4172 134 2 34)ωex(a 34)ωex(a NUM ejpam-4172 134 3 ) ) PUNCT ejpam-4172 134 4 eq eq NOUN ejpam-4172 134 5 . . PUNCT ejpam-4172 135 1 ( ( PUNCT ejpam-4172 135 2 30)rel 30)rel PROPN ejpam-4172 135 3 . . PUNCT ejpam-4172 136 1 error error NOUN ejpam-4172 136 2 ( ( PUNCT ejpam-4172 136 3 % % INTJ ejpam-4172 136 4 ) ) PUNCT ejpam-4172 136 5 0.01 0.01 NUM ejpam-4172 136 6 0.866031 0.866031 NUM ejpam-4172 136 7 0.866031 0.866031 NUM ejpam-4172 136 8 0.00 0.00 NUM ejpam-4172 136 9 % % NOUN ejpam-4172 136 10 0.500028 0.500028 NUM ejpam-4172 136 11 0.500028 0.500028 NUM ejpam-4172 136 12 0.00 0.00 NUM ejpam-4172 136 13 % % NOUN ejpam-4172 136 14 0.1 0.1 NUM ejpam-4172 136 15 0.866563 0.866563 NUM ejpam-4172 136 16 0.866563 0.866563 NUM ejpam-4172 136 17 0.00 0.00 NUM ejpam-4172 136 18 % % NOUN ejpam-4172 136 19 0.502786 0.502786 NUM ejpam-4172 136 20 0.502786 0.502786 NUM ejpam-4172 136 21 0.00 0.00 NUM ejpam-4172 136 22 % % NOUN ejpam-4172 136 23 0.5 0.5 NUM ejpam-4172 136 24 0.877786 0.877786 NUM ejpam-4172 136 25 0.877676 0.877676 NUM ejpam-4172 136 26 0.01 0.01 NUM ejpam-4172 136 27 % % NOUN ejpam-4172 136 28 0.558091 0.558091 NUM ejpam-4172 136 29 0.557297 0.557297 NUM ejpam-4172 136 30 0.14 0.14 NUM ejpam-4172 136 31 % % NOUN ejpam-4172 136 32 1 1 NUM ejpam-4172 136 33 0.900299 0.900299 NUM ejpam-4172 136 34 0.899558 0.899558 NUM ejpam-4172 136 35 0.08 0.08 NUM ejpam-4172 136 36 % % NOUN ejpam-4172 136 37 0.656870 0.656870 NUM ejpam-4172 136 38 0.652771 0.652771 NUM ejpam-4172 136 39 0.62 0.62 NUM ejpam-4172 136 40 % % NOUN ejpam-4172 136 41 5 5 NUM ejpam-4172 136 42 0.971089 0.971089 NUM ejpam-4172 136 43 0.969220 0.969220 NUM ejpam-4172 136 44 0.19 0.19 NUM ejpam-4172 136 45 % % NOUN ejpam-4172 136 46 0.910461 0.910461 NUM ejpam-4172 136 47 0.904141 0.904141 NUM ejpam-4172 136 48 0.69 0.69 NUM ejpam-4172 136 49 % % NOUN ejpam-4172 136 50 10 10 NUM ejpam-4172 136 51 0.985352 0.985352 NUM ejpam-4172 136 52 0.984197 0.984197 NUM ejpam-4172 136 53 0.11 0.11 NUM ejpam-4172 136 54 % % NOUN ejpam-4172 136 55 0.955353 0.955353 NUM ejpam-4172 136 56 0.951696 0.951696 NUM ejpam-4172 136 57 0.38 0.38 NUM ejpam-4172 136 58 % % NOUN ejpam-4172 136 59 50 50 NUM ejpam-4172 136 60 0.997068 0.997068 NUM ejpam-4172 136 61 0.996814 0.996814 NUM ejpam-4172 136 62 0.02 0.02 NUM ejpam-4172 136 63 % % NOUN ejpam-4172 136 64 0.991172 0.991172 NUM ejpam-4172 136 65 0.990407 0.990407 NUM ejpam-4172 136 66 0.07 0.07 NUM ejpam-4172 136 67 % % NOUN ejpam-4172 136 68 100 100 NUM ejpam-4172 137 1 0.998535 0.998535 NUM ejpam-4172 137 2 0.998407 0.998407 NUM ejpam-4172 137 3 0.01 0.01 NUM ejpam-4172 137 4 % % NOUN ejpam-4172 137 5 0.995595 0.995595 NUM ejpam-4172 137 6 0.995214 0.995214 NUM ejpam-4172 137 7 0.03 0.03 NUM ejpam-4172 137 8 % % NOUN ejpam-4172 137 9 1000 1000 NUM ejpam-4172 137 10 0.999854 0.999854 NUM ejpam-4172 137 11 0.999841 0.999841 NUM ejpam-4172 137 12 0.00 0.00 NUM ejpam-4172 137 13 % % NOUN ejpam-4172 137 14 0.999560 0.999560 NUM ejpam-4172 137 15 0.999522 0.999522 NUM ejpam-4172 137 16 0.00 0.00 NUM ejpam-4172 137 17 % % NOUN ejpam-4172 137 18 10000 10000 NUM ejpam-4172 137 19 0.999985 0.999985 NUM ejpam-4172 137 20 0.999984 0.999984 NUM ejpam-4172 137 21 0.00 0.00 NUM ejpam-4172 137 22 % % NOUN ejpam-4172 137 23 0.999956 0.999956 NUM ejpam-4172 137 24 0.999952 0.999952 NUM ejpam-4172 137 25 0.00 0.00 NUM ejpam-4172 137 26 % % NOUN ejpam-4172 137 27 100000 100000 NUM ejpam-4172 137 28 0.999999 0.999999 NUM ejpam-4172 137 29 0.999998 0.999998 NUM ejpam-4172 137 30 0.00 0.00 NUM ejpam-4172 137 31 % % NOUN ejpam-4172 137 32 0.999996 0.999996 NUM ejpam-4172 137 33 0.999995 0.999995 NUM ejpam-4172 137 34 0.00 0.00 NUM ejpam-4172 137 35 % % NOUN ejpam-4172 137 36 table table NOUN ejpam-4172 137 37 2 2 NUM ejpam-4172 137 38 : : PUNCT ejpam-4172 137 39 comparison comparison NOUN ejpam-4172 137 40 between between ADP ejpam-4172 137 41 frequencies frequency NOUN ejpam-4172 137 42 ωapp(a ωapp(a NUM ejpam-4172 137 43 ) ) PUNCT ejpam-4172 137 44 and and CCONJ ejpam-4172 137 45 ωex(a ωex(a NOUN ejpam-4172 137 46 ) ) PUNCT ejpam-4172 137 47 for for ADP ejpam-4172 137 48 different different ADJ ejpam-4172 137 49 values value NOUN ejpam-4172 137 50 of of ADP ejpam-4172 137 51 a. a. NOUN ejpam-4172 137 52 case case NOUN ejpam-4172 137 53 γ γ X ejpam-4172 137 54 = = PROPN ejpam-4172 137 55 0.75 0.75 NUM ejpam-4172 137 56 here here ADV ejpam-4172 137 57 we we PRON ejpam-4172 137 58 propose propose VERB ejpam-4172 137 59 to to PART ejpam-4172 137 60 consider consider VERB ejpam-4172 137 61 the the DET ejpam-4172 137 62 jacobi jacobi PROPN ejpam-4172 137 63 polynomial polynomial PROPN ejpam-4172 137 64 p p PROPN ejpam-4172 137 65 ( ( PUNCT ejpam-4172 137 66 457,763 457,763 NUM ejpam-4172 137 67 ) ) PUNCT ejpam-4172 137 68 1 1 NUM ejpam-4172 137 69 ( ( PUNCT ejpam-4172 137 70 x x X ejpam-4172 137 71 ) ) PUNCT ejpam-4172 137 72 whose whose DET ejpam-4172 137 73 root root NOUN ejpam-4172 137 74 is be AUX ejpam-4172 137 75 x0 x0 PROPN ejpam-4172 137 76 = = PUNCT ejpam-4172 137 77 0.250409 0.250409 NUM ejpam-4172 137 78 . . PUNCT ejpam-4172 138 1 besides besides SCONJ ejpam-4172 138 2 we we PRON ejpam-4172 138 3 we we PRON ejpam-4172 138 4 choose choose VERB ejpam-4172 138 5 the the DET ejpam-4172 138 6 collocation collocation NOUN ejpam-4172 138 7 point point NOUN ejpam-4172 138 8 ωt ωt ADP ejpam-4172 138 9 = = SYM ejpam-4172 138 10 0.250409π 0.250409π PROPN ejpam-4172 138 11 , , PUNCT ejpam-4172 138 12 the the DET ejpam-4172 138 13 approximation approximation NOUN ejpam-4172 138 14 to to ADP ejpam-4172 138 15 periodic periodic ADJ ejpam-4172 138 16 solution solution NOUN ejpam-4172 138 17 of of ADP ejpam-4172 138 18 the the DET ejpam-4172 138 19 nonlinear nonlinear ADJ ejpam-4172 138 20 oscillator oscillator NOUN ejpam-4172 138 21 is be AUX ejpam-4172 138 22 u(t u(t NOUN ejpam-4172 138 23 ) ) PUNCT ejpam-4172 138 24 = = PUNCT ejpam-4172 139 1 a a DET ejpam-4172 139 2 cos(ωapp(a)t cos(ωapp(a)t PROPN ejpam-4172 139 3 ) ) PUNCT ejpam-4172 139 4 , , PUNCT ejpam-4172 139 5 ( ( PUNCT ejpam-4172 139 6 33 33 NUM ejpam-4172 139 7 ) ) PUNCT ejpam-4172 139 8 where where SCONJ ejpam-4172 139 9 ωapp(a ωapp(a NUM ejpam-4172 139 10 ) ) PUNCT ejpam-4172 139 11 = = SYM ejpam-4172 139 12 √ √ PROPN ejpam-4172 139 13 a2 a2 PROPN ejpam-4172 139 14 sin2(0.250409π sin2(0.250409π PROPN ejpam-4172 139 15 ) ) PUNCT ejpam-4172 139 16 + + CCONJ ejpam-4172 139 17 ( ( PUNCT ejpam-4172 139 18 1.5 1.5 NUM ejpam-4172 139 19 ) ) PUNCT ejpam-4172 139 20 [ [ PUNCT ejpam-4172 140 1 √ √ ADP ejpam-4172 140 2 1 1 NUM ejpam-4172 140 3 + + NOUN ejpam-4172 140 4 a2 a2 PROPN ejpam-4172 140 5 cos2(0.250409π)− cos2(0.250409π)− VERB ejpam-4172 140 6 √ √ NOUN ejpam-4172 140 7 1 1 NUM ejpam-4172 140 8 + + SYM ejpam-4172 140 9 a2 a2 PROPN ejpam-4172 140 10 ] ] X ejpam-4172 140 11 a a DET ejpam-4172 140 12 sin(0.250409π sin(0.250409π NOUN ejpam-4172 140 13 ) ) PUNCT ejpam-4172 140 14 . . PUNCT ejpam-4172 141 1 ( ( PUNCT ejpam-4172 141 2 34 34 NUM ejpam-4172 141 3 ) ) PUNCT ejpam-4172 141 4 in in ADP ejpam-4172 141 5 order order NOUN ejpam-4172 141 6 to to PART ejpam-4172 141 7 illustrate illustrate VERB ejpam-4172 141 8 that that SCONJ ejpam-4172 141 9 our our PRON ejpam-4172 141 10 obtained obtain VERB ejpam-4172 141 11 result result NOUN ejpam-4172 141 12 is be AUX ejpam-4172 141 13 a a DET ejpam-4172 141 14 nice nice ADJ ejpam-4172 141 15 approximation approximation NOUN ejpam-4172 141 16 to to ADP ejpam-4172 141 17 the the DET ejpam-4172 141 18 exact exact ADJ ejpam-4172 141 19 solution solution NOUN ejpam-4172 141 20 , , PUNCT ejpam-4172 141 21 we we PRON ejpam-4172 141 22 compute compute VERB ejpam-4172 141 23 numerical numerical ADJ ejpam-4172 141 24 examples example NOUN ejpam-4172 141 25 assuming assume VERB ejpam-4172 141 26 a a DET ejpam-4172 141 27 = = SYM ejpam-4172 141 28 50 50 NUM ejpam-4172 141 29 and and CCONJ ejpam-4172 141 30 γ γ X ejpam-4172 141 31 = = SYM ejpam-4172 141 32 0.25 0.25 NUM ejpam-4172 141 33 , , PUNCT ejpam-4172 141 34 0.75 0.75 NUM ejpam-4172 141 35 , , PUNCT ejpam-4172 141 36 see see VERB ejpam-4172 141 37 fig fig NOUN ejpam-4172 141 38 . . PUNCT ejpam-4172 142 1 3 3 NUM ejpam-4172 142 2 and and CCONJ ejpam-4172 142 3 fig fig NOUN ejpam-4172 142 4 . . PUNCT ejpam-4172 143 1 4 4 NUM ejpam-4172 143 2 , , PUNCT ejpam-4172 143 3 respectively respectively ADV ejpam-4172 143 4 , , PUNCT ejpam-4172 143 5 and and CCONJ ejpam-4172 143 6 also also ADV ejpam-4172 143 7 table table VERB ejpam-4172 143 8 2 2 NUM ejpam-4172 143 9 . . PUNCT ejpam-4172 144 1 o. o. PROPN ejpam-4172 144 2 gonzález gonzález PROPN ejpam-4172 144 3 - - PUNCT ejpam-4172 144 4 gaxiola gaxiola PROPN ejpam-4172 144 5 , , PUNCT ejpam-4172 144 6 j. j. PROPN ejpam-4172 144 7 ruiz ruiz PROPN ejpam-4172 144 8 de de PROPN ejpam-4172 144 9 chávez chávez PROPN ejpam-4172 144 10 / / SYM ejpam-4172 144 11 eur eur PROPN ejpam-4172 144 12 . . PUNCT ejpam-4172 145 1 j. j. PROPN ejpam-4172 145 2 pure pure PROPN ejpam-4172 145 3 appl appl PROPN ejpam-4172 145 4 . . PROPN ejpam-4172 145 5 math math PROPN ejpam-4172 145 6 , , PUNCT ejpam-4172 145 7 15 15 NUM ejpam-4172 145 8 ( ( PUNCT ejpam-4172 145 9 1 1 NUM ejpam-4172 145 10 ) ) PUNCT ejpam-4172 145 11 ( ( PUNCT ejpam-4172 145 12 2022 2022 NUM ejpam-4172 145 13 ) ) PUNCT ejpam-4172 145 14 , , PUNCT ejpam-4172 145 15 82 82 NUM ejpam-4172 145 16 - - SYM ejpam-4172 145 17 99 99 NUM ejpam-4172 146 1 91 91 NUM ejpam-4172 146 2 0 0 NUM ejpam-4172 146 3 1 1 NUM ejpam-4172 146 4 2 2 NUM ejpam-4172 146 5 3 3 NUM ejpam-4172 146 6 4 4 NUM ejpam-4172 146 7 5 5 NUM ejpam-4172 146 8 6 6 NUM ejpam-4172 146 9 -4 -4 NOUN ejpam-4172 146 10 -2 -2 NOUN ejpam-4172 146 11 0 0 NUM ejpam-4172 146 12 2 2 NUM ejpam-4172 146 13 4 4 NUM ejpam-4172 146 14 t t NOUN ejpam-4172 146 15 u(t u(t NOUN ejpam-4172 146 16 ) ) PUNCT ejpam-4172 146 17 figure figure NOUN ejpam-4172 146 18 3 3 NUM ejpam-4172 146 19 : : PUNCT ejpam-4172 146 20 example example NOUN ejpam-4172 146 21 2 2 NUM ejpam-4172 146 22 : : PUNCT ejpam-4172 146 23 comparison comparison NOUN ejpam-4172 146 24 of of ADP ejpam-4172 146 25 the the DET ejpam-4172 146 26 approximate approximate ADJ ejpam-4172 146 27 periodic periodic ADJ ejpam-4172 146 28 solution solution NOUN ejpam-4172 146 29 ( ( PUNCT ejpam-4172 146 30 dashed dash VERB ejpam-4172 146 31 ) ) PUNCT ejpam-4172 146 32 with with ADP ejpam-4172 146 33 exact exact ADJ ejpam-4172 146 34 solution solution NOUN ejpam-4172 146 35 ( ( PUNCT ejpam-4172 146 36 green green NOUN ejpam-4172 146 37 ) ) PUNCT ejpam-4172 146 38 for for ADP ejpam-4172 146 39 a a DET ejpam-4172 146 40 = = SYM ejpam-4172 146 41 50 50 NUM ejpam-4172 146 42 and and CCONJ ejpam-4172 146 43 γ γ X ejpam-4172 146 44 = = NOUN ejpam-4172 146 45 0.25 0.25 NUM ejpam-4172 146 46 . . PUNCT ejpam-4172 146 47 0 0 NUM ejpam-4172 147 1 1 1 NUM ejpam-4172 147 2 2 2 NUM ejpam-4172 147 3 3 3 NUM ejpam-4172 147 4 4 4 NUM ejpam-4172 147 5 5 5 NUM ejpam-4172 147 6 6 6 NUM ejpam-4172 147 7 7 7 NUM ejpam-4172 147 8 -4 -4 INTJ ejpam-4172 147 9 -2 -2 NOUN ejpam-4172 147 10 0 0 NUM ejpam-4172 147 11 2 2 NUM ejpam-4172 147 12 4 4 NUM ejpam-4172 147 13 t t NOUN ejpam-4172 147 14 u(t u(t NOUN ejpam-4172 147 15 ) ) PUNCT ejpam-4172 147 16 figure figure NOUN ejpam-4172 147 17 4 4 NUM ejpam-4172 147 18 : : PUNCT ejpam-4172 147 19 example example NOUN ejpam-4172 147 20 2 2 NUM ejpam-4172 147 21 : : PUNCT ejpam-4172 147 22 comparison comparison NOUN ejpam-4172 147 23 of of ADP ejpam-4172 147 24 the the DET ejpam-4172 147 25 approximate approximate ADJ ejpam-4172 147 26 periodic periodic ADJ ejpam-4172 147 27 solution solution NOUN ejpam-4172 147 28 ( ( PUNCT ejpam-4172 147 29 dashed dash VERB ejpam-4172 147 30 ) ) PUNCT ejpam-4172 147 31 with with ADP ejpam-4172 147 32 exact exact ADJ ejpam-4172 147 33 solution solution NOUN ejpam-4172 147 34 ( ( PUNCT ejpam-4172 147 35 green green NOUN ejpam-4172 147 36 ) ) PUNCT ejpam-4172 147 37 for for ADP ejpam-4172 147 38 a a DET ejpam-4172 147 39 = = SYM ejpam-4172 147 40 50 50 NUM ejpam-4172 147 41 and and CCONJ ejpam-4172 147 42 γ γ X ejpam-4172 147 43 = = SYM ejpam-4172 147 44 0.75 0.75 NUM ejpam-4172 147 45 . . PUNCT ejpam-4172 148 1 we we PRON ejpam-4172 148 2 consider consider VERB ejpam-4172 148 3 our our PRON ejpam-4172 148 4 approximation approximation NOUN ejpam-4172 148 5 to to PART ejpam-4172 148 6 be be AUX ejpam-4172 148 7 acceptable acceptable ADJ ejpam-4172 148 8 , , PUNCT ejpam-4172 148 9 since since SCONJ ejpam-4172 148 10 the the DET ejpam-4172 148 11 best good ADJ ejpam-4172 148 12 approximation approximation NOUN ejpam-4172 148 13 to to ADP ejpam-4172 148 14 this this DET ejpam-4172 148 15 problem problem NOUN ejpam-4172 148 16 had have AUX ejpam-4172 148 17 shown show VERB ejpam-4172 148 18 a a DET ejpam-4172 148 19 maximum maximum ADJ ejpam-4172 148 20 error error NOUN ejpam-4172 148 21 of of ADP ejpam-4172 148 22 0.8 0.8 NUM ejpam-4172 148 23 % % NOUN ejpam-4172 148 24 for for ADP ejpam-4172 148 25 all all DET ejpam-4172 148 26 the the DET ejpam-4172 148 27 range range NOUN ejpam-4172 148 28 of of ADP ejpam-4172 148 29 values value NOUN ejpam-4172 148 30 of of ADP ejpam-4172 148 31 amplitude amplitude NOUN ejpam-4172 148 32 of of ADP ejpam-4172 148 33 oscillation oscillation NOUN ejpam-4172 149 1 a a DET ejpam-4172 149 2 [ [ X ejpam-4172 149 3 4 4 NUM ejpam-4172 149 4 ] ] PUNCT ejpam-4172 149 5 . . PUNCT ejpam-4172 150 1 example example NOUN ejpam-4172 150 2 3 3 NUM ejpam-4172 150 3 o. o. PROPN ejpam-4172 150 4 gonzález gonzález PROPN ejpam-4172 150 5 - - PUNCT ejpam-4172 150 6 gaxiola gaxiola PROPN ejpam-4172 150 7 , , PUNCT ejpam-4172 150 8 j. j. PROPN ejpam-4172 150 9 ruiz ruiz PROPN ejpam-4172 150 10 de de PROPN ejpam-4172 150 11 chávez chávez PROPN ejpam-4172 150 12 / / SYM ejpam-4172 150 13 eur eur PROPN ejpam-4172 150 14 . . PUNCT ejpam-4172 151 1 j. j. PROPN ejpam-4172 151 2 pure pure PROPN ejpam-4172 151 3 appl appl PROPN ejpam-4172 151 4 . . PROPN ejpam-4172 151 5 math math PROPN ejpam-4172 151 6 , , PUNCT ejpam-4172 151 7 15 15 NUM ejpam-4172 151 8 ( ( PUNCT ejpam-4172 151 9 1 1 NUM ejpam-4172 151 10 ) ) PUNCT ejpam-4172 151 11 ( ( PUNCT ejpam-4172 151 12 2022 2022 NUM ejpam-4172 151 13 ) ) PUNCT ejpam-4172 151 14 , , PUNCT ejpam-4172 151 15 82 82 NUM ejpam-4172 151 16 - - SYM ejpam-4172 151 17 99 99 NUM ejpam-4172 151 18 92 92 NUM ejpam-4172 151 19 consider consider VERB ejpam-4172 151 20 the the DET ejpam-4172 151 21 duffing duffing NOUN ejpam-4172 151 22 - - PUNCT ejpam-4172 151 23 relativistic relativistic ADJ ejpam-4172 151 24 oscillator oscillator NOUN ejpam-4172 151 25 [ [ X ejpam-4172 151 26 45 45 NUM ejpam-4172 151 27 ] ] X ejpam-4172 151 28 : : PUNCT ejpam-4172 151 29 ü+ ü+ VERB ejpam-4172 151 30 u+ u+ NOUN ejpam-4172 151 31 u3 u3 NOUN ejpam-4172 151 32 − − NOUN ejpam-4172 151 33 λu√ λu√ NOUN ejpam-4172 151 34 1 1 NUM ejpam-4172 151 35 + + CCONJ ejpam-4172 151 36 u2 u2 NOUN ejpam-4172 151 37 = = SYM ejpam-4172 151 38 0 0 NUM ejpam-4172 151 39 , , PUNCT ejpam-4172 151 40 u(0 u(0 NOUN ejpam-4172 151 41 ) ) PUNCT ejpam-4172 151 42 = = SYM ejpam-4172 151 43 a a PRON ejpam-4172 151 44 , , PUNCT ejpam-4172 151 45 u̇(0 u̇(0 NOUN ejpam-4172 151 46 ) ) PUNCT ejpam-4172 151 47 = = SYM ejpam-4172 151 48 0 0 NUM ejpam-4172 151 49 , , PUNCT ejpam-4172 151 50 ( ( PUNCT ejpam-4172 151 51 35 35 NUM ejpam-4172 151 52 ) ) PUNCT ejpam-4172 151 53 where where SCONJ ejpam-4172 151 54 0 0 NUM ejpam-4172 151 55 < < X ejpam-4172 151 56 λ λ X ejpam-4172 151 57 < < X ejpam-4172 151 58 1 1 NUM ejpam-4172 151 59 is be AUX ejpam-4172 151 60 a a DET ejpam-4172 151 61 constant constant ADJ ejpam-4172 151 62 . . PUNCT ejpam-4172 152 1 since since SCONJ ejpam-4172 152 2 f(u f(u PROPN ejpam-4172 152 3 ) ) PUNCT ejpam-4172 152 4 = = PUNCT ejpam-4172 152 5 u+ u+ NOUN ejpam-4172 152 6 u3 u3 NOUN ejpam-4172 152 7 − − PROPN ejpam-4172 152 8 λu√ λu√ NOUN ejpam-4172 152 9 1+u2 1+u2 NUM ejpam-4172 152 10 , , PUNCT ejpam-4172 152 11 we we PRON ejpam-4172 152 12 get get VERB ejpam-4172 152 13 f f PROPN ejpam-4172 152 14 ( ( PUNCT ejpam-4172 152 15 u u NOUN ejpam-4172 152 16 ) ) PUNCT ejpam-4172 152 17 = = SYM ejpam-4172 153 1 ∫ ∫ PROPN ejpam-4172 153 2 f(u)du f(u)du PROPN ejpam-4172 153 3 = = PROPN ejpam-4172 153 4 u2 u2 PROPN ejpam-4172 153 5 2 2 NUM ejpam-4172 153 6 + + CCONJ ejpam-4172 153 7 u4 u4 PROPN ejpam-4172 153 8 4 4 NUM ejpam-4172 153 9 − − NOUN ejpam-4172 153 10 λ λ NOUN ejpam-4172 153 11 √ √ ADV ejpam-4172 153 12 1 1 NUM ejpam-4172 153 13 + + CCONJ ejpam-4172 153 14 u2 u2 NOUN ejpam-4172 153 15 . . PUNCT ejpam-4172 154 1 ( ( PUNCT ejpam-4172 154 2 36 36 NUM ejpam-4172 154 3 ) ) PUNCT ejpam-4172 154 4 as as ADP ejpam-4172 154 5 a a DET ejpam-4172 154 6 consequence consequence NOUN ejpam-4172 154 7 , , PUNCT ejpam-4172 154 8 we we PRON ejpam-4172 154 9 have have VERB ejpam-4172 154 10 f f X ejpam-4172 154 11 ( ( PUNCT ejpam-4172 154 12 a a NOUN ejpam-4172 154 13 ) ) PUNCT ejpam-4172 154 14 = = SYM ejpam-4172 154 15 a2 a2 NOUN ejpam-4172 154 16 2 2 NUM ejpam-4172 154 17 + + NUM ejpam-4172 154 18 a4 a4 NOUN ejpam-4172 154 19 4 4 NUM ejpam-4172 154 20 − − NOUN ejpam-4172 155 1 λ λ NOUN ejpam-4172 155 2 √ √ NOUN ejpam-4172 155 3 1 1 NUM ejpam-4172 155 4 + + SYM ejpam-4172 155 5 a2 a2 NOUN ejpam-4172 155 6 , , PUNCT ejpam-4172 155 7 ( ( PUNCT ejpam-4172 155 8 37 37 NUM ejpam-4172 155 9 ) ) PUNCT ejpam-4172 155 10 f f NOUN ejpam-4172 155 11 ( ( PUNCT ejpam-4172 155 12 a a DET ejpam-4172 155 13 cos(πx0 cos(πx0 NOUN ejpam-4172 155 14 ) ) PUNCT ejpam-4172 155 15 ) ) PUNCT ejpam-4172 156 1 = = SYM ejpam-4172 156 2 a2 a2 PROPN ejpam-4172 156 3 cos2(πx0 cos2(πx0 NOUN ejpam-4172 156 4 ) ) PUNCT ejpam-4172 156 5 2 2 NUM ejpam-4172 156 6 + + NUM ejpam-4172 156 7 a4 a4 NOUN ejpam-4172 156 8 cos4(πx0 cos4(πx0 NOUN ejpam-4172 156 9 ) ) PUNCT ejpam-4172 156 10 4 4 NUM ejpam-4172 156 11 − − NOUN ejpam-4172 156 12 λ λ NOUN ejpam-4172 156 13 √ √ NOUN ejpam-4172 156 14 1 1 NUM ejpam-4172 156 15 + + NOUN ejpam-4172 156 16 a2 a2 NOUN ejpam-4172 156 17 cos2(πx0 cos2(πx0 NOUN ejpam-4172 156 18 ) ) PUNCT ejpam-4172 156 19 . . PUNCT ejpam-4172 157 1 ( ( PUNCT ejpam-4172 157 2 38 38 NUM ejpam-4172 157 3 ) ) PUNCT ejpam-4172 157 4 we we PRON ejpam-4172 157 5 remark remark VERB ejpam-4172 157 6 that that SCONJ ejpam-4172 157 7 eq eq ADP ejpam-4172 157 8 . . PUNCT ejpam-4172 158 1 ( ( PUNCT ejpam-4172 158 2 35 35 NUM ejpam-4172 158 3 ) ) PUNCT ejpam-4172 158 4 is be AUX ejpam-4172 158 5 a a DET ejpam-4172 158 6 numerical numerical ADJ ejpam-4172 158 7 model model NOUN ejpam-4172 158 8 of of ADP ejpam-4172 158 9 a a DET ejpam-4172 158 10 conservative conservative ADJ ejpam-4172 158 11 system system NOUN ejpam-4172 158 12 . . PUNCT ejpam-4172 159 1 the the DET ejpam-4172 159 2 exact exact ADJ ejpam-4172 159 3 frequency frequency NOUN ejpam-4172 159 4 ωex ωex NOUN ejpam-4172 159 5 of of ADP ejpam-4172 159 6 this this DET ejpam-4172 159 7 oscillator oscillator NOUN ejpam-4172 159 8 is be AUX ejpam-4172 159 9 given give VERB ejpam-4172 159 10 as as ADP ejpam-4172 159 11 [ [ X ejpam-4172 159 12 45 45 NUM ejpam-4172 159 13 ] ] NUM ejpam-4172 159 14 : : PUNCT ejpam-4172 159 15 ωex(a ωex(a PROPN ejpam-4172 159 16 ) ) PUNCT ejpam-4172 159 17 = = PUNCT ejpam-4172 160 1 π π PROPN ejpam-4172 161 1 2 2 PROPN ejpam-4172 161 2 ∫ ∫ PROPN ejpam-4172 161 3 a a DET ejpam-4172 161 4 0 0 NUM ejpam-4172 161 5 dy√ dy√ PROPN ejpam-4172 161 6 ( ( PUNCT ejpam-4172 161 7 a2 a2 PROPN ejpam-4172 161 8 − − PROPN ejpam-4172 161 9 y2 y2 PROPN ejpam-4172 161 10 ) ) PUNCT ejpam-4172 162 1 + + CCONJ ejpam-4172 162 2 1 1 NUM ejpam-4172 162 3 2(a 2(a NUM ejpam-4172 162 4 4 4 NUM ejpam-4172 162 5 − − NOUN ejpam-4172 163 1 y4)− y4)− PROPN ejpam-4172 163 2 2λ 2λ PROPN ejpam-4172 163 3 ( ( PUNCT ejpam-4172 163 4 √ √ NUM ejpam-4172 163 5 a2 a2 PROPN ejpam-4172 163 6 + + CCONJ ejpam-4172 163 7 1− 1− NUM ejpam-4172 163 8 √ √ NUM ejpam-4172 163 9 y2 y2 INTJ ejpam-4172 164 1 + + CCONJ ejpam-4172 164 2 1 1 X ejpam-4172 164 3 ) ) PUNCT ejpam-4172 164 4 −1 −1 NOUN ejpam-4172 164 5 . . PUNCT ejpam-4172 165 1 ( ( PUNCT ejpam-4172 165 2 39 39 NUM ejpam-4172 165 3 ) ) PUNCT ejpam-4172 165 4 case case NOUN ejpam-4172 165 5 λ λ X ejpam-4172 165 6 = = SYM ejpam-4172 165 7 0.5 0.5 NUM ejpam-4172 165 8 for for ADP ejpam-4172 165 9 this this DET ejpam-4172 165 10 value value NOUN ejpam-4172 165 11 , , PUNCT ejpam-4172 165 12 we we PRON ejpam-4172 165 13 consider consider VERB ejpam-4172 165 14 the the DET ejpam-4172 165 15 jacobi jacobi PROPN ejpam-4172 165 16 polynomial polynomial PROPN ejpam-4172 165 17 p p PROPN ejpam-4172 165 18 ( ( PUNCT ejpam-4172 165 19 891,1487 891,1487 NUM ejpam-4172 165 20 ) ) PUNCT ejpam-4172 165 21 1 1 NUM ejpam-4172 165 22 ( ( PUNCT ejpam-4172 165 23 x x X ejpam-4172 165 24 ) ) PUNCT ejpam-4172 165 25 whose whose DET ejpam-4172 165 26 root root NOUN ejpam-4172 165 27 is be AUX ejpam-4172 165 28 x0 x0 PROPN ejpam-4172 165 29 = = SYM ejpam-4172 165 30 0.25042 0.25042 NUM ejpam-4172 165 31 . . PUNCT ejpam-4172 166 1 moreover moreover ADV ejpam-4172 166 2 , , PUNCT ejpam-4172 166 3 we we PRON ejpam-4172 166 4 take take VERB ejpam-4172 166 5 ωt ωt PRON ejpam-4172 166 6 = = PUNCT ejpam-4172 166 7 0.25042π 0.25042π NOUN ejpam-4172 166 8 and and CCONJ ejpam-4172 166 9 consider consider VERB ejpam-4172 166 10 the the DET ejpam-4172 166 11 proposed propose VERB ejpam-4172 166 12 approach approach NOUN ejpam-4172 166 13 in in ADP ejpam-4172 166 14 eq eq ADP ejpam-4172 166 15 . . PUNCT ejpam-4172 167 1 ( ( PUNCT ejpam-4172 167 2 19 19 NUM ejpam-4172 167 3 ) ) PUNCT ejpam-4172 167 4 . . PUNCT ejpam-4172 168 1 hence hence ADV ejpam-4172 168 2 one one PRON ejpam-4172 168 3 can can AUX ejpam-4172 168 4 assume assume VERB ejpam-4172 168 5 for for ADP ejpam-4172 168 6 the the DET ejpam-4172 168 7 frequency frequency NOUN ejpam-4172 168 8 - - PUNCT ejpam-4172 168 9 amplitude amplitude NOUN ejpam-4172 168 10 formulation formulation NOUN ejpam-4172 168 11 ωapp(a ωapp(a NUM ejpam-4172 168 12 ) ) PUNCT ejpam-4172 168 13 = = PUNCT ejpam-4172 168 14 1 1 NUM ejpam-4172 168 15 a a DET ejpam-4172 168 16 sin(0.25042π sin(0.25042π NOUN ejpam-4172 168 17 ) ) PUNCT ejpam-4172 168 18 ( ( PUNCT ejpam-4172 168 19 a2 a2 PROPN ejpam-4172 168 20 sin2(0.25042π sin2(0.25042π PROPN ejpam-4172 168 21 ) ) PUNCT ejpam-4172 168 22 + + NUM ejpam-4172 168 23 a4 a4 NOUN ejpam-4172 168 24 2 2 NUM ejpam-4172 168 25 − − NOUN ejpam-4172 168 26 √ √ NUM ejpam-4172 168 27 a2 a2 PROPN ejpam-4172 168 28 + + CCONJ ejpam-4172 168 29 1 1 NUM ejpam-4172 168 30 − − NUM ejpam-4172 168 31 a4 a4 NOUN ejpam-4172 168 32 2 2 NUM ejpam-4172 168 33 cos4(0.25042π cos4(0.25042π NOUN ejpam-4172 168 34 ) ) PUNCT ejpam-4172 169 1 + + CCONJ ejpam-4172 169 2 √ √ NUM ejpam-4172 169 3 1 1 NUM ejpam-4172 169 4 + + PROPN ejpam-4172 169 5 a2 a2 PROPN ejpam-4172 169 6 cos2(0.25042π cos2(0.25042π PROPN ejpam-4172 169 7 ) ) PUNCT ejpam-4172 169 8 ) ) PUNCT ejpam-4172 169 9 1/2 1/2 NUM ejpam-4172 169 10 ( ( PUNCT ejpam-4172 169 11 40 40 NUM ejpam-4172 169 12 ) ) PUNCT ejpam-4172 169 13 on on ADP ejpam-4172 169 14 the the DET ejpam-4172 169 15 other other ADJ ejpam-4172 169 16 hand hand NOUN ejpam-4172 169 17 , , PUNCT ejpam-4172 169 18 if if SCONJ ejpam-4172 169 19 we we PRON ejpam-4172 169 20 consider consider VERB ejpam-4172 169 21 the the DET ejpam-4172 169 22 approximation approximation NOUN ejpam-4172 169 23 given give VERB ejpam-4172 169 24 by by ADP ejpam-4172 169 25 ( ( PUNCT ejpam-4172 169 26 20 20 NUM ejpam-4172 169 27 ) ) PUNCT ejpam-4172 169 28 , , PUNCT ejpam-4172 169 29 we we PRON ejpam-4172 169 30 have have AUX ejpam-4172 169 31 obtain obtain VERB ejpam-4172 169 32 the the DET ejpam-4172 169 33 following follow VERB ejpam-4172 169 34 periodic periodic ADJ ejpam-4172 169 35 solution solution NOUN ejpam-4172 169 36 of of ADP ejpam-4172 169 37 the the DET ejpam-4172 169 38 equation equation NOUN ejpam-4172 169 39 ( ( PUNCT ejpam-4172 169 40 35 35 NUM ejpam-4172 169 41 ) ) PUNCT ejpam-4172 169 42 uapp(t uapp(t PROPN ejpam-4172 169 43 ) ) PUNCT ejpam-4172 169 44 = = PUNCT ejpam-4172 170 1 a a DET ejpam-4172 170 2 cos(ωapp(a)t cos(ωapp(a)t PROPN ejpam-4172 170 3 ) ) PUNCT ejpam-4172 170 4 , , PUNCT ejpam-4172 170 5 ( ( PUNCT ejpam-4172 170 6 41 41 NUM ejpam-4172 170 7 ) ) PUNCT ejpam-4172 170 8 where where SCONJ ejpam-4172 170 9 ωapp(a ωapp(a NUM ejpam-4172 170 10 ) ) PUNCT ejpam-4172 170 11 is be AUX ejpam-4172 170 12 given give VERB ejpam-4172 170 13 explicitly explicitly ADV ejpam-4172 170 14 by by ADP ejpam-4172 170 15 the the DET ejpam-4172 170 16 eq eq NOUN ejpam-4172 170 17 . . PUNCT ejpam-4172 171 1 ( ( PUNCT ejpam-4172 171 2 40 40 NUM ejpam-4172 171 3 ) ) PUNCT ejpam-4172 171 4 . . PUNCT ejpam-4172 172 1 case case NOUN ejpam-4172 172 2 λ λ NOUN ejpam-4172 172 3 = = SYM ejpam-4172 172 4 0.9 0.9 NUM ejpam-4172 172 5 o. o. PROPN ejpam-4172 172 6 gonzález gonzález PROPN ejpam-4172 172 7 - - PUNCT ejpam-4172 172 8 gaxiola gaxiola PROPN ejpam-4172 172 9 , , PUNCT ejpam-4172 172 10 j. j. PROPN ejpam-4172 172 11 ruiz ruiz PROPN ejpam-4172 172 12 de de PROPN ejpam-4172 172 13 chávez chávez PROPN ejpam-4172 172 14 / / SYM ejpam-4172 172 15 eur eur PROPN ejpam-4172 172 16 . . PUNCT ejpam-4172 173 1 j. j. PROPN ejpam-4172 173 2 pure pure PROPN ejpam-4172 173 3 appl appl PROPN ejpam-4172 173 4 . . PROPN ejpam-4172 173 5 math math PROPN ejpam-4172 173 6 , , PUNCT ejpam-4172 173 7 15 15 NUM ejpam-4172 173 8 ( ( PUNCT ejpam-4172 173 9 1 1 NUM ejpam-4172 173 10 ) ) PUNCT ejpam-4172 173 11 ( ( PUNCT ejpam-4172 173 12 2022 2022 NUM ejpam-4172 173 13 ) ) PUNCT ejpam-4172 173 14 , , PUNCT ejpam-4172 173 15 82 82 NUM ejpam-4172 173 16 - - SYM ejpam-4172 173 17 99 99 NUM ejpam-4172 173 18 93 93 NUM ejpam-4172 173 19 here here ADV ejpam-4172 173 20 , , PUNCT ejpam-4172 173 21 we we PRON ejpam-4172 173 22 consider consider VERB ejpam-4172 173 23 the the DET ejpam-4172 173 24 jacobi jacobi PROPN ejpam-4172 173 25 polynomial polynomial PROPN ejpam-4172 173 26 p p PROPN ejpam-4172 173 27 ( ( PUNCT ejpam-4172 173 28 925,1549 925,1549 NUM ejpam-4172 173 29 ) ) PUNCT ejpam-4172 173 30 1 1 NUM ejpam-4172 173 31 ( ( PUNCT ejpam-4172 173 32 x x X ejpam-4172 173 33 ) ) PUNCT ejpam-4172 173 34 whose whose DET ejpam-4172 173 35 root root NOUN ejpam-4172 173 36 is be AUX ejpam-4172 173 37 x0 x0 PROPN ejpam-4172 173 38 = = PUNCT ejpam-4172 173 39 0.252019 0.252019 NUM ejpam-4172 173 40 . . PUNCT ejpam-4172 174 1 choosing choose VERB ejpam-4172 174 2 the the DET ejpam-4172 174 3 collocation collocation NOUN ejpam-4172 174 4 point point NOUN ejpam-4172 174 5 ωt ωt ADP ejpam-4172 174 6 = = SYM ejpam-4172 174 7 0.252019π 0.252019π PROPN ejpam-4172 174 8 , , PUNCT ejpam-4172 174 9 the the DET ejpam-4172 174 10 analytical analytical ADJ ejpam-4172 174 11 approximation approximation NOUN ejpam-4172 174 12 to to ADP ejpam-4172 174 13 periodic periodic ADJ ejpam-4172 174 14 solution solution NOUN ejpam-4172 174 15 of of ADP ejpam-4172 174 16 the the DET ejpam-4172 174 17 nonlinear nonlinear ADJ ejpam-4172 174 18 oscillator oscillator NOUN ejpam-4172 174 19 is be AUX ejpam-4172 174 20 u(t u(t NOUN ejpam-4172 174 21 ) ) PUNCT ejpam-4172 174 22 = = PUNCT ejpam-4172 175 1 a a DET ejpam-4172 175 2 cos(ωapp(a)t cos(ωapp(a)t PROPN ejpam-4172 175 3 ) ) PUNCT ejpam-4172 175 4 , , PUNCT ejpam-4172 175 5 ( ( PUNCT ejpam-4172 175 6 42 42 NUM ejpam-4172 175 7 ) ) PUNCT ejpam-4172 175 8 where where SCONJ ejpam-4172 175 9 ωapp(a ωapp(a NUM ejpam-4172 175 10 ) ) PUNCT ejpam-4172 175 11 = = PUNCT ejpam-4172 176 1 1 1 NUM ejpam-4172 176 2 a a DET ejpam-4172 176 3 sin(0.252019π sin(0.252019π NOUN ejpam-4172 176 4 ) ) PUNCT ejpam-4172 176 5 ( ( PUNCT ejpam-4172 176 6 a2 a2 PROPN ejpam-4172 176 7 sin2(0.252019π sin2(0.252019π PROPN ejpam-4172 176 8 ) ) PUNCT ejpam-4172 176 9 + + NUM ejpam-4172 177 1 a4 a4 NOUN ejpam-4172 177 2 2 2 NUM ejpam-4172 177 3 − − NOUN ejpam-4172 177 4 a4 a4 NOUN ejpam-4172 177 5 2 2 NUM ejpam-4172 177 6 cos4(0.25042π cos4(0.25042π PROPN ejpam-4172 177 7 ) ) PUNCT ejpam-4172 178 1 + + CCONJ ejpam-4172 178 2 9 9 NUM ejpam-4172 178 3 5 5 NUM ejpam-4172 178 4 [ [ PUNCT ejpam-4172 178 5 √ √ NUM ejpam-4172 178 6 1 1 NUM ejpam-4172 178 7 + + PROPN ejpam-4172 178 8 a2 a2 PROPN ejpam-4172 178 9 cos2(0.252019π)− cos2(0.252019π)− PROPN ejpam-4172 178 10 √ √ PROPN ejpam-4172 178 11 a2 a2 PROPN ejpam-4172 178 12 + + CCONJ ejpam-4172 178 13 1 1 NUM ejpam-4172 178 14 ] ] PUNCT ejpam-4172 178 15 ) ) PUNCT ejpam-4172 178 16 1/2 1/2 NUM ejpam-4172 178 17 ( ( PUNCT ejpam-4172 178 18 43 43 NUM ejpam-4172 178 19 ) ) PUNCT ejpam-4172 178 20 now now ADV ejpam-4172 178 21 , , PUNCT ejpam-4172 178 22 by by ADP ejpam-4172 178 23 taking take VERB ejpam-4172 178 24 the the DET ejpam-4172 178 25 approximation approximation NOUN ejpam-4172 178 26 given give VERB ejpam-4172 178 27 by by ADP ejpam-4172 178 28 eq eq PROPN ejpam-4172 178 29 . . PUNCT ejpam-4172 179 1 ( ( PUNCT ejpam-4172 179 2 20 20 NUM ejpam-4172 179 3 ) ) PUNCT ejpam-4172 179 4 , , PUNCT ejpam-4172 179 5 we we PRON ejpam-4172 179 6 have have AUX ejpam-4172 179 7 obtained obtain VERB ejpam-4172 179 8 the the DET ejpam-4172 179 9 following follow VERB ejpam-4172 179 10 periodic periodic ADJ ejpam-4172 179 11 solution solution NOUN ejpam-4172 179 12 , , PUNCT ejpam-4172 179 13 for for ADP ejpam-4172 179 14 the the DET ejpam-4172 179 15 case case NOUN ejpam-4172 179 16 λ λ X ejpam-4172 179 17 = = NOUN ejpam-4172 179 18 0.9 0.9 NUM ejpam-4172 179 19 of of ADP ejpam-4172 179 20 the the DET ejpam-4172 179 21 eq eq NOUN ejpam-4172 179 22 . . PUNCT ejpam-4172 180 1 ( ( PUNCT ejpam-4172 180 2 35 35 NUM ejpam-4172 180 3 ) ) PUNCT ejpam-4172 180 4 uapp(t uapp(t PROPN ejpam-4172 180 5 ) ) PUNCT ejpam-4172 180 6 = = PUNCT ejpam-4172 181 1 a a DET ejpam-4172 181 2 cos(ωapp(a)t cos(ωapp(a)t PROPN ejpam-4172 181 3 ) ) PUNCT ejpam-4172 181 4 , , PUNCT ejpam-4172 181 5 ( ( PUNCT ejpam-4172 181 6 44 44 NUM ejpam-4172 181 7 ) ) PUNCT ejpam-4172 181 8 where where SCONJ ejpam-4172 181 9 ωapp(a ωapp(a NUM ejpam-4172 181 10 ) ) PUNCT ejpam-4172 181 11 is be AUX ejpam-4172 181 12 explicitly explicitly ADV ejpam-4172 181 13 given give VERB ejpam-4172 181 14 by by ADP ejpam-4172 181 15 the the DET ejpam-4172 181 16 eq eq NOUN ejpam-4172 181 17 . . PUNCT ejpam-4172 182 1 ( ( PUNCT ejpam-4172 182 2 43 43 NUM ejpam-4172 182 3 ) ) PUNCT ejpam-4172 182 4 . . PUNCT ejpam-4172 183 1 the the DET ejpam-4172 183 2 numerical numerical ADJ ejpam-4172 183 3 results result NOUN ejpam-4172 183 4 are be AUX ejpam-4172 183 5 illustrated illustrate VERB ejpam-4172 183 6 in in ADP ejpam-4172 183 7 the the DET ejpam-4172 183 8 figs fig NOUN ejpam-4172 183 9 . . PUNCT ejpam-4172 184 1 5 5 NUM ejpam-4172 184 2 and and CCONJ ejpam-4172 184 3 6 6 NUM ejpam-4172 184 4 . . PUNCT ejpam-4172 185 1 in in ADP ejpam-4172 185 2 table table NOUN ejpam-4172 185 3 3 3 NUM ejpam-4172 185 4 compares compare VERB ejpam-4172 185 5 the the DET ejpam-4172 185 6 approximate approximate ADJ ejpam-4172 185 7 frequency frequency NOUN ejpam-4172 185 8 with with ADP ejpam-4172 185 9 the the DET ejpam-4172 185 10 exact exact ADJ ejpam-4172 185 11 one one NUM ejpam-4172 185 12 . . PUNCT ejpam-4172 186 1 λ λ X ejpam-4172 186 2 = = SYM ejpam-4172 186 3 0.5 0.5 NUM ejpam-4172 186 4 λ λ NOUN ejpam-4172 186 5 = = NOUN ejpam-4172 186 6 0.9 0.9 NUM ejpam-4172 186 7 a a DET ejpam-4172 186 8 ωapp(a ωapp(a PROPN ejpam-4172 186 9 ) ) PUNCT ejpam-4172 186 10 eq eq NOUN ejpam-4172 186 11 . . PUNCT ejpam-4172 187 1 ( ( PUNCT ejpam-4172 187 2 40)ωex(a 40)ωex(a X ejpam-4172 187 3 ) ) PUNCT ejpam-4172 187 4 eq eq NOUN ejpam-4172 187 5 . . PUNCT ejpam-4172 188 1 ( ( PUNCT ejpam-4172 188 2 39)rel 39)rel NUM ejpam-4172 188 3 . . PUNCT ejpam-4172 189 1 error error NOUN ejpam-4172 189 2 ( ( PUNCT ejpam-4172 189 3 % % INTJ ejpam-4172 189 4 ) ) PUNCT ejpam-4172 189 5 ωapp(a ωapp(a NUM ejpam-4172 189 6 ) ) PUNCT ejpam-4172 189 7 eq eq NOUN ejpam-4172 189 8 . . PUNCT ejpam-4172 189 9 ( ( PUNCT ejpam-4172 189 10 43)ωex(a 43)ωex(a NOUN ejpam-4172 189 11 ) ) PUNCT ejpam-4172 189 12 eq eq NOUN ejpam-4172 189 13 . . PUNCT ejpam-4172 189 14 ( ( PUNCT ejpam-4172 189 15 39)rel 39)rel NUM ejpam-4172 189 16 . . PUNCT ejpam-4172 190 1 error error NOUN ejpam-4172 190 2 ( ( PUNCT ejpam-4172 190 3 % % INTJ ejpam-4172 190 4 ) ) PUNCT ejpam-4172 190 5 0.01 0.01 NUM ejpam-4172 191 1 0.707173 0.707173 NUM ejpam-4172 191 2 0.707173 0.707173 NUM ejpam-4172 191 3 0.00 0.00 NUM ejpam-4172 191 4 % % NOUN ejpam-4172 191 5 0.316399 0.316399 NUM ejpam-4172 191 6 0.316400 0.316400 NUM ejpam-4172 191 7 0.00 0.00 NUM ejpam-4172 191 8 % % NOUN ejpam-4172 191 9 0.1 0.1 NUM ejpam-4172 191 10 0.713692 0.713692 NUM ejpam-4172 191 11 0.713692 0.713692 NUM ejpam-4172 191 12 0.00 0.00 NUM ejpam-4172 191 13 % % NOUN ejpam-4172 191 14 0.332881 0.332881 NUM ejpam-4172 191 15 0.332881 0.332881 NUM ejpam-4172 191 16 0.00 0.00 NUM ejpam-4172 191 17 % % NOUN ejpam-4172 191 18 0.5 0.5 NUM ejpam-4172 191 19 0.853412 0.853412 NUM ejpam-4172 191 20 0.851625 0.851625 NUM ejpam-4172 191 21 0.20 0.20 NUM ejpam-4172 191 22 % % NOUN ejpam-4172 191 23 0.600200 0.600200 NUM ejpam-4172 191 24 0.594202 0.594202 NUM ejpam-4172 191 25 1.00 1.00 NUM ejpam-4172 191 26 % % NOUN ejpam-4172 191 27 1 1 NUM ejpam-4172 191 28 1.170609 1.170609 NUM ejpam-4172 191 29 1.160531 1.160531 NUM ejpam-4172 191 30 0.86 0.86 NUM ejpam-4172 191 31 % % NOUN ejpam-4172 191 32 1.031538 1.031538 NUM ejpam-4172 191 33 1.014874 1.014874 NUM ejpam-4172 191 34 1.64 1.64 NUM ejpam-4172 191 35 % % NOUN ejpam-4172 191 36 5 5 NUM ejpam-4172 191 37 4.429387 4.429387 NUM ejpam-4172 191 38 4.342080 4.342080 NUM ejpam-4172 191 39 2.01 2.01 NUM ejpam-4172 191 40 % % NOUN ejpam-4172 191 41 4.411920 4.411920 NUM ejpam-4172 191 42 4.329715 4.329715 NUM ejpam-4172 191 43 1.89 1.89 NUM ejpam-4172 191 44 % % NOUN ejpam-4172 191 45 10 10 NUM ejpam-4172 191 46 8.710673 8.710673 NUM ejpam-4172 191 47 8.529475 8.529475 NUM ejpam-4172 191 48 2.12 2.12 NUM ejpam-4172 191 49 % % NOUN ejpam-4172 191 50 8.693554 8.693554 NUM ejpam-4172 191 51 8.526183 8.526183 NUM ejpam-4172 191 52 1.96 1.96 NUM ejpam-4172 191 53 % % NOUN ejpam-4172 191 54 50 50 NUM ejpam-4172 191 55 43.29364 43.29364 NUM ejpam-4172 191 56 42.37283 42.37283 NUM ejpam-4172 191 57 2.17 2.17 NUM ejpam-4172 191 58 % % NOUN ejpam-4172 191 59 43.22095 43.22095 NUM ejpam-4172 191 60 42.37269 42.37269 NUM ejpam-4172 191 61 2.00 2.00 NUM ejpam-4172 191 62 % % NOUN ejpam-4172 191 63 100 100 NUM ejpam-4172 191 64 86.57018 86.57018 NUM ejpam-4172 191 65 84.72744 84.72744 NUM ejpam-4172 191 66 2.17 2.17 NUM ejpam-4172 191 67 % % NOUN ejpam-4172 191 68 86.42497 86.42497 NUM ejpam-4172 191 69 84.72740 84.72740 NUM ejpam-4172 191 70 2.00 2.00 NUM ejpam-4172 191 71 % % NOUN ejpam-4172 191 72 1000 1000 NUM ejpam-4172 191 73 865.6450 865.6450 NUM ejpam-4172 191 74 847.2137 847.2137 NUM ejpam-4172 191 75 2.17 2.17 NUM ejpam-4172 191 76 % % NOUN ejpam-4172 191 77 864.1931 864.1931 NUM ejpam-4172 191 78 847.2137 847.2137 NUM ejpam-4172 191 79 2.00 2.00 NUM ejpam-4172 191 80 % % NOUN ejpam-4172 191 81 10000 10000 NUM ejpam-4172 191 82 8656.444 8656.444 NUM ejpam-4172 191 83 8472.131 8472.131 NUM ejpam-4172 191 84 2.17 2.17 NUM ejpam-4172 191 85 % % NOUN ejpam-4172 191 86 8641.925 8641.925 NUM ejpam-4172 191 87 8472.131 8472.131 NUM ejpam-4172 191 88 2.00 2.00 NUM ejpam-4172 191 89 % % NOUN ejpam-4172 191 90 100000 100000 NUM ejpam-4172 191 91 86564.44 86564.44 NUM ejpam-4172 191 92 84721.31 84721.31 NUM ejpam-4172 191 93 2.17 2.17 NUM ejpam-4172 191 94 % % NOUN ejpam-4172 191 95 86419.25 86419.25 NUM ejpam-4172 191 96 84721.31 84721.31 NUM ejpam-4172 191 97 2.00 2.00 NUM ejpam-4172 191 98 % % NOUN ejpam-4172 191 99 table table NOUN ejpam-4172 191 100 3 3 NUM ejpam-4172 191 101 : : PUNCT ejpam-4172 191 102 comparison comparison NOUN ejpam-4172 191 103 between between ADP ejpam-4172 191 104 frequencies frequency NOUN ejpam-4172 191 105 ωapp(a ωapp(a NUM ejpam-4172 191 106 ) ) PUNCT ejpam-4172 191 107 and and CCONJ ejpam-4172 191 108 ωex(a ωex(a NOUN ejpam-4172 191 109 ) ) PUNCT ejpam-4172 191 110 for for ADP ejpam-4172 191 111 different different ADJ ejpam-4172 191 112 values value NOUN ejpam-4172 191 113 of of ADP ejpam-4172 191 114 a. a. PROPN ejpam-4172 191 115 o. o. PROPN ejpam-4172 191 116 gonzález gonzález PROPN ejpam-4172 191 117 - - PUNCT ejpam-4172 191 118 gaxiola gaxiola PROPN ejpam-4172 191 119 , , PUNCT ejpam-4172 191 120 j. j. PROPN ejpam-4172 191 121 ruiz ruiz PROPN ejpam-4172 191 122 de de PROPN ejpam-4172 191 123 chávez chávez PROPN ejpam-4172 191 124 / / SYM ejpam-4172 191 125 eur eur PROPN ejpam-4172 191 126 . . PUNCT ejpam-4172 192 1 j. j. PROPN ejpam-4172 192 2 pure pure PROPN ejpam-4172 192 3 appl appl PROPN ejpam-4172 192 4 . . PROPN ejpam-4172 192 5 math math PROPN ejpam-4172 192 6 , , PUNCT ejpam-4172 192 7 15 15 NUM ejpam-4172 192 8 ( ( PUNCT ejpam-4172 192 9 1 1 NUM ejpam-4172 192 10 ) ) PUNCT ejpam-4172 192 11 ( ( PUNCT ejpam-4172 192 12 2022 2022 NUM ejpam-4172 192 13 ) ) PUNCT ejpam-4172 192 14 , , PUNCT ejpam-4172 192 15 82 82 NUM ejpam-4172 192 16 - - SYM ejpam-4172 192 17 99 99 NUM ejpam-4172 192 18 94 94 NUM ejpam-4172 192 19 0 0 NUM ejpam-4172 192 20 1 1 NUM ejpam-4172 192 21 2 2 NUM ejpam-4172 192 22 3 3 NUM ejpam-4172 192 23 4 4 NUM ejpam-4172 192 24 5 5 NUM ejpam-4172 192 25 -1.0 -1.0 PROPN ejpam-4172 192 26 -0.5 -0.5 X ejpam-4172 192 27 0.0 0.0 NUM ejpam-4172 192 28 0.5 0.5 NUM ejpam-4172 192 29 1.0 1.0 NUM ejpam-4172 192 30 t t NOUN ejpam-4172 192 31 u(t u(t NOUN ejpam-4172 192 32 ) ) PUNCT ejpam-4172 192 33 figure figure NOUN ejpam-4172 192 34 5 5 NUM ejpam-4172 192 35 : : PUNCT ejpam-4172 192 36 comparison comparison NOUN ejpam-4172 192 37 of of ADP ejpam-4172 192 38 the the DET ejpam-4172 192 39 approximate approximate ADJ ejpam-4172 192 40 periodic periodic ADJ ejpam-4172 192 41 solution solution NOUN ejpam-4172 192 42 ( ( PUNCT ejpam-4172 192 43 dashed dash VERB ejpam-4172 192 44 ) ) PUNCT ejpam-4172 192 45 with with ADP ejpam-4172 192 46 exact exact ADJ ejpam-4172 192 47 solution solution NOUN ejpam-4172 192 48 ( ( PUNCT ejpam-4172 192 49 red red NOUN ejpam-4172 192 50 ) ) PUNCT ejpam-4172 192 51 for for ADP ejpam-4172 192 52 a a DET ejpam-4172 192 53 = = SYM ejpam-4172 192 54 1 1 NUM ejpam-4172 192 55 and and CCONJ ejpam-4172 192 56 λ λ X ejpam-4172 192 57 = = NOUN ejpam-4172 192 58 0.5 0.5 NUM ejpam-4172 192 59 . . NOUN ejpam-4172 192 60 0 0 NUM ejpam-4172 193 1 2 2 NUM ejpam-4172 193 2 4 4 NUM ejpam-4172 193 3 6 6 NUM ejpam-4172 193 4 8 8 NUM ejpam-4172 193 5 10 10 NUM ejpam-4172 193 6 -0.4 -0.4 NUM ejpam-4172 193 7 -0.2 -0.2 PROPN ejpam-4172 193 8 0.0 0.0 NUM ejpam-4172 193 9 0.2 0.2 NUM ejpam-4172 193 10 0.4 0.4 NUM ejpam-4172 193 11 t t NOUN ejpam-4172 193 12 u(t u(t PROPN ejpam-4172 193 13 ) ) PUNCT ejpam-4172 193 14 figure figure NOUN ejpam-4172 193 15 6 6 NUM ejpam-4172 193 16 : : PUNCT ejpam-4172 193 17 comparison comparison NOUN ejpam-4172 193 18 of of ADP ejpam-4172 193 19 the the DET ejpam-4172 193 20 approximate approximate ADJ ejpam-4172 193 21 periodic periodic ADJ ejpam-4172 193 22 solution solution NOUN ejpam-4172 193 23 ( ( PUNCT ejpam-4172 193 24 dashed dash VERB ejpam-4172 193 25 ) ) PUNCT ejpam-4172 193 26 with with ADP ejpam-4172 193 27 exact exact ADJ ejpam-4172 193 28 solution solution NOUN ejpam-4172 193 29 ( ( PUNCT ejpam-4172 193 30 red red NOUN ejpam-4172 193 31 ) ) PUNCT ejpam-4172 193 32 for for ADP ejpam-4172 193 33 a a DET ejpam-4172 193 34 = = SYM ejpam-4172 193 35 0.5 0.5 NUM ejpam-4172 193 36 and and CCONJ ejpam-4172 193 37 λ λ X ejpam-4172 193 38 = = NOUN ejpam-4172 193 39 0.9 0.9 NUM ejpam-4172 193 40 . . PUNCT ejpam-4172 194 1 we we PRON ejpam-4172 194 2 consider consider VERB ejpam-4172 194 3 our our PRON ejpam-4172 194 4 approximation approximation NOUN ejpam-4172 194 5 to to PART ejpam-4172 194 6 be be AUX ejpam-4172 194 7 acceptable acceptable ADJ ejpam-4172 194 8 , , PUNCT ejpam-4172 194 9 since since SCONJ ejpam-4172 194 10 the the DET ejpam-4172 194 11 best good ADJ ejpam-4172 194 12 approximation approximation NOUN ejpam-4172 194 13 to to ADP ejpam-4172 194 14 this this DET ejpam-4172 194 15 problem problem NOUN ejpam-4172 194 16 had have AUX ejpam-4172 194 17 shown show VERB ejpam-4172 194 18 a a DET ejpam-4172 194 19 maximum maximum ADJ ejpam-4172 194 20 error error NOUN ejpam-4172 194 21 of of ADP ejpam-4172 194 22 2.2 2.2 NUM ejpam-4172 194 23 % % NOUN ejpam-4172 194 24 for for ADP ejpam-4172 194 25 all all DET ejpam-4172 194 26 the the DET ejpam-4172 194 27 range range NOUN ejpam-4172 194 28 of of ADP ejpam-4172 194 29 values value NOUN ejpam-4172 194 30 of of ADP ejpam-4172 194 31 amplitude amplitude NOUN ejpam-4172 194 32 of of ADP ejpam-4172 194 33 oscillation oscillation NOUN ejpam-4172 195 1 a a PRON ejpam-4172 196 1 [ [ X ejpam-4172 196 2 39 39 NUM ejpam-4172 196 3 ] ] PUNCT ejpam-4172 196 4 . . PUNCT ejpam-4172 197 1 references reference NOUN ejpam-4172 197 2 95 95 NUM ejpam-4172 197 3 4 4 NUM ejpam-4172 197 4 . . PUNCT ejpam-4172 198 1 conclusions conclusion NOUN ejpam-4172 198 2 another another DET ejpam-4172 198 3 amplitude amplitude NOUN ejpam-4172 198 4 - - PUNCT ejpam-4172 198 5 frequency frequency NOUN ejpam-4172 198 6 relationship relationship NOUN ejpam-4172 198 7 as as ADP ejpam-4172 198 8 of of ADP ejpam-4172 198 9 late late ADV ejpam-4172 198 10 established establish VERB ejpam-4172 198 11 in in ADP ejpam-4172 198 12 [ [ X ejpam-4172 198 13 21 21 NUM ejpam-4172 198 14 ] ] PUNCT ejpam-4172 198 15 is be AUX ejpam-4172 198 16 proved prove VERB ejpam-4172 198 17 being be AUX ejpam-4172 198 18 a a DET ejpam-4172 198 19 ground ground NOUN ejpam-4172 198 20 - - PUNCT ejpam-4172 198 21 breaking break VERB ejpam-4172 198 22 numerical numerical ADJ ejpam-4172 198 23 tool tool NOUN ejpam-4172 198 24 for for ADP ejpam-4172 198 25 use use NOUN ejpam-4172 198 26 in in ADP ejpam-4172 198 27 the the DET ejpam-4172 198 28 quest quest NOUN ejpam-4172 198 29 for for ADP ejpam-4172 198 30 periodic periodic ADJ ejpam-4172 198 31 solutions solution NOUN ejpam-4172 198 32 of of ADP ejpam-4172 198 33 nonlinear nonlinear ADJ ejpam-4172 198 34 oscillators oscillator NOUN ejpam-4172 198 35 . . PUNCT ejpam-4172 199 1 it it PRON ejpam-4172 199 2 is be AUX ejpam-4172 199 3 basic basic ADJ ejpam-4172 199 4 , , PUNCT ejpam-4172 199 5 clear clear ADJ ejpam-4172 199 6 and and CCONJ ejpam-4172 199 7 effective effective ADJ ejpam-4172 199 8 . . PUNCT ejpam-4172 200 1 also also ADV ejpam-4172 200 2 , , PUNCT ejpam-4172 200 3 the the DET ejpam-4172 200 4 approximate approximate ADJ ejpam-4172 200 5 analytical analytical ADJ ejpam-4172 200 6 solutions solution NOUN ejpam-4172 200 7 are be AUX ejpam-4172 200 8 suitable suitable ADJ ejpam-4172 200 9 for for ADP ejpam-4172 200 10 both both CCONJ ejpam-4172 200 11 small small ADJ ejpam-4172 200 12 and and CCONJ ejpam-4172 200 13 large large ADJ ejpam-4172 200 14 estimations estimation NOUN ejpam-4172 200 15 of of ADP ejpam-4172 200 16 oscillation oscillation NOUN ejpam-4172 200 17 amplitude amplitude NOUN ejpam-4172 200 18 and and CCONJ ejpam-4172 200 19 parameter parameter NOUN ejpam-4172 200 20 . . PUNCT ejpam-4172 201 1 it it PRON ejpam-4172 201 2 is be AUX ejpam-4172 201 3 worth worth ADJ ejpam-4172 201 4 mentioning mention VERB ejpam-4172 201 5 that that SCONJ ejpam-4172 201 6 the the DET ejpam-4172 201 7 problem problem NOUN ejpam-4172 201 8 of of ADP ejpam-4172 201 9 finding find VERB ejpam-4172 201 10 a a DET ejpam-4172 201 11 way way NOUN ejpam-4172 201 12 to to PART ejpam-4172 201 13 find find VERB ejpam-4172 201 14 the the DET ejpam-4172 201 15 point point NOUN ejpam-4172 201 16 x0 x0 PROPN ejpam-4172 201 17 as as ADP ejpam-4172 201 18 a a DET ejpam-4172 201 19 function function NOUN ejpam-4172 201 20 of of ADP ejpam-4172 201 21 the the DET ejpam-4172 201 22 parameters parameter NOUN ejpam-4172 201 23 of of ADP ejpam-4172 201 24 the the DET ejpam-4172 201 25 jacobi jacobi PROPN ejpam-4172 201 26 polynomial polynomial PROPN ejpam-4172 201 27 remains remain VERB ejpam-4172 201 28 open open ADJ ejpam-4172 201 29 . . PUNCT ejpam-4172 202 1 at at ADP ejpam-4172 202 2 last last ADJ ejpam-4172 202 3 , , PUNCT ejpam-4172 202 4 three three NUM ejpam-4172 202 5 physical physical ADJ ejpam-4172 202 6 models model NOUN ejpam-4172 202 7 have have AUX ejpam-4172 202 8 been be AUX ejpam-4172 202 9 resolved resolve VERB ejpam-4172 202 10 to to PART ejpam-4172 202 11 outline outline VERB ejpam-4172 202 12 the the DET ejpam-4172 202 13 effectiveness effectiveness NOUN ejpam-4172 202 14 and and CCONJ ejpam-4172 202 15 applicability applicability NOUN ejpam-4172 202 16 of of ADP ejpam-4172 202 17 the the DET ejpam-4172 202 18 proposed propose VERB ejpam-4172 202 19 solution solution NOUN ejpam-4172 202 20 approach approach NOUN ejpam-4172 202 21 and and CCONJ ejpam-4172 202 22 the the DET ejpam-4172 202 23 outcomes outcome NOUN ejpam-4172 202 24 in in ADP ejpam-4172 202 25 tables table NOUN ejpam-4172 202 26 1 1 NUM ejpam-4172 202 27 , , PUNCT ejpam-4172 202 28 2 2 NUM ejpam-4172 202 29 and and CCONJ ejpam-4172 202 30 3 3 NUM ejpam-4172 202 31 and and CCONJ ejpam-4172 202 32 figs fig NOUN ejpam-4172 202 33 . . PUNCT ejpam-4172 203 1 1 1 NUM ejpam-4172 203 2 , , PUNCT ejpam-4172 203 3 2 2 NUM ejpam-4172 203 4 , , PUNCT ejpam-4172 203 5 3 3 NUM ejpam-4172 203 6 , , PUNCT ejpam-4172 203 7 4 4 NUM ejpam-4172 203 8 , , PUNCT ejpam-4172 203 9 5 5 NUM ejpam-4172 203 10 and and CCONJ ejpam-4172 203 11 6 6 NUM ejpam-4172 203 12 uncover uncover NOUN ejpam-4172 203 13 that that SCONJ ejpam-4172 203 14 this this DET ejpam-4172 203 15 strategy strategy NOUN ejpam-4172 203 16 can can AUX ejpam-4172 203 17 be be AUX ejpam-4172 203 18 considered consider VERB ejpam-4172 203 19 as as ADP ejpam-4172 203 20 a a DET ejpam-4172 203 21 practical practical ADJ ejpam-4172 203 22 option option NOUN ejpam-4172 203 23 for for ADP ejpam-4172 203 24 traditional traditional ADJ ejpam-4172 203 25 techniques technique NOUN ejpam-4172 203 26 which which PRON ejpam-4172 203 27 can can AUX ejpam-4172 203 28 resolve resolve VERB ejpam-4172 203 29 exceptionally exceptionally ADV ejpam-4172 203 30 nonlinear nonlinear ADJ ejpam-4172 203 31 oscillatory oscillatory ADJ ejpam-4172 203 32 frameworks framework NOUN ejpam-4172 203 33 . . PUNCT ejpam-4172 204 1 all all DET ejpam-4172 204 2 numerical numerical ADJ ejpam-4172 204 3 work work NOUN ejpam-4172 204 4 and and CCONJ ejpam-4172 204 5 illustrations illustration NOUN ejpam-4172 204 6 were be AUX ejpam-4172 204 7 performed perform VERB ejpam-4172 204 8 with with ADP ejpam-4172 204 9 the the DET ejpam-4172 204 10 mathematica mathematica PROPN ejpam-4172 204 11 software software PROPN ejpam-4172 204 12 package package NOUN ejpam-4172 204 13 . . PUNCT ejpam-4172 205 1 the the DET ejpam-4172 205 2 present present ADJ ejpam-4172 205 3 method method NOUN ejpam-4172 205 4 can can AUX ejpam-4172 205 5 be be AUX ejpam-4172 205 6 used use VERB ejpam-4172 205 7 to to PART ejpam-4172 205 8 introduce introduce VERB ejpam-4172 205 9 the the DET ejpam-4172 205 10 study study NOUN ejpam-4172 205 11 of of ADP ejpam-4172 205 12 nonlinear nonlinear ADJ ejpam-4172 205 13 vibrations vibration NOUN ejpam-4172 205 14 and and CCONJ ejpam-4172 205 15 oscillators oscillator NOUN ejpam-4172 205 16 in in ADP ejpam-4172 205 17 undergraduate undergraduate ADJ ejpam-4172 205 18 sciences science NOUN ejpam-4172 205 19 and and CCONJ ejpam-4172 205 20 engineering engineering NOUN ejpam-4172 205 21 courses course NOUN ejpam-4172 205 22 due due ADJ ejpam-4172 205 23 to to ADP ejpam-4172 205 24 its its PRON ejpam-4172 205 25 simplicity simplicity NOUN ejpam-4172 205 26 and and CCONJ ejpam-4172 205 27 accuracy accuracy NOUN ejpam-4172 205 28 for for ADP ejpam-4172 205 29 a a DET ejpam-4172 205 30 wide wide ADJ ejpam-4172 205 31 range range NOUN ejpam-4172 205 32 of of ADP ejpam-4172 205 33 problems problem NOUN ejpam-4172 205 34 . . PUNCT ejpam-4172 206 1 it it PRON ejpam-4172 206 2 can can AUX ejpam-4172 206 3 also also ADV ejpam-4172 206 4 add add VERB ejpam-4172 206 5 pedagogical pedagogical ADJ ejpam-4172 206 6 value value NOUN ejpam-4172 206 7 to to ADP ejpam-4172 206 8 undergrads undergrad NOUN ejpam-4172 206 9 from from ADP ejpam-4172 206 10 various various ADJ ejpam-4172 206 11 branches branch NOUN ejpam-4172 206 12 of of ADP ejpam-4172 206 13 natural natural ADJ ejpam-4172 206 14 sciences science NOUN ejpam-4172 206 15 and and CCONJ ejpam-4172 206 16 engineering engineering NOUN ejpam-4172 206 17 , , PUNCT ejpam-4172 206 18 in in ADP ejpam-4172 206 19 particular particular ADJ ejpam-4172 206 20 mechanical mechanical ADJ ejpam-4172 206 21 engineering engineering NOUN ejpam-4172 206 22 students student NOUN ejpam-4172 206 23 by by ADP ejpam-4172 206 24 introducing introduce VERB ejpam-4172 206 25 the the DET ejpam-4172 206 26 jacobi jacobi PROPN ejpam-4172 206 27 polynomials polynomial NOUN ejpam-4172 206 28 in in ADP ejpam-4172 206 29 a a DET ejpam-4172 206 30 pleasant pleasant ADJ ejpam-4172 206 31 manner manner NOUN ejpam-4172 206 32 . . PUNCT ejpam-4172 207 1 this this PRON ejpam-4172 207 2 is be AUX ejpam-4172 207 3 because because SCONJ ejpam-4172 207 4 the the DET ejpam-4172 207 5 present present ADJ ejpam-4172 207 6 methodology methodology NOUN ejpam-4172 207 7 applies apply VERB ejpam-4172 207 8 these these DET ejpam-4172 207 9 polynomials polynomial NOUN ejpam-4172 207 10 in in ADP ejpam-4172 207 11 a a DET ejpam-4172 207 12 simple simple ADJ ejpam-4172 207 13 scheme scheme NOUN ejpam-4172 207 14 that that PRON ejpam-4172 207 15 does do AUX ejpam-4172 207 16 not not PART ejpam-4172 207 17 make make VERB ejpam-4172 207 18 it it PRON ejpam-4172 207 19 seem seem VERB ejpam-4172 207 20 of of ADP ejpam-4172 207 21 great great ADJ ejpam-4172 207 22 mathematical mathematical ADJ ejpam-4172 207 23 difficulty difficulty NOUN ejpam-4172 207 24 . . PUNCT ejpam-4172 208 1 references reference NOUN ejpam-4172 208 2 [ [ X ejpam-4172 208 3 1 1 NUM ejpam-4172 208 4 ] ] X ejpam-4172 208 5 ronald ronald PROPN ejpam-4172 208 6 e e PROPN ejpam-4172 208 7 mickens mickens PROPN ejpam-4172 208 8 . . PUNCT ejpam-4172 209 1 oscilations oscilation NOUN ejpam-4172 209 2 in in ADP ejpam-4172 209 3 planar planar ADJ ejpam-4172 209 4 dynamic dynamic ADJ ejpam-4172 209 5 systems system NOUN ejpam-4172 209 6 . . PUNCT ejpam-4172 210 1 world world NOUN ejpam-4172 210 2 scientific scientific ADJ ejpam-4172 210 3 , , PUNCT ejpam-4172 210 4 cop cop NOUN ejpam-4172 210 5 , , PUNCT ejpam-4172 210 6 1996 1996 NUM ejpam-4172 210 7 . . PUNCT ejpam-4172 211 1 [ [ X ejpam-4172 211 2 2 2 NUM ejpam-4172 211 3 ] ] PUNCT ejpam-4172 211 4 ali ali PROPN ejpam-4172 211 5 hasan hasan PROPN ejpam-4172 211 6 nayfeh nayfeh PROPN ejpam-4172 211 7 . . PUNCT ejpam-4172 212 1 problems problem NOUN ejpam-4172 212 2 in in ADP ejpam-4172 212 3 perturbation perturbation NOUN ejpam-4172 212 4 . . PUNCT ejpam-4172 213 1 wiley wiley PROPN ejpam-4172 213 2 , , PUNCT ejpam-4172 213 3 repr repr ADJ ejpam-4172 213 4 , , PUNCT ejpam-4172 213 5 1993 1993 NUM ejpam-4172 213 6 . . PUNCT ejpam-4172 214 1 [ [ X ejpam-4172 214 2 3 3 X ejpam-4172 214 3 ] ] X ejpam-4172 214 4 m.k m.k PROPN ejpam-4172 214 5 . . PROPN ejpam-4172 214 6 yazdi yazdi PROPN ejpam-4172 214 7 . . PUNCT ejpam-4172 215 1 “ " PUNCT ejpam-4172 215 2 approximate approximate ADJ ejpam-4172 215 3 solutions solution NOUN ejpam-4172 215 4 to to ADP ejpam-4172 215 5 nonlinear nonlinear ADJ ejpam-4172 215 6 oscillations oscillation NOUN ejpam-4172 215 7 via via ADP ejpam-4172 215 8 an an DET ejpam-4172 215 9 improved improve VERB ejpam-4172 215 10 he he PRON ejpam-4172 215 11 ’s ’ VERB ejpam-4172 215 12 variational variational ADJ ejpam-4172 215 13 approach approach NOUN ejpam-4172 215 14 ” " PUNCT ejpam-4172 215 15 . . PUNCT ejpam-4172 216 1 in in ADP ejpam-4172 216 2 : : PUNCT ejpam-4172 216 3 karbala karbala PROPN ejpam-4172 216 4 international international PROPN ejpam-4172 216 5 journal journal PROPN ejpam-4172 216 6 of of ADP ejpam-4172 216 7 modern modern ADJ ejpam-4172 216 8 science science NOUN ejpam-4172 216 9 2 2 NUM ejpam-4172 216 10 ( ( PUNCT ejpam-4172 216 11 2016 2016 NUM ejpam-4172 216 12 ) ) PUNCT ejpam-4172 216 13 , , PUNCT ejpam-4172 216 14 pp pp ADJ ejpam-4172 216 15 . . PUNCT ejpam-4172 217 1 289–297 289–297 NUM ejpam-4172 217 2 . . PUNCT ejpam-4172 218 1 doi doi NOUN ejpam-4172 218 2 : : PUNCT ejpam-4172 218 3 10.1016 10.1016 NUM ejpam-4172 218 4 / / SYM ejpam-4172 218 5 j.kijoms.2016.11.003 j.kijoms.2016.11.003 PROPN ejpam-4172 218 6 . . PUNCT ejpam-4172 219 1 [ [ X ejpam-4172 219 2 4 4 NUM ejpam-4172 219 3 ] ] PUNCT ejpam-4172 219 4 akuro akuro NOUN ejpam-4172 219 5 big big ADJ ejpam-4172 219 6 - - PUNCT ejpam-4172 219 7 alabo alabo NOUN ejpam-4172 219 8 and and CCONJ ejpam-4172 219 9 collins collins PROPN ejpam-4172 219 10 onyinyechukwu onyinyechukwu PROPN ejpam-4172 219 11 ogbodo ogbodo PROPN ejpam-4172 219 12 . . PUNCT ejpam-4172 220 1 “ " PUNCT ejpam-4172 220 2 dynamic dynamic ADJ ejpam-4172 220 3 analysis analysis NOUN ejpam-4172 220 4 of of ADP ejpam-4172 220 5 crank crank NOUN ejpam-4172 220 6 mechanism mechanism NOUN ejpam-4172 220 7 with with ADP ejpam-4172 220 8 complex complex ADJ ejpam-4172 220 9 trigonometric trigonometric ADJ ejpam-4172 220 10 nonlinearity nonlinearity NOUN ejpam-4172 220 11 : : PUNCT ejpam-4172 220 12 a a DET ejpam-4172 220 13 comparative comparative ADJ ejpam-4172 220 14 study study NOUN ejpam-4172 220 15 of of ADP ejpam-4172 220 16 approximate approximate ADJ ejpam-4172 220 17 analytical analytical ADJ ejpam-4172 220 18 methods method NOUN ejpam-4172 220 19 ” " PUNCT ejpam-4172 220 20 . . PUNCT ejpam-4172 221 1 in in ADP ejpam-4172 221 2 : : PUNCT ejpam-4172 221 3 sn sn PROPN ejpam-4172 221 4 applied apply VERB ejpam-4172 221 5 sciences science NOUN ejpam-4172 221 6 1 1 NUM ejpam-4172 221 7 ( ( PUNCT ejpam-4172 221 8 2019 2019 NUM ejpam-4172 221 9 ) ) PUNCT ejpam-4172 221 10 . . PUNCT ejpam-4172 222 1 doi doi NOUN ejpam-4172 222 2 : : PUNCT ejpam-4172 222 3 10.1007 10.1007 NUM ejpam-4172 222 4 / / SYM ejpam-4172 222 5 s42452019 s42452019 NOUN ejpam-4172 222 6 - - PUNCT ejpam-4172 222 7 0673 0673 NUM ejpam-4172 222 8 - - PUNCT ejpam-4172 222 9 3 3 NUM ejpam-4172 222 10 . . PUNCT ejpam-4172 222 11 ( ( PUNCT ejpam-4172 222 12 visited visit VERB ejpam-4172 222 13 on on ADP ejpam-4172 222 14 11/19/2021 11/19/2021 NUM ejpam-4172 222 15 ) ) PUNCT ejpam-4172 222 16 . . PUNCT ejpam-4172 223 1 [ [ X ejpam-4172 223 2 5 5 X ejpam-4172 223 3 ] ] X ejpam-4172 223 4 y. y. PROPN ejpam-4172 223 5 khan khan PROPN ejpam-4172 223 6 and and CCONJ ejpam-4172 223 7 a. a. NOUN ejpam-4172 223 8 mirzabeigy mirzabeigy NOUN ejpam-4172 223 9 . . PUNCT ejpam-4172 224 1 “ " PUNCT ejpam-4172 224 2 improved improved ADJ ejpam-4172 224 3 accuracy accuracy NOUN ejpam-4172 224 4 of of ADP ejpam-4172 224 5 he he PRON ejpam-4172 224 6 ’s ’ VERB ejpam-4172 224 7 energy energy NOUN ejpam-4172 224 8 balance balance NOUN ejpam-4172 224 9 method method NOUN ejpam-4172 224 10 for for ADP ejpam-4172 224 11 analysis analysis NOUN ejpam-4172 224 12 of of ADP ejpam-4172 224 13 conservative conservative ADJ ejpam-4172 224 14 nonlinear nonlinear ADJ ejpam-4172 224 15 oscillator oscillator NOUN ejpam-4172 224 16 ” " PUNCT ejpam-4172 224 17 . . PUNCT ejpam-4172 225 1 in in ADP ejpam-4172 225 2 : : PUNCT ejpam-4172 225 3 neural neural ADJ ejpam-4172 225 4 computing computing NOUN ejpam-4172 225 5 and and CCONJ ejpam-4172 225 6 applications application NOUN ejpam-4172 225 7 25 25 NUM ejpam-4172 225 8 ( ( PUNCT ejpam-4172 225 9 2014 2014 NUM ejpam-4172 225 10 ) ) PUNCT ejpam-4172 225 11 , , PUNCT ejpam-4172 225 12 pp pp ADP ejpam-4172 225 13 . . PUNCT ejpam-4172 226 1 889–895 889–895 NUM ejpam-4172 226 2 . . PUNCT ejpam-4172 226 3 doi doi NOUN ejpam-4172 226 4 : : PUNCT ejpam-4172 226 5 10.1007 10.1007 NUM ejpam-4172 226 6 / / SYM ejpam-4172 226 7 s00521 s00521 NOUN ejpam-4172 226 8 - - PUNCT ejpam-4172 226 9 014 014 NUM ejpam-4172 226 10 - - PUNCT ejpam-4172 226 11 1576 1576 NUM ejpam-4172 226 12 - - SYM ejpam-4172 226 13 2 2 NUM ejpam-4172 226 14 . . PUNCT ejpam-4172 226 15 ( ( PUNCT ejpam-4172 226 16 visited visit VERB ejpam-4172 226 17 on on ADP ejpam-4172 226 18 11/19/2021 11/19/2021 NUM ejpam-4172 226 19 ) ) PUNCT ejpam-4172 226 20 . . PUNCT ejpam-4172 227 1 [ [ X ejpam-4172 227 2 6 6 NUM ejpam-4172 227 3 ] ] PUNCT ejpam-4172 227 4 a. a. NOUN ejpam-4172 227 5 yildirim yildirim PROPN ejpam-4172 227 6 et et PROPN ejpam-4172 227 7 al al PROPN ejpam-4172 227 8 . . PUNCT ejpam-4172 228 1 “ " PUNCT ejpam-4172 228 2 analysis analysis NOUN ejpam-4172 228 3 of of ADP ejpam-4172 228 4 nonlinear nonlinear ADJ ejpam-4172 228 5 oscillations oscillation NOUN ejpam-4172 228 6 of of ADP ejpam-4172 228 7 a a DET ejpam-4172 228 8 punctual punctual ADJ ejpam-4172 228 9 charge charge NOUN ejpam-4172 228 10 in in ADP ejpam-4172 228 11 the the DET ejpam-4172 228 12 electric electric ADJ ejpam-4172 228 13 field field NOUN ejpam-4172 228 14 of of ADP ejpam-4172 228 15 a a DET ejpam-4172 228 16 charged charge VERB ejpam-4172 228 17 ring ring NOUN ejpam-4172 228 18 via via ADP ejpam-4172 228 19 a a DET ejpam-4172 228 20 hamiltonian hamiltonian ADJ ejpam-4172 228 21 approach approach NOUN ejpam-4172 228 22 and and CCONJ ejpam-4172 228 23 the the DET ejpam-4172 228 24 energy energy NOUN ejpam-4172 228 25 balance balance NOUN ejpam-4172 228 26 method method NOUN ejpam-4172 228 27 ” " PUNCT ejpam-4172 228 28 . . PUNCT ejpam-4172 229 1 in in ADP ejpam-4172 229 2 : : PUNCT ejpam-4172 229 3 computers computer NOUN ejpam-4172 229 4 & & CCONJ ejpam-4172 229 5 mathematics mathematic NOUN ejpam-4172 229 6 with with ADP ejpam-4172 229 7 applications application NOUN ejpam-4172 229 8 62 62 NUM ejpam-4172 229 9 ( ( PUNCT ejpam-4172 229 10 2011 2011 NUM ejpam-4172 229 11 ) ) PUNCT ejpam-4172 229 12 , , PUNCT ejpam-4172 229 13 pp pp ADP ejpam-4172 229 14 . . PUNCT ejpam-4172 230 1 486–490 486–490 NUM ejpam-4172 230 2 . . PUNCT ejpam-4172 231 1 doi doi NOUN ejpam-4172 231 2 : : PUNCT ejpam-4172 231 3 10.1016 10.1016 NUM ejpam-4172 231 4 / / SYM ejpam-4172 231 5 j.camwa.2011.05.029 j.camwa.2011.05.029 PROPN ejpam-4172 231 6 . . PUNCT ejpam-4172 232 1 ( ( PUNCT ejpam-4172 232 2 visited visit VERB ejpam-4172 232 3 on on ADP ejpam-4172 232 4 11/19/2021 11/19/2021 NUM ejpam-4172 232 5 ) ) PUNCT ejpam-4172 232 6 . . PUNCT ejpam-4172 233 1 references reference NOUN ejpam-4172 233 2 96 96 NUM ejpam-4172 234 1 [ [ X ejpam-4172 234 2 7 7 NUM ejpam-4172 234 3 ] ] X ejpam-4172 234 4 ji ji PROPN ejpam-4172 234 5 - - PROPN ejpam-4172 234 6 huan huan PROPN ejpam-4172 234 7 he he PRON ejpam-4172 234 8 . . PUNCT ejpam-4172 235 1 “ " PUNCT ejpam-4172 235 2 hamiltonian hamiltonian ADJ ejpam-4172 235 3 approach approach NOUN ejpam-4172 235 4 to to ADP ejpam-4172 235 5 nonlinear nonlinear ADJ ejpam-4172 235 6 oscillators oscillator NOUN ejpam-4172 235 7 ” " PUNCT ejpam-4172 235 8 . . PUNCT ejpam-4172 236 1 in in ADP ejpam-4172 236 2 : : PUNCT ejpam-4172 236 3 physics physics NOUN ejpam-4172 236 4 letters letter NOUN ejpam-4172 236 5 a a DET ejpam-4172 236 6 374 374 NUM ejpam-4172 236 7 ( ( PUNCT ejpam-4172 236 8 2010 2010 NUM ejpam-4172 236 9 ) ) PUNCT ejpam-4172 236 10 , , PUNCT ejpam-4172 236 11 pp pp ADP ejpam-4172 236 12 . . PUNCT ejpam-4172 236 13 2312–2314 2312–2314 NUM ejpam-4172 236 14 . . PUNCT ejpam-4172 237 1 doi doi NOUN ejpam-4172 237 2 : : PUNCT ejpam-4172 237 3 10.1016 10.1016 NUM ejpam-4172 237 4 / / SYM ejpam-4172 237 5 j.physleta.2010.03.064 j.physleta.2010.03.064 PROPN ejpam-4172 237 6 . . PUNCT ejpam-4172 238 1 ( ( PUNCT ejpam-4172 238 2 visited visit VERB ejpam-4172 238 3 on on ADP ejpam-4172 238 4 11/19/2021 11/19/2021 NUM ejpam-4172 238 5 ) ) PUNCT ejpam-4172 238 6 . . PUNCT ejpam-4172 239 1 [ [ X ejpam-4172 239 2 8 8 X ejpam-4172 239 3 ] ] X ejpam-4172 239 4 ahmet ahmet PROPN ejpam-4172 239 5 yildirim yildirim PROPN ejpam-4172 239 6 , , PUNCT ejpam-4172 239 7 z. z. PROPN ejpam-4172 239 8 saadatnia saadatnia PROPN ejpam-4172 239 9 , , PUNCT ejpam-4172 239 10 and and CCONJ ejpam-4172 239 11 h. h. PROPN ejpam-4172 239 12 askari askari PROPN ejpam-4172 239 13 . . PUNCT ejpam-4172 240 1 “ " PUNCT ejpam-4172 240 2 application application NOUN ejpam-4172 240 3 of of ADP ejpam-4172 240 4 the the DET ejpam-4172 240 5 hamiltonian hamiltonian ADJ ejpam-4172 240 6 approach approach NOUN ejpam-4172 240 7 to to ADP ejpam-4172 240 8 nonlinear nonlinear ADJ ejpam-4172 240 9 oscillators oscillator NOUN ejpam-4172 240 10 with with ADP ejpam-4172 240 11 rational rational ADJ ejpam-4172 240 12 and and CCONJ ejpam-4172 240 13 irrational irrational ADJ ejpam-4172 240 14 elastic elastic ADJ ejpam-4172 240 15 terms term NOUN ejpam-4172 240 16 ” " PUNCT ejpam-4172 240 17 . . PUNCT ejpam-4172 241 1 in in ADP ejpam-4172 241 2 : : PUNCT ejpam-4172 241 3 mathematical mathematical ADJ ejpam-4172 241 4 and and CCONJ ejpam-4172 241 5 computer computer NOUN ejpam-4172 241 6 modelling model VERB ejpam-4172 241 7 54 54 NUM ejpam-4172 241 8 ( ( PUNCT ejpam-4172 241 9 2011 2011 NUM ejpam-4172 241 10 ) ) PUNCT ejpam-4172 241 11 , , PUNCT ejpam-4172 241 12 pp pp ADP ejpam-4172 241 13 . . PUNCT ejpam-4172 242 1 697–703 697–703 NUM ejpam-4172 242 2 . . PUNCT ejpam-4172 243 1 doi doi NOUN ejpam-4172 243 2 : : PUNCT ejpam-4172 243 3 10.1016 10.1016 NUM ejpam-4172 243 4 / / SYM ejpam-4172 243 5 j.mcm j.mcm PROPN ejpam-4172 243 6 . . PUNCT ejpam-4172 244 1 2011.03.012 2011.03.012 NUM ejpam-4172 244 2 . . PUNCT ejpam-4172 245 1 ( ( PUNCT ejpam-4172 245 2 visited visit VERB ejpam-4172 245 3 on on ADP ejpam-4172 245 4 11/19/2021 11/19/2021 NUM ejpam-4172 245 5 ) ) PUNCT ejpam-4172 245 6 . . PUNCT ejpam-4172 246 1 [ [ X ejpam-4172 246 2 9 9 NUM ejpam-4172 246 3 ] ] PUNCT ejpam-4172 246 4 alex alex PROPN ejpam-4172 246 5 eĺıas eĺıas PROPN ejpam-4172 246 6 - - PROPN ejpam-4172 246 7 zúñiga zúñiga NUM ejpam-4172 246 8 . . PUNCT ejpam-4172 247 1 “ " PUNCT ejpam-4172 247 2 exact exact ADJ ejpam-4172 247 3 solution solution NOUN ejpam-4172 247 4 of of ADP ejpam-4172 247 5 the the DET ejpam-4172 247 6 cubic cubic ADJ ejpam-4172 247 7 - - PUNCT ejpam-4172 247 8 quintic quintic ADJ ejpam-4172 247 9 duffing duffing NOUN ejpam-4172 247 10 oscillator oscillator NOUN ejpam-4172 247 11 ” " PUNCT ejpam-4172 247 12 . . PUNCT ejpam-4172 248 1 in in ADP ejpam-4172 248 2 : : PUNCT ejpam-4172 248 3 applied apply VERB ejpam-4172 248 4 mathematical mathematical ADJ ejpam-4172 248 5 modelling model VERB ejpam-4172 248 6 37 37 NUM ejpam-4172 248 7 ( ( PUNCT ejpam-4172 248 8 2013 2013 NUM ejpam-4172 248 9 ) ) PUNCT ejpam-4172 248 10 , , PUNCT ejpam-4172 248 11 pp pp ADJ ejpam-4172 248 12 . . PUNCT ejpam-4172 249 1 2574–2579 2574–2579 NUM ejpam-4172 249 2 . . PUNCT ejpam-4172 250 1 doi doi NOUN ejpam-4172 250 2 : : PUNCT ejpam-4172 250 3 10.1016 10.1016 NUM ejpam-4172 250 4 / / SYM ejpam-4172 250 5 j.apm.2012 j.apm.2012 PROPN ejpam-4172 250 6 . . PUNCT ejpam-4172 251 1 04.005 04.005 NUM ejpam-4172 251 2 . . PUNCT ejpam-4172 252 1 ( ( PUNCT ejpam-4172 252 2 visited visit VERB ejpam-4172 252 3 on on ADP ejpam-4172 252 4 11/19/2021 11/19/2021 NUM ejpam-4172 252 5 ) ) PUNCT ejpam-4172 252 6 . . PUNCT ejpam-4172 253 1 [ [ X ejpam-4172 253 2 10 10 NUM ejpam-4172 253 3 ] ] X ejpam-4172 253 4 alex alex PROPN ejpam-4172 253 5 eĺıas eĺıas PROPN ejpam-4172 253 6 - - PROPN ejpam-4172 253 7 zúñiga zúñiga NUM ejpam-4172 253 8 . . PUNCT ejpam-4172 254 1 “ " PUNCT ejpam-4172 254 2 solution solution NOUN ejpam-4172 254 3 of of ADP ejpam-4172 254 4 the the DET ejpam-4172 254 5 damped damped ADJ ejpam-4172 254 6 cubic cubic ADJ ejpam-4172 254 7 – – PUNCT ejpam-4172 254 8 quintic quintic ADJ ejpam-4172 254 9 duffing duffing NOUN ejpam-4172 254 10 oscillator oscillator NOUN ejpam-4172 254 11 by by ADP ejpam-4172 254 12 using use VERB ejpam-4172 254 13 jacobi jacobi PROPN ejpam-4172 254 14 elliptic elliptic ADJ ejpam-4172 254 15 functions function NOUN ejpam-4172 254 16 ” " PUNCT ejpam-4172 254 17 . . PUNCT ejpam-4172 255 1 in in ADP ejpam-4172 255 2 : : PUNCT ejpam-4172 255 3 applied applied ADJ ejpam-4172 255 4 mathematics mathematic NOUN ejpam-4172 255 5 and and CCONJ ejpam-4172 255 6 computation computation NOUN ejpam-4172 255 7 246 246 NUM ejpam-4172 255 8 ( ( PUNCT ejpam-4172 255 9 2014 2014 NUM ejpam-4172 255 10 ) ) PUNCT ejpam-4172 255 11 , , PUNCT ejpam-4172 255 12 pp pp ADP ejpam-4172 255 13 . . PUNCT ejpam-4172 256 1 474–481 474–481 NUM ejpam-4172 256 2 . . PUNCT ejpam-4172 257 1 doi doi NOUN ejpam-4172 257 2 : : PUNCT ejpam-4172 257 3 10.1016 10.1016 NUM ejpam-4172 257 4 / / SYM ejpam-4172 257 5 j.amc.2014.07.110 j.amc.2014.07.110 NOUN ejpam-4172 257 6 . . PUNCT ejpam-4172 258 1 [ [ X ejpam-4172 258 2 11 11 NUM ejpam-4172 258 3 ] ] X ejpam-4172 258 4 j.h j.h PROPN ejpam-4172 258 5 . . PUNCT ejpam-4172 259 1 he he PRON ejpam-4172 259 2 . . PUNCT ejpam-4172 260 1 “ " PUNCT ejpam-4172 260 2 max max PROPN ejpam-4172 260 3 - - PUNCT ejpam-4172 260 4 min min NOUN ejpam-4172 260 5 approach approach NOUN ejpam-4172 260 6 to to ADP ejpam-4172 260 7 nonlinear nonlinear ADJ ejpam-4172 260 8 oscillators oscillator NOUN ejpam-4172 260 9 ” " PUNCT ejpam-4172 260 10 . . PUNCT ejpam-4172 261 1 in in ADP ejpam-4172 261 2 : : PUNCT ejpam-4172 261 3 international international ADJ ejpam-4172 261 4 journal journal NOUN ejpam-4172 261 5 of of ADP ejpam-4172 261 6 nonlinear nonlinear PROPN ejpam-4172 261 7 sciences sciences PROPN ejpam-4172 261 8 and and CCONJ ejpam-4172 261 9 numerical numerical PROPN ejpam-4172 261 10 simulation simulation NOUN ejpam-4172 261 11 9 9 NUM ejpam-4172 261 12 ( ( PUNCT ejpam-4172 261 13 2008 2008 NUM ejpam-4172 261 14 ) ) PUNCT ejpam-4172 261 15 . . PUNCT ejpam-4172 262 1 doi doi NOUN ejpam-4172 262 2 : : PUNCT ejpam-4172 262 3 10.1515 10.1515 NUM ejpam-4172 262 4 / / SYM ejpam-4172 262 5 ijnsns ijnsns NOUN ejpam-4172 262 6 . . PUNCT ejpam-4172 263 1 2008.9.2.207 2008.9.2.207 NUM ejpam-4172 263 2 . . PUNCT ejpam-4172 263 3 ( ( PUNCT ejpam-4172 263 4 visited visit VERB ejpam-4172 263 5 on on ADP ejpam-4172 263 6 11/12/2020 11/12/2020 NUM ejpam-4172 263 7 ) ) PUNCT ejpam-4172 263 8 . . PUNCT ejpam-4172 264 1 [ [ X ejpam-4172 264 2 12 12 NUM ejpam-4172 264 3 ] ] PUNCT ejpam-4172 264 4 m. m. NOUN ejpam-4172 264 5 kalami kalami PROPN ejpam-4172 264 6 yazdi yazdi PROPN ejpam-4172 264 7 et et PROPN ejpam-4172 264 8 al al PROPN ejpam-4172 264 9 . . PUNCT ejpam-4172 265 1 “ " PUNCT ejpam-4172 265 2 dynamic dynamic ADJ ejpam-4172 265 3 analysis analysis NOUN ejpam-4172 265 4 of of ADP ejpam-4172 265 5 vibrating vibrate VERB ejpam-4172 265 6 systems system NOUN ejpam-4172 265 7 with with ADP ejpam-4172 265 8 nonlinearities nonlinearitie NOUN ejpam-4172 265 9 ” " PUNCT ejpam-4172 265 10 . . PUNCT ejpam-4172 266 1 in in ADP ejpam-4172 266 2 : : PUNCT ejpam-4172 266 3 communications communication NOUN ejpam-4172 266 4 in in ADP ejpam-4172 266 5 theoretical theoretical ADJ ejpam-4172 266 6 physics physics NOUN ejpam-4172 266 7 57 57 NUM ejpam-4172 266 8 ( ( PUNCT ejpam-4172 266 9 2012 2012 NUM ejpam-4172 266 10 ) ) PUNCT ejpam-4172 266 11 , , PUNCT ejpam-4172 266 12 pp pp ADP ejpam-4172 266 13 . . PUNCT ejpam-4172 267 1 183–187 183–187 NUM ejpam-4172 267 2 . . PUNCT ejpam-4172 268 1 doi doi NOUN ejpam-4172 268 2 : : PUNCT ejpam-4172 268 3 10 10 NUM ejpam-4172 268 4 . . PUNCT ejpam-4172 269 1 1088/0253 1088/0253 NUM ejpam-4172 269 2 - - SYM ejpam-4172 269 3 6102/57/2/03 6102/57/2/03 NOUN ejpam-4172 269 4 . . PUNCT ejpam-4172 270 1 ( ( PUNCT ejpam-4172 270 2 visited visit VERB ejpam-4172 270 3 on on ADP ejpam-4172 270 4 01/03/2020 01/03/2020 NUM ejpam-4172 270 5 ) ) PUNCT ejpam-4172 270 6 . . PUNCT ejpam-4172 271 1 [ [ X ejpam-4172 271 2 13 13 NUM ejpam-4172 271 3 ] ] X ejpam-4172 271 4 o. o. PROPN ejpam-4172 271 5 gonzález gonzález PROPN ejpam-4172 271 6 - - PUNCT ejpam-4172 271 7 gaxiola gaxiola PROPN ejpam-4172 271 8 , , PUNCT ejpam-4172 271 9 j. j. PROPN ejpam-4172 271 10 a. a. PROPN ejpam-4172 271 11 santiago santiago PROPN ejpam-4172 271 12 , , PUNCT ejpam-4172 271 13 and and CCONJ ejpam-4172 271 14 j. j. PROPN ejpam-4172 271 15 ruiz ruiz PROPN ejpam-4172 271 16 de de PROPN ejpam-4172 271 17 chávez chávez PROPN ejpam-4172 271 18 . . PUNCT ejpam-4172 272 1 “ " PUNCT ejpam-4172 272 2 solution solution NOUN ejpam-4172 272 3 for for ADP ejpam-4172 272 4 the the DET ejpam-4172 272 5 nonlinear nonlinear ADJ ejpam-4172 272 6 relativistic relativistic ADJ ejpam-4172 272 7 harmonic harmonic ADJ ejpam-4172 272 8 oscillator oscillator NOUN ejpam-4172 272 9 via via ADP ejpam-4172 272 10 laplace laplace NOUN ejpam-4172 272 11 - - PUNCT ejpam-4172 272 12 adomian adomian NOUN ejpam-4172 272 13 decomposition decomposition NOUN ejpam-4172 272 14 method method NOUN ejpam-4172 272 15 ” " PUNCT ejpam-4172 272 16 . . PUNCT ejpam-4172 273 1 in in ADP ejpam-4172 273 2 : : PUNCT ejpam-4172 273 3 international international ADJ ejpam-4172 273 4 journal journal NOUN ejpam-4172 273 5 of of ADP ejpam-4172 273 6 applied applied ADJ ejpam-4172 273 7 and and CCONJ ejpam-4172 273 8 computational computational ADJ ejpam-4172 273 9 mathematics mathematic NOUN ejpam-4172 273 10 3 3 NUM ejpam-4172 273 11 ( ( PUNCT ejpam-4172 273 12 2016 2016 NUM ejpam-4172 273 13 ) ) PUNCT ejpam-4172 273 14 , , PUNCT ejpam-4172 273 15 pp pp PROPN ejpam-4172 273 16 . . PUNCT ejpam-4172 273 17 2627–2638 2627–2638 NUM ejpam-4172 273 18 . . PUNCT ejpam-4172 274 1 doi doi NOUN ejpam-4172 274 2 : : PUNCT ejpam-4172 274 3 10.1007 10.1007 NUM ejpam-4172 274 4 / / SYM ejpam-4172 274 5 s40819 s40819 NOUN ejpam-4172 274 6 - - PUNCT ejpam-4172 274 7 016 016 NUM ejpam-4172 274 8 - - PUNCT ejpam-4172 274 9 0267 0267 NUM ejpam-4172 274 10 - - PUNCT ejpam-4172 274 11 3 3 NUM ejpam-4172 274 12 . . PUNCT ejpam-4172 275 1 ( ( PUNCT ejpam-4172 275 2 visited visit VERB ejpam-4172 275 3 on on ADP ejpam-4172 275 4 11/19/2021 11/19/2021 NUM ejpam-4172 275 5 ) ) PUNCT ejpam-4172 275 6 . . PUNCT ejpam-4172 276 1 [ [ X ejpam-4172 276 2 14 14 NUM ejpam-4172 276 3 ] ] X ejpam-4172 276 4 ji ji PROPN ejpam-4172 276 5 - - PROPN ejpam-4172 276 6 huan huan PROPN ejpam-4172 276 7 he he PRON ejpam-4172 276 8 . . PUNCT ejpam-4172 277 1 “ " PUNCT ejpam-4172 277 2 variational variational ADJ ejpam-4172 277 3 approach approach NOUN ejpam-4172 277 4 for for ADP ejpam-4172 277 5 nonlinear nonlinear ADJ ejpam-4172 277 6 oscillators oscillator NOUN ejpam-4172 277 7 ” " PUNCT ejpam-4172 277 8 . . PUNCT ejpam-4172 278 1 in in ADP ejpam-4172 278 2 : : PUNCT ejpam-4172 278 3 chaos chaos NOUN ejpam-4172 278 4 , , PUNCT ejpam-4172 278 5 solitons soliton NOUN ejpam-4172 278 6 & & CCONJ ejpam-4172 278 7 fractals fractal NOUN ejpam-4172 278 8 34 34 NUM ejpam-4172 278 9 ( ( PUNCT ejpam-4172 278 10 2007 2007 NUM ejpam-4172 278 11 ) ) PUNCT ejpam-4172 278 12 , , PUNCT ejpam-4172 278 13 pp pp ADP ejpam-4172 278 14 . . PUNCT ejpam-4172 278 15 1430–1439 1430–1439 NUM ejpam-4172 278 16 . . PUNCT ejpam-4172 279 1 doi doi NOUN ejpam-4172 279 2 : : PUNCT ejpam-4172 279 3 10.1016 10.1016 NUM ejpam-4172 279 4 / / SYM ejpam-4172 279 5 j.chaos.2006.10.026 j.chaos.2006.10.026 PROPN ejpam-4172 279 6 . . PUNCT ejpam-4172 280 1 ( ( PUNCT ejpam-4172 280 2 visited visit VERB ejpam-4172 280 3 on on ADP ejpam-4172 280 4 11/19/2021 11/19/2021 NUM ejpam-4172 280 5 ) ) PUNCT ejpam-4172 280 6 . . PUNCT ejpam-4172 281 1 [ [ X ejpam-4172 281 2 15 15 NUM ejpam-4172 281 3 ] ] PUNCT ejpam-4172 281 4 a. a. NOUN ejpam-4172 281 5 mirzabeigy mirzabeigy NOUN ejpam-4172 281 6 , , PUNCT ejpam-4172 281 7 m. m. NOUN ejpam-4172 281 8 kalami kalami PROPN ejpam-4172 281 9 - - PUNCT ejpam-4172 281 10 yazdi yazdi NOUN ejpam-4172 281 11 , , PUNCT ejpam-4172 281 12 and and CCONJ ejpam-4172 281 13 a. a. PROPN ejpam-4172 281 14 yildirim yildirim PROPN ejpam-4172 281 15 . . PUNCT ejpam-4172 282 1 “ " PUNCT ejpam-4172 282 2 analytical analytical ADJ ejpam-4172 282 3 approximations approximation NOUN ejpam-4172 282 4 for for ADP ejpam-4172 282 5 a a DET ejpam-4172 282 6 conservative conservative ADJ ejpam-4172 282 7 nonlinear nonlinear ADJ ejpam-4172 282 8 singular singular ADJ ejpam-4172 282 9 oscillator oscillator NOUN ejpam-4172 282 10 in in ADP ejpam-4172 282 11 plasma plasma NOUN ejpam-4172 282 12 physics physic NOUN ejpam-4172 282 13 ” " PUNCT ejpam-4172 282 14 . . PUNCT ejpam-4172 283 1 in in ADP ejpam-4172 283 2 : : PUNCT ejpam-4172 283 3 journal journal NOUN ejpam-4172 283 4 of of ADP ejpam-4172 283 5 the the DET ejpam-4172 283 6 egyptian egyptian PROPN ejpam-4172 283 7 mathematical mathematical ADJ ejpam-4172 283 8 society society NOUN ejpam-4172 283 9 20 20 NUM ejpam-4172 283 10 ( ( PUNCT ejpam-4172 283 11 2012 2012 NUM ejpam-4172 283 12 ) ) PUNCT ejpam-4172 283 13 , , PUNCT ejpam-4172 283 14 pp pp ADP ejpam-4172 283 15 . . PUNCT ejpam-4172 284 1 163–166 163–166 NUM ejpam-4172 284 2 . . PUNCT ejpam-4172 284 3 doi doi NOUN ejpam-4172 284 4 : : PUNCT ejpam-4172 284 5 10.1016 10.1016 NUM ejpam-4172 284 6 / / SYM ejpam-4172 284 7 j.joems j.joem NOUN ejpam-4172 284 8 . . PUNCT ejpam-4172 285 1 2012.05.001 2012.05.001 NUM ejpam-4172 285 2 . . PUNCT ejpam-4172 286 1 ( ( PUNCT ejpam-4172 286 2 visited visit VERB ejpam-4172 286 3 on on ADP ejpam-4172 286 4 01/03/2020 01/03/2020 NUM ejpam-4172 286 5 ) ) PUNCT ejpam-4172 286 6 . . PUNCT ejpam-4172 287 1 [ [ X ejpam-4172 287 2 16 16 NUM ejpam-4172 287 3 ] ] PUNCT ejpam-4172 287 4 m. m. NOUN ejpam-4172 287 5 rafei rafei PROPN ejpam-4172 287 6 et et PROPN ejpam-4172 287 7 al al PROPN ejpam-4172 287 8 . . PUNCT ejpam-4172 288 1 “ " PUNCT ejpam-4172 288 2 the the DET ejpam-4172 288 3 variational variational ADJ ejpam-4172 288 4 iteration iteration NOUN ejpam-4172 288 5 method method NOUN ejpam-4172 288 6 for for ADP ejpam-4172 288 7 nonlinear nonlinear ADJ ejpam-4172 288 8 oscillators oscillator NOUN ejpam-4172 288 9 with with ADP ejpam-4172 288 10 discontinuities discontinuity NOUN ejpam-4172 288 11 ” " PUNCT ejpam-4172 288 12 . . PUNCT ejpam-4172 289 1 in in ADP ejpam-4172 289 2 : : PUNCT ejpam-4172 289 3 journal journal NOUN ejpam-4172 289 4 of of ADP ejpam-4172 289 5 sound sound NOUN ejpam-4172 289 6 and and CCONJ ejpam-4172 289 7 vibration vibration NOUN ejpam-4172 289 8 305 305 NUM ejpam-4172 289 9 ( ( PUNCT ejpam-4172 289 10 2007 2007 NUM ejpam-4172 289 11 ) ) PUNCT ejpam-4172 289 12 , , PUNCT ejpam-4172 289 13 pp pp ADP ejpam-4172 289 14 . . PUNCT ejpam-4172 290 1 614–620 614–620 NUM ejpam-4172 290 2 . . PUNCT ejpam-4172 291 1 doi doi NOUN ejpam-4172 291 2 : : PUNCT ejpam-4172 292 1 10.1016 10.1016 NUM ejpam-4172 292 2 / / SYM ejpam-4172 292 3 j.jsv.2007.04.020 j.jsv.2007.04.020 PROPN ejpam-4172 292 4 . . PUNCT ejpam-4172 293 1 ( ( PUNCT ejpam-4172 293 2 visited visit VERB ejpam-4172 293 3 on on ADP ejpam-4172 293 4 11/19/2021 11/19/2021 NUM ejpam-4172 293 5 ) ) PUNCT ejpam-4172 293 6 . . PUNCT ejpam-4172 294 1 [ [ X ejpam-4172 294 2 17 17 NUM ejpam-4172 294 3 ] ] X ejpam-4172 294 4 ji ji PROPN ejpam-4172 294 5 - - PROPN ejpam-4172 294 6 huan huan PROPN ejpam-4172 294 7 he he PRON ejpam-4172 294 8 . . PUNCT ejpam-4172 295 1 “ " PUNCT ejpam-4172 295 2 homotopy homotopy VERB ejpam-4172 295 3 perturbation perturbation NOUN ejpam-4172 295 4 method method NOUN ejpam-4172 295 5 for for ADP ejpam-4172 295 6 solving solve VERB ejpam-4172 295 7 boundary boundary ADJ ejpam-4172 295 8 value value NOUN ejpam-4172 295 9 problems problem NOUN ejpam-4172 295 10 ” " PUNCT ejpam-4172 295 11 . . PUNCT ejpam-4172 296 1 in in ADP ejpam-4172 296 2 : : PUNCT ejpam-4172 296 3 physics physics NOUN ejpam-4172 296 4 letters letter NOUN ejpam-4172 296 5 a a DET ejpam-4172 296 6 350 350 NUM ejpam-4172 296 7 ( ( PUNCT ejpam-4172 296 8 2006 2006 NUM ejpam-4172 296 9 ) ) PUNCT ejpam-4172 296 10 , , PUNCT ejpam-4172 296 11 pp pp PROPN ejpam-4172 296 12 . . PUNCT ejpam-4172 296 13 87–88 87–88 NUM ejpam-4172 296 14 . . PUNCT ejpam-4172 297 1 doi doi NOUN ejpam-4172 297 2 : : PUNCT ejpam-4172 297 3 10.1016 10.1016 NUM ejpam-4172 297 4 / / SYM ejpam-4172 297 5 j.physleta.2005.10 j.physleta.2005.10 PROPN ejpam-4172 297 6 . . PUNCT ejpam-4172 298 1 005 005 NUM ejpam-4172 298 2 . . PUNCT ejpam-4172 299 1 ( ( PUNCT ejpam-4172 299 2 visited visit VERB ejpam-4172 299 3 on on ADP ejpam-4172 299 4 11/19/2021 11/19/2021 NUM ejpam-4172 299 5 ) ) PUNCT ejpam-4172 299 6 . . PUNCT ejpam-4172 300 1 [ [ X ejpam-4172 300 2 18 18 NUM ejpam-4172 300 3 ] ] X ejpam-4172 300 4 ji ji PROPN ejpam-4172 300 5 - - PROPN ejpam-4172 300 6 huan huan PROPN ejpam-4172 300 7 he he PRON ejpam-4172 300 8 . . PUNCT ejpam-4172 301 1 “ " PUNCT ejpam-4172 301 2 homotopy homotopy VERB ejpam-4172 301 3 perturbation perturbation NOUN ejpam-4172 301 4 method method NOUN ejpam-4172 301 5 for for ADP ejpam-4172 301 6 bifurcation bifurcation NOUN ejpam-4172 301 7 of of ADP ejpam-4172 301 8 nonlinear nonlinear ADJ ejpam-4172 301 9 problems problem NOUN ejpam-4172 301 10 ” " PUNCT ejpam-4172 301 11 . . PUNCT ejpam-4172 302 1 in in ADP ejpam-4172 302 2 : : PUNCT ejpam-4172 302 3 international international ADJ ejpam-4172 302 4 journal journal NOUN ejpam-4172 302 5 of of ADP ejpam-4172 302 6 nonlinear nonlinear PROPN ejpam-4172 302 7 sciences sciences PROPN ejpam-4172 302 8 and and CCONJ ejpam-4172 302 9 numerical numerical PROPN ejpam-4172 302 10 simulation simulation NOUN ejpam-4172 302 11 6 6 NUM ejpam-4172 302 12 ( ( PUNCT ejpam-4172 302 13 2005 2005 NUM ejpam-4172 302 14 ) ) PUNCT ejpam-4172 302 15 . . PUNCT ejpam-4172 303 1 doi doi NOUN ejpam-4172 303 2 : : PUNCT ejpam-4172 303 3 10.1515 10.1515 NUM ejpam-4172 303 4 / / SYM ejpam-4172 303 5 ijnsns.2005.6.2.207 ijnsns.2005.6.2.207 NOUN ejpam-4172 303 6 . . PUNCT ejpam-4172 304 1 ( ( PUNCT ejpam-4172 304 2 visited visit VERB ejpam-4172 304 3 on on ADP ejpam-4172 304 4 11/19/2021 11/19/2021 NUM ejpam-4172 304 5 ) ) PUNCT ejpam-4172 304 6 . . PUNCT ejpam-4172 305 1 references reference NOUN ejpam-4172 305 2 97 97 NUM ejpam-4172 306 1 [ [ X ejpam-4172 306 2 19 19 NUM ejpam-4172 306 3 ] ] PUNCT ejpam-4172 306 4 l. l. PROPN ejpam-4172 306 5 cveticanin cveticanin PROPN ejpam-4172 306 6 , , PUNCT ejpam-4172 306 7 m. m. NOUN ejpam-4172 306 8 kalami kalami PROPN ejpam-4172 306 9 - - PUNCT ejpam-4172 306 10 yazdi yazdi NOUN ejpam-4172 306 11 , , PUNCT ejpam-4172 306 12 and and CCONJ ejpam-4172 306 13 h. h. PROPN ejpam-4172 306 14 askari askari PROPN ejpam-4172 306 15 . . PUNCT ejpam-4172 307 1 “ " PUNCT ejpam-4172 307 2 analytical analytical ADJ ejpam-4172 307 3 approximations approximation NOUN ejpam-4172 307 4 to to ADP ejpam-4172 307 5 the the DET ejpam-4172 307 6 solutions solution NOUN ejpam-4172 307 7 for for ADP ejpam-4172 307 8 a a DET ejpam-4172 307 9 generalized generalized ADJ ejpam-4172 307 10 oscillator oscillator NOUN ejpam-4172 307 11 with with ADP ejpam-4172 307 12 strong strong ADJ ejpam-4172 307 13 nonlinear nonlinear ADJ ejpam-4172 307 14 terms term NOUN ejpam-4172 307 15 ” " PUNCT ejpam-4172 307 16 . . PUNCT ejpam-4172 308 1 in in ADP ejpam-4172 308 2 : : PUNCT ejpam-4172 308 3 journal journal NOUN ejpam-4172 308 4 of of ADP ejpam-4172 308 5 engineering engineering NOUN ejpam-4172 308 6 mathematics mathematic NOUN ejpam-4172 308 7 77 77 NUM ejpam-4172 308 8 ( ( PUNCT ejpam-4172 308 9 2012 2012 NUM ejpam-4172 308 10 ) ) PUNCT ejpam-4172 308 11 , , PUNCT ejpam-4172 308 12 pp pp ADP ejpam-4172 308 13 . . PUNCT ejpam-4172 309 1 211–223 211–223 NUM ejpam-4172 309 2 . . PUNCT ejpam-4172 310 1 doi doi NOUN ejpam-4172 310 2 : : PUNCT ejpam-4172 310 3 10.1007 10.1007 NUM ejpam-4172 310 4 / / SYM ejpam-4172 310 5 s106650129542 s106650129542 NOUN ejpam-4172 310 6 - - PUNCT ejpam-4172 310 7 4 4 NUM ejpam-4172 310 8 . . PUNCT ejpam-4172 311 1 ( ( PUNCT ejpam-4172 311 2 visited visit VERB ejpam-4172 311 3 on on ADP ejpam-4172 311 4 01/03/2020 01/03/2020 NUM ejpam-4172 311 5 ) ) PUNCT ejpam-4172 311 6 . . PUNCT ejpam-4172 312 1 [ [ X ejpam-4172 312 2 20 20 NUM ejpam-4172 312 3 ] ] X ejpam-4172 312 4 ji ji PROPN ejpam-4172 312 5 - - PROPN ejpam-4172 312 6 huan huan PROPN ejpam-4172 312 7 he he PRON ejpam-4172 312 8 . . PUNCT ejpam-4172 313 1 “ " PUNCT ejpam-4172 313 2 some some DET ejpam-4172 313 3 asymptotic asymptotic ADJ ejpam-4172 313 4 methods method NOUN ejpam-4172 313 5 for for ADP ejpam-4172 313 6 strongly strongly ADV ejpam-4172 313 7 nonlinear nonlinear ADJ ejpam-4172 313 8 equations equation NOUN ejpam-4172 313 9 ” " PUNCT ejpam-4172 313 10 . . PUNCT ejpam-4172 314 1 in in ADP ejpam-4172 314 2 : : PUNCT ejpam-4172 314 3 international international ADJ ejpam-4172 314 4 journal journal NOUN ejpam-4172 314 5 of of ADP ejpam-4172 314 6 modern modern ADJ ejpam-4172 314 7 physics physics PROPN ejpam-4172 314 8 b b PROPN ejpam-4172 314 9 20 20 NUM ejpam-4172 314 10 ( ( PUNCT ejpam-4172 314 11 2006 2006 NUM ejpam-4172 314 12 ) ) PUNCT ejpam-4172 314 13 , , PUNCT ejpam-4172 314 14 pp pp ADJ ejpam-4172 314 15 . . PUNCT ejpam-4172 314 16 1141–1199 1141–1199 NUM ejpam-4172 314 17 . . PUNCT ejpam-4172 315 1 doi doi NOUN ejpam-4172 315 2 : : PUNCT ejpam-4172 315 3 10.1142/ 10.1142/ NUM ejpam-4172 315 4 s0217979206033796 s0217979206033796 PROPN ejpam-4172 315 5 . . PUNCT ejpam-4172 316 1 ( ( PUNCT ejpam-4172 316 2 visited visit VERB ejpam-4172 316 3 on on ADP ejpam-4172 316 4 11/19/2021 11/19/2021 NUM ejpam-4172 316 5 ) ) PUNCT ejpam-4172 316 6 . . PUNCT ejpam-4172 317 1 [ [ X ejpam-4172 317 2 21 21 NUM ejpam-4172 317 3 ] ] X ejpam-4172 317 4 m.k m.k PROPN ejpam-4172 317 5 . . PROPN ejpam-4172 317 6 yazdi yazdi PROPN ejpam-4172 317 7 and and CCONJ ejpam-4172 317 8 p.h p.h PROPN ejpam-4172 317 9 . . PROPN ejpam-4172 317 10 tehrani tehrani PROPN ejpam-4172 317 11 . . PUNCT ejpam-4172 318 1 “ " PUNCT ejpam-4172 318 2 the the DET ejpam-4172 318 3 energy energy NOUN ejpam-4172 318 4 balance balance NOUN ejpam-4172 318 5 to to AUX ejpam-4172 318 6 nonlinear nonlinear ADJ ejpam-4172 318 7 oscillations oscillation NOUN ejpam-4172 318 8 via via ADP ejpam-4172 318 9 jacobi jacobi PROPN ejpam-4172 318 10 collocation collocation NOUN ejpam-4172 318 11 method method NOUN ejpam-4172 318 12 ” " PUNCT ejpam-4172 318 13 . . PUNCT ejpam-4172 319 1 in in ADP ejpam-4172 319 2 : : PUNCT ejpam-4172 319 3 alexandria alexandria PROPN ejpam-4172 319 4 engineering engineering PROPN ejpam-4172 319 5 journal journal PROPN ejpam-4172 319 6 54 54 NUM ejpam-4172 319 7 ( ( PUNCT ejpam-4172 319 8 2015 2015 NUM ejpam-4172 319 9 ) ) PUNCT ejpam-4172 319 10 , , PUNCT ejpam-4172 319 11 pp pp PROPN ejpam-4172 319 12 . . PUNCT ejpam-4172 319 13 99 99 NUM ejpam-4172 319 14 – – PUNCT ejpam-4172 319 15 103 103 NUM ejpam-4172 319 16 . . PUNCT ejpam-4172 319 17 doi doi NOUN ejpam-4172 319 18 : : PUNCT ejpam-4172 319 19 10.1016 10.1016 NUM ejpam-4172 319 20 / / SYM ejpam-4172 319 21 j.aej.2015.03.016 j.aej.2015.03.016 PROPN ejpam-4172 319 22 . . PUNCT ejpam-4172 320 1 ( ( PUNCT ejpam-4172 320 2 visited visit VERB ejpam-4172 320 3 on on ADP ejpam-4172 320 4 11/19/2021 11/19/2021 NUM ejpam-4172 320 5 ) ) PUNCT ejpam-4172 320 6 . . PUNCT ejpam-4172 321 1 [ [ X ejpam-4172 321 2 22 22 NUM ejpam-4172 321 3 ] ] PUNCT ejpam-4172 321 4 akuro akuro NOUN ejpam-4172 321 5 big big ADJ ejpam-4172 321 6 - - PUNCT ejpam-4172 321 7 alabo alabo NOUN ejpam-4172 321 8 . . PUNCT ejpam-4172 322 1 “ " PUNCT ejpam-4172 322 2 a a DET ejpam-4172 322 3 simple simple ADJ ejpam-4172 322 4 cubication cubication NOUN ejpam-4172 322 5 method method NOUN ejpam-4172 322 6 for for ADP ejpam-4172 322 7 approximate approximate ADJ ejpam-4172 322 8 solution solution NOUN ejpam-4172 322 9 of of ADP ejpam-4172 322 10 nonlinear nonlinear ADJ ejpam-4172 322 11 hamiltonian hamiltonian ADJ ejpam-4172 322 12 oscillators oscillator NOUN ejpam-4172 322 13 ” " PUNCT ejpam-4172 322 14 . . PUNCT ejpam-4172 323 1 in in ADP ejpam-4172 323 2 : : PUNCT ejpam-4172 323 3 international international ADJ ejpam-4172 323 4 journal journal NOUN ejpam-4172 323 5 of of ADP ejpam-4172 323 6 mechanical mechanical ADJ ejpam-4172 323 7 engineering engineering NOUN ejpam-4172 323 8 education education NOUN ejpam-4172 323 9 48 48 NUM ejpam-4172 323 10 ( ( PUNCT ejpam-4172 323 11 2019 2019 NUM ejpam-4172 323 12 ) ) PUNCT ejpam-4172 323 13 , , PUNCT ejpam-4172 323 14 pp pp ADP ejpam-4172 323 15 . . PUNCT ejpam-4172 324 1 241–254 241–254 NUM ejpam-4172 324 2 . . PUNCT ejpam-4172 325 1 doi doi NOUN ejpam-4172 325 2 : : PUNCT ejpam-4172 325 3 10.1177/0306419018822489 10.1177/0306419018822489 NUM ejpam-4172 325 4 . . PUNCT ejpam-4172 326 1 ( ( PUNCT ejpam-4172 326 2 visited visit VERB ejpam-4172 326 3 on on ADP ejpam-4172 326 4 11/19/2021 11/19/2021 NUM ejpam-4172 326 5 ) ) PUNCT ejpam-4172 326 6 . . PUNCT ejpam-4172 327 1 [ [ X ejpam-4172 327 2 23 23 NUM ejpam-4172 327 3 ] ] PUNCT ejpam-4172 327 4 akuro akuro NOUN ejpam-4172 327 5 big big ADJ ejpam-4172 327 6 - - PUNCT ejpam-4172 327 7 alabo alabo NOUN ejpam-4172 327 8 . . PUNCT ejpam-4172 328 1 “ " PUNCT ejpam-4172 328 2 approximate approximate ADJ ejpam-4172 328 3 periodic periodic ADJ ejpam-4172 328 4 solution solution NOUN ejpam-4172 328 5 for for ADP ejpam-4172 328 6 the the DET ejpam-4172 328 7 large large ADJ ejpam-4172 328 8 - - PUNCT ejpam-4172 328 9 amplitude amplitude NOUN ejpam-4172 328 10 oscillations oscillation NOUN ejpam-4172 328 11 of of ADP ejpam-4172 328 12 a a DET ejpam-4172 328 13 simple simple ADJ ejpam-4172 328 14 pendulum pendulum NOUN ejpam-4172 328 15 ” " PUNCT ejpam-4172 328 16 . . PUNCT ejpam-4172 329 1 in in ADP ejpam-4172 329 2 : : PUNCT ejpam-4172 329 3 international international ADJ ejpam-4172 329 4 journal journal NOUN ejpam-4172 329 5 of of ADP ejpam-4172 329 6 mechanical mechanical ADJ ejpam-4172 329 7 engineering engineering NOUN ejpam-4172 329 8 education education NOUN ejpam-4172 329 9 48 48 NUM ejpam-4172 329 10 ( ( PUNCT ejpam-4172 329 11 2019 2019 NUM ejpam-4172 329 12 ) ) PUNCT ejpam-4172 329 13 , , PUNCT ejpam-4172 329 14 pp pp ADP ejpam-4172 329 15 . . PUNCT ejpam-4172 330 1 335–350 335–350 NUM ejpam-4172 330 2 . . PUNCT ejpam-4172 331 1 doi doi NOUN ejpam-4172 331 2 : : PUNCT ejpam-4172 331 3 10.1177/0306419019842298 10.1177/0306419019842298 NUM ejpam-4172 331 4 . . PUNCT ejpam-4172 332 1 ( ( PUNCT ejpam-4172 332 2 visited visit VERB ejpam-4172 332 3 on on ADP ejpam-4172 332 4 07/08/2021 07/08/2021 NUM ejpam-4172 332 5 ) ) PUNCT ejpam-4172 332 6 . . PUNCT ejpam-4172 333 1 [ [ X ejpam-4172 333 2 24 24 NUM ejpam-4172 333 3 ] ] X ejpam-4172 333 4 ji ji PROPN ejpam-4172 333 5 - - PROPN ejpam-4172 333 6 huan huan NOUN ejpam-4172 333 7 he he PROPN ejpam-4172 333 8 et et PROPN ejpam-4172 333 9 al al PROPN ejpam-4172 333 10 . . PUNCT ejpam-4172 334 1 “ " PUNCT ejpam-4172 334 2 hamiltonian hamiltonian ADJ ejpam-4172 334 3 - - PUNCT ejpam-4172 334 4 based base VERB ejpam-4172 334 5 frequency frequency NOUN ejpam-4172 334 6 - - PUNCT ejpam-4172 334 7 amplitude amplitude NOUN ejpam-4172 334 8 formulation formulation NOUN ejpam-4172 334 9 for for ADP ejpam-4172 334 10 nonlinear nonlinear ADJ ejpam-4172 334 11 oscillators oscillator NOUN ejpam-4172 334 12 ” " PUNCT ejpam-4172 334 13 . . PUNCT ejpam-4172 335 1 in in ADP ejpam-4172 335 2 : : PUNCT ejpam-4172 335 3 facta facta PROPN ejpam-4172 335 4 universitatis universitatis PROPN ejpam-4172 335 5 , , PUNCT ejpam-4172 335 6 series series NOUN ejpam-4172 335 7 : : PUNCT ejpam-4172 335 8 mechanical mechanical ADJ ejpam-4172 335 9 engineering engineering NOUN ejpam-4172 335 10 19 19 NUM ejpam-4172 335 11 ( ( PUNCT ejpam-4172 335 12 2021 2021 NUM ejpam-4172 335 13 ) ) PUNCT ejpam-4172 335 14 , , PUNCT ejpam-4172 335 15 p. p. NOUN ejpam-4172 335 16 199 199 NUM ejpam-4172 335 17 . . PUNCT ejpam-4172 336 1 doi doi NOUN ejpam-4172 336 2 : : PUNCT ejpam-4172 336 3 10.22190 10.22190 NUM ejpam-4172 336 4 / / SYM ejpam-4172 336 5 fume201205002h fume201205002h PROPN ejpam-4172 336 6 . . PUNCT ejpam-4172 337 1 ( ( PUNCT ejpam-4172 337 2 visited visit VERB ejpam-4172 337 3 on on ADP ejpam-4172 337 4 11/19/2021 11/19/2021 NUM ejpam-4172 337 5 ) ) PUNCT ejpam-4172 337 6 . . PUNCT ejpam-4172 338 1 [ [ X ejpam-4172 338 2 25 25 NUM ejpam-4172 338 3 ] ] X ejpam-4172 338 4 ji ji PROPN ejpam-4172 338 5 - - PROPN ejpam-4172 338 6 huan huan PROPN ejpam-4172 338 7 he he PRON ejpam-4172 338 8 . . PUNCT ejpam-4172 339 1 “ " PUNCT ejpam-4172 339 2 the the DET ejpam-4172 339 3 simpler simple ADJ ejpam-4172 339 4 , , PUNCT ejpam-4172 339 5 the the PRON ejpam-4172 339 6 better well ADJ ejpam-4172 339 7 : : PUNCT ejpam-4172 339 8 analytical analytical ADJ ejpam-4172 339 9 methods method NOUN ejpam-4172 339 10 for for ADP ejpam-4172 339 11 nonlinear nonlinear ADJ ejpam-4172 339 12 oscillators oscillator NOUN ejpam-4172 339 13 and and CCONJ ejpam-4172 339 14 fractional fractional ADJ ejpam-4172 339 15 oscillators oscillator NOUN ejpam-4172 339 16 ” " PUNCT ejpam-4172 339 17 . . PUNCT ejpam-4172 340 1 in in ADP ejpam-4172 340 2 : : PUNCT ejpam-4172 340 3 journal journal NOUN ejpam-4172 340 4 of of ADP ejpam-4172 340 5 low low ADJ ejpam-4172 340 6 frequency frequency NOUN ejpam-4172 340 7 noise noise NOUN ejpam-4172 340 8 , , PUNCT ejpam-4172 340 9 vibration vibration NOUN ejpam-4172 340 10 and and CCONJ ejpam-4172 340 11 active active ADJ ejpam-4172 340 12 control control NOUN ejpam-4172 340 13 38 38 NUM ejpam-4172 340 14 ( ( PUNCT ejpam-4172 340 15 2019 2019 NUM ejpam-4172 340 16 ) ) PUNCT ejpam-4172 340 17 , , PUNCT ejpam-4172 340 18 pp pp PROPN ejpam-4172 340 19 . . PUNCT ejpam-4172 340 20 1252–1260 1252–1260 NUM ejpam-4172 340 21 . . PUNCT ejpam-4172 340 22 doi doi NOUN ejpam-4172 340 23 : : PUNCT ejpam-4172 340 24 10.1177/1461348419844145 10.1177/1461348419844145 NUM ejpam-4172 340 25 . . PUNCT ejpam-4172 341 1 ( ( PUNCT ejpam-4172 341 2 visited visit VERB ejpam-4172 341 3 on on ADP ejpam-4172 341 4 11/19/2021 11/19/2021 NUM ejpam-4172 341 5 ) ) PUNCT ejpam-4172 341 6 . . PUNCT ejpam-4172 342 1 [ [ X ejpam-4172 342 2 26 26 NUM ejpam-4172 342 3 ] ] X ejpam-4172 342 4 ji ji PROPN ejpam-4172 342 5 - - PROPN ejpam-4172 342 6 huan huan PROPN ejpam-4172 342 7 he he PRON ejpam-4172 342 8 . . PUNCT ejpam-4172 343 1 “ " PUNCT ejpam-4172 343 2 the the DET ejpam-4172 343 3 simplest simple ADJ ejpam-4172 343 4 approach approach NOUN ejpam-4172 343 5 to to ADP ejpam-4172 343 6 nonlinear nonlinear ADJ ejpam-4172 343 7 oscillators oscillator NOUN ejpam-4172 343 8 ” " PUNCT ejpam-4172 343 9 . . PUNCT ejpam-4172 344 1 in in ADP ejpam-4172 344 2 : : PUNCT ejpam-4172 344 3 results result NOUN ejpam-4172 344 4 in in ADP ejpam-4172 344 5 physics physics NOUN ejpam-4172 344 6 ( ( PUNCT ejpam-4172 344 7 2019 2019 NUM ejpam-4172 344 8 ) ) PUNCT ejpam-4172 344 9 , , PUNCT ejpam-4172 344 10 p. p. NOUN ejpam-4172 344 11 102546 102546 NUM ejpam-4172 344 12 . . PUNCT ejpam-4172 345 1 doi doi NOUN ejpam-4172 345 2 : : PUNCT ejpam-4172 345 3 10.1016 10.1016 NUM ejpam-4172 345 4 / / SYM ejpam-4172 345 5 j.rinp.2019.102546 j.rinp.2019.102546 ADJ ejpam-4172 345 6 . . PUNCT ejpam-4172 346 1 [ [ X ejpam-4172 346 2 27 27 NUM ejpam-4172 346 3 ] ] X ejpam-4172 346 4 yue yue PROPN ejpam-4172 346 5 wu wu PROPN ejpam-4172 346 6 and and CCONJ ejpam-4172 346 7 ji ji PROPN ejpam-4172 346 8 - - PROPN ejpam-4172 346 9 huan huan PROPN ejpam-4172 346 10 he he PRON ejpam-4172 346 11 . . PUNCT ejpam-4172 347 1 “ " PUNCT ejpam-4172 347 2 homotopy homotopy VERB ejpam-4172 347 3 perturbation perturbation NOUN ejpam-4172 347 4 method method NOUN ejpam-4172 347 5 for for ADP ejpam-4172 347 6 nonlinear nonlinear ADJ ejpam-4172 347 7 oscillators oscillator NOUN ejpam-4172 347 8 with with ADP ejpam-4172 347 9 coordinate coordinate NOUN ejpam-4172 347 10 - - PUNCT ejpam-4172 347 11 dependent dependent ADJ ejpam-4172 347 12 mass mass NOUN ejpam-4172 347 13 ” " PUNCT ejpam-4172 347 14 . . PUNCT ejpam-4172 348 1 in in ADP ejpam-4172 348 2 : : PUNCT ejpam-4172 348 3 results result NOUN ejpam-4172 348 4 in in ADP ejpam-4172 348 5 physics physics NOUN ejpam-4172 348 6 10 10 NUM ejpam-4172 348 7 ( ( PUNCT ejpam-4172 348 8 2018 2018 NUM ejpam-4172 348 9 ) ) PUNCT ejpam-4172 348 10 , , PUNCT ejpam-4172 348 11 pp pp ADP ejpam-4172 348 12 . . PUNCT ejpam-4172 349 1 270–271 270–271 NUM ejpam-4172 349 2 . . PUNCT ejpam-4172 350 1 doi doi NOUN ejpam-4172 350 2 : : PUNCT ejpam-4172 350 3 10.1016 10.1016 NUM ejpam-4172 350 4 / / SYM ejpam-4172 350 5 j.rinp.2018.06.015 j.rinp.2018.06.015 PROPN ejpam-4172 350 6 . . PUNCT ejpam-4172 351 1 [ [ X ejpam-4172 351 2 28 28 NUM ejpam-4172 351 3 ] ] X ejpam-4172 351 4 ji ji PROPN ejpam-4172 351 5 - - PROPN ejpam-4172 351 6 huan huan PROPN ejpam-4172 351 7 he he PRON ejpam-4172 351 8 . . PUNCT ejpam-4172 352 1 “ " PUNCT ejpam-4172 352 2 amplitude amplitude NOUN ejpam-4172 352 3 - - PUNCT ejpam-4172 352 4 frequency frequency NOUN ejpam-4172 352 5 relationship relationship NOUN ejpam-4172 352 6 for for ADP ejpam-4172 352 7 conservative conservative ADJ ejpam-4172 352 8 nonlinear nonlinear ADJ ejpam-4172 352 9 oscillators oscillator NOUN ejpam-4172 352 10 with with ADP ejpam-4172 352 11 odd odd ADJ ejpam-4172 352 12 nonlinearities nonlinearitie NOUN ejpam-4172 352 13 ” " PUNCT ejpam-4172 352 14 . . PUNCT ejpam-4172 353 1 in in ADP ejpam-4172 353 2 : : PUNCT ejpam-4172 353 3 international international ADJ ejpam-4172 353 4 journal journal NOUN ejpam-4172 353 5 of of ADP ejpam-4172 353 6 applied applied ADJ ejpam-4172 353 7 and and CCONJ ejpam-4172 353 8 computational computational ADJ ejpam-4172 353 9 mathematics mathematic NOUN ejpam-4172 353 10 3 3 NUM ejpam-4172 353 11 ( ( PUNCT ejpam-4172 353 12 2016 2016 NUM ejpam-4172 353 13 ) ) PUNCT ejpam-4172 353 14 , , PUNCT ejpam-4172 353 15 pp pp ADP ejpam-4172 353 16 . . PUNCT ejpam-4172 353 17 1557–1560 1557–1560 X ejpam-4172 353 18 . . PUNCT ejpam-4172 354 1 doi doi NOUN ejpam-4172 354 2 : : PUNCT ejpam-4172 354 3 10.1007 10.1007 NUM ejpam-4172 354 4 / / SYM ejpam-4172 354 5 s40819 s40819 NOUN ejpam-4172 354 6 - - PUNCT ejpam-4172 354 7 016 016 NUM ejpam-4172 354 8 - - PUNCT ejpam-4172 354 9 0160 0160 NUM ejpam-4172 354 10 - - PUNCT ejpam-4172 354 11 0 0 NUM ejpam-4172 354 12 . . PUNCT ejpam-4172 355 1 ( ( PUNCT ejpam-4172 355 2 visited visit VERB ejpam-4172 355 3 on on ADP ejpam-4172 355 4 11/19/2021 11/19/2021 NUM ejpam-4172 355 5 ) ) PUNCT ejpam-4172 355 6 . . PUNCT ejpam-4172 356 1 [ [ X ejpam-4172 356 2 29 29 NUM ejpam-4172 356 3 ] ] X ejpam-4172 356 4 ji ji PROPN ejpam-4172 356 5 - - PROPN ejpam-4172 356 6 huan huan PROPN ejpam-4172 356 7 he he PRON ejpam-4172 356 8 . . PUNCT ejpam-4172 357 1 “ " PUNCT ejpam-4172 357 2 an an DET ejpam-4172 357 3 elementary elementary ADJ ejpam-4172 357 4 introduction introduction NOUN ejpam-4172 357 5 to to ADP ejpam-4172 357 6 the the DET ejpam-4172 357 7 homotopy homotopy NOUN ejpam-4172 357 8 perturbation perturbation NOUN ejpam-4172 357 9 method method NOUN ejpam-4172 357 10 ” " PUNCT ejpam-4172 357 11 . . PUNCT ejpam-4172 358 1 in in ADP ejpam-4172 358 2 : : PUNCT ejpam-4172 358 3 computers computer NOUN ejpam-4172 358 4 & & CCONJ ejpam-4172 358 5 mathematics mathematic NOUN ejpam-4172 358 6 with with ADP ejpam-4172 358 7 applications application NOUN ejpam-4172 358 8 57 57 NUM ejpam-4172 358 9 ( ( PUNCT ejpam-4172 358 10 2009 2009 NUM ejpam-4172 358 11 ) ) PUNCT ejpam-4172 358 12 , , PUNCT ejpam-4172 358 13 pp pp ADP ejpam-4172 358 14 . . PUNCT ejpam-4172 359 1 410–412 410–412 NUM ejpam-4172 359 2 . . PUNCT ejpam-4172 360 1 doi doi NOUN ejpam-4172 360 2 : : PUNCT ejpam-4172 360 3 10 10 NUM ejpam-4172 360 4 . . X ejpam-4172 361 1 1016 1016 NUM ejpam-4172 361 2 / / SYM ejpam-4172 361 3 j.camwa.2008.06.003 j.camwa.2008.06.003 PROPN ejpam-4172 361 4 . . PUNCT ejpam-4172 362 1 ( ( PUNCT ejpam-4172 362 2 visited visit VERB ejpam-4172 362 3 on on ADP ejpam-4172 362 4 11/19/2021 11/19/2021 NUM ejpam-4172 362 5 ) ) PUNCT ejpam-4172 362 6 . . PUNCT ejpam-4172 363 1 [ [ X ejpam-4172 363 2 30 30 NUM ejpam-4172 363 3 ] ] X ejpam-4172 363 4 ji ji PROPN ejpam-4172 363 5 - - PROPN ejpam-4172 363 6 huan huan PROPN ejpam-4172 363 7 he he PRON ejpam-4172 363 8 , , PUNCT ejpam-4172 363 9 yusry yusry PROPN ejpam-4172 363 10 o. o. PROPN ejpam-4172 363 11 el el PROPN ejpam-4172 363 12 - - PROPN ejpam-4172 363 13 dib dib PROPN ejpam-4172 363 14 , , PUNCT ejpam-4172 363 15 and and CCONJ ejpam-4172 363 16 amal amal PROPN ejpam-4172 363 17 a. a. PROPN ejpam-4172 363 18 mady mady PROPN ejpam-4172 363 19 . . PUNCT ejpam-4172 364 1 “ " PUNCT ejpam-4172 364 2 homotopy homotopy VERB ejpam-4172 364 3 perturbation perturbation NOUN ejpam-4172 364 4 method method NOUN ejpam-4172 364 5 for for ADP ejpam-4172 364 6 the the DET ejpam-4172 364 7 fractal fractal ADJ ejpam-4172 364 8 toda toda PROPN ejpam-4172 364 9 oscillator oscillator PROPN ejpam-4172 364 10 ” " PUNCT ejpam-4172 364 11 . . PUNCT ejpam-4172 365 1 in in ADP ejpam-4172 365 2 : : PUNCT ejpam-4172 365 3 fractal fractal ADJ ejpam-4172 365 4 and and CCONJ ejpam-4172 365 5 fractional fractional ADJ ejpam-4172 365 6 5 5 NUM ejpam-4172 365 7 ( ( PUNCT ejpam-4172 365 8 2021 2021 NUM ejpam-4172 365 9 ) ) PUNCT ejpam-4172 365 10 , , PUNCT ejpam-4172 365 11 p. p. NOUN ejpam-4172 365 12 93 93 NUM ejpam-4172 365 13 . . PUNCT ejpam-4172 366 1 doi doi NOUN ejpam-4172 366 2 : : PUNCT ejpam-4172 366 3 10.3390 10.3390 NUM ejpam-4172 366 4 / / SYM ejpam-4172 366 5 fractalfract5030093 fractalfract5030093 PROPN ejpam-4172 366 6 . . PUNCT ejpam-4172 367 1 ( ( PUNCT ejpam-4172 367 2 visited visit VERB ejpam-4172 367 3 on on ADP ejpam-4172 367 4 11/19/2021 11/19/2021 NUM ejpam-4172 367 5 ) ) PUNCT ejpam-4172 367 6 . . PUNCT ejpam-4172 368 1 references reference NOUN ejpam-4172 368 2 98 98 NUM ejpam-4172 369 1 [ [ X ejpam-4172 369 2 31 31 NUM ejpam-4172 369 3 ] ] X ejpam-4172 369 4 ji ji PROPN ejpam-4172 369 5 - - PROPN ejpam-4172 369 6 huan huan PROPN ejpam-4172 369 7 he he PRON ejpam-4172 369 8 . . PUNCT ejpam-4172 370 1 “ " PUNCT ejpam-4172 370 2 comment comment NOUN ejpam-4172 370 3 on on ADP ejpam-4172 370 4 ‘ ' PUNCT ejpam-4172 370 5 he he PRON ejpam-4172 370 6 ’s ’s PART ejpam-4172 370 7 frequency frequency NOUN ejpam-4172 370 8 formulation formulation NOUN ejpam-4172 370 9 for for ADP ejpam-4172 370 10 nonlinear nonlinear ADJ ejpam-4172 370 11 oscillators oscillator NOUN ejpam-4172 370 12 ’ ' PUNCT ejpam-4172 370 13 ” " PUNCT ejpam-4172 370 14 . . PUNCT ejpam-4172 371 1 in in ADP ejpam-4172 371 2 : : PUNCT ejpam-4172 371 3 european european ADJ ejpam-4172 371 4 journal journal PROPN ejpam-4172 371 5 of of ADP ejpam-4172 371 6 physics physics PROPN ejpam-4172 371 7 29 29 NUM ejpam-4172 371 8 ( ( PUNCT ejpam-4172 371 9 2008 2008 NUM ejpam-4172 371 10 ) ) PUNCT ejpam-4172 371 11 , , PUNCT ejpam-4172 371 12 pp pp PROPN ejpam-4172 371 13 . . PUNCT ejpam-4172 371 14 l19 l19 NOUN ejpam-4172 371 15 – – PUNCT ejpam-4172 371 16 l22 l22 NOUN ejpam-4172 371 17 . . PUNCT ejpam-4172 372 1 doi doi NOUN ejpam-4172 372 2 : : PUNCT ejpam-4172 372 3 10.1088/01430807/29/4 10.1088/01430807/29/4 NUM ejpam-4172 372 4 / / SYM ejpam-4172 372 5 l02 l02 NOUN ejpam-4172 372 6 . . PUNCT ejpam-4172 373 1 ( ( PUNCT ejpam-4172 373 2 visited visit VERB ejpam-4172 373 3 on on ADP ejpam-4172 373 4 11/19/2021 11/19/2021 NUM ejpam-4172 373 5 ) ) PUNCT ejpam-4172 373 6 . . PUNCT ejpam-4172 374 1 [ [ X ejpam-4172 374 2 32 32 NUM ejpam-4172 374 3 ] ] PUNCT ejpam-4172 374 4 ben ben PROPN ejpam-4172 374 5 - - PUNCT ejpam-4172 374 6 yu yu PROPN ejpam-4172 374 7 guo guo PROPN ejpam-4172 374 8 , , PUNCT ejpam-4172 374 9 jie jie PROPN ejpam-4172 374 10 shen shen PROPN ejpam-4172 374 11 , , PUNCT ejpam-4172 374 12 and and CCONJ ejpam-4172 374 13 li li PROPN ejpam-4172 374 14 - - PROPN ejpam-4172 374 15 lianwang lianwang PROPN ejpam-4172 374 16 . . PUNCT ejpam-4172 375 1 “ " PUNCT ejpam-4172 375 2 generalized generalized ADJ ejpam-4172 375 3 jacobi jacobi NOUN ejpam-4172 375 4 polynomials polynomial NOUN ejpam-4172 375 5 / / SYM ejpam-4172 375 6 functions function NOUN ejpam-4172 375 7 and and CCONJ ejpam-4172 375 8 their their PRON ejpam-4172 375 9 applications application NOUN ejpam-4172 375 10 ” " PUNCT ejpam-4172 375 11 . . PUNCT ejpam-4172 376 1 in in ADP ejpam-4172 376 2 : : PUNCT ejpam-4172 376 3 applied apply VERB ejpam-4172 376 4 numerical numerical ADJ ejpam-4172 376 5 mathematics mathematic NOUN ejpam-4172 376 6 59 59 NUM ejpam-4172 376 7 ( ( PUNCT ejpam-4172 376 8 2009 2009 NUM ejpam-4172 376 9 ) ) PUNCT ejpam-4172 376 10 , , PUNCT ejpam-4172 376 11 pp pp ADP ejpam-4172 376 12 . . PUNCT ejpam-4172 376 13 1011 1011 NUM ejpam-4172 376 14 – – PUNCT ejpam-4172 376 15 1028 1028 NUM ejpam-4172 376 16 . . PUNCT ejpam-4172 377 1 doi doi NOUN ejpam-4172 377 2 : : PUNCT ejpam-4172 377 3 10.1016 10.1016 NUM ejpam-4172 377 4 / / SYM ejpam-4172 377 5 j.apnum.2008.04.003 j.apnum.2008.04.003 PROPN ejpam-4172 377 6 . . PROPN ejpam-4172 377 7 ( ( PUNCT ejpam-4172 377 8 visited visit VERB ejpam-4172 377 9 on on ADP ejpam-4172 377 10 11/19/2021 11/19/2021 NUM ejpam-4172 377 11 ) ) PUNCT ejpam-4172 377 12 . . PUNCT ejpam-4172 378 1 [ [ X ejpam-4172 378 2 33 33 NUM ejpam-4172 378 3 ] ] X ejpam-4172 378 4 o. o. PROPN ejpam-4172 378 5 gonzález gonzález PROPN ejpam-4172 378 6 - - PUNCT ejpam-4172 378 7 gaxiola gaxiola PROPN ejpam-4172 378 8 . . PUNCT ejpam-4172 379 1 “ " PUNCT ejpam-4172 379 2 periodic periodic ADJ ejpam-4172 379 3 solution solution NOUN ejpam-4172 379 4 for for ADP ejpam-4172 379 5 strongly strongly ADV ejpam-4172 379 6 nonlinear nonlinear ADJ ejpam-4172 379 7 oscillators oscillator NOUN ejpam-4172 379 8 by by ADP ejpam-4172 379 9 he he PRON ejpam-4172 379 10 ’s ’ VERB ejpam-4172 379 11 new new ADJ ejpam-4172 379 12 amplitude amplitude NOUN ejpam-4172 379 13 – – PUNCT ejpam-4172 379 14 frequency frequency NOUN ejpam-4172 379 15 relationship relationship NOUN ejpam-4172 379 16 ” " PUNCT ejpam-4172 379 17 . . PUNCT ejpam-4172 380 1 in in ADP ejpam-4172 380 2 : : PUNCT ejpam-4172 380 3 international international ADJ ejpam-4172 380 4 journal journal NOUN ejpam-4172 380 5 of of ADP ejpam-4172 380 6 applied applied ADJ ejpam-4172 380 7 and and CCONJ ejpam-4172 380 8 computational computational ADJ ejpam-4172 380 9 mathematics mathematic NOUN ejpam-4172 380 10 3 3 NUM ejpam-4172 380 11 ( ( PUNCT ejpam-4172 380 12 2017 2017 NUM ejpam-4172 380 13 ) ) PUNCT ejpam-4172 380 14 , , PUNCT ejpam-4172 380 15 pp pp ADJ ejpam-4172 380 16 . . PUNCT ejpam-4172 380 17 1249–1259 1249–1259 NUM ejpam-4172 380 18 . . PUNCT ejpam-4172 381 1 doi doi NOUN ejpam-4172 381 2 : : PUNCT ejpam-4172 381 3 10.1007 10.1007 NUM ejpam-4172 381 4 / / SYM ejpam-4172 381 5 s40819 s40819 NOUN ejpam-4172 381 6 - - PUNCT ejpam-4172 381 7 0170414 0170414 NUM ejpam-4172 381 8 - - PUNCT ejpam-4172 381 9 5 5 NUM ejpam-4172 381 10 . . PUNCT ejpam-4172 381 11 ( ( PUNCT ejpam-4172 381 12 visited visit VERB ejpam-4172 381 13 on on ADP ejpam-4172 381 14 11/19/2021 11/19/2021 NUM ejpam-4172 381 15 ) ) PUNCT ejpam-4172 381 16 . . PUNCT ejpam-4172 382 1 [ [ X ejpam-4172 382 2 34 34 NUM ejpam-4172 382 3 ] ] X ejpam-4172 382 4 xu xu PROPN ejpam-4172 382 5 - - PUNCT ejpam-4172 382 6 chu chu PROPN ejpam-4172 382 7 cai cai PROPN ejpam-4172 382 8 and and CCONJ ejpam-4172 382 9 jun jun PROPN ejpam-4172 382 10 - - PUNCT ejpam-4172 382 11 fang fang PROPN ejpam-4172 382 12 liu liu PROPN ejpam-4172 382 13 . . PUNCT ejpam-4172 383 1 “ " PUNCT ejpam-4172 383 2 application application NOUN ejpam-4172 383 3 of of ADP ejpam-4172 383 4 the the DET ejpam-4172 383 5 modified modify VERB ejpam-4172 383 6 frequency frequency NOUN ejpam-4172 383 7 formulation formulation NOUN ejpam-4172 383 8 to to ADP ejpam-4172 383 9 a a DET ejpam-4172 383 10 nonlinear nonlinear ADJ ejpam-4172 383 11 oscillator oscillator NOUN ejpam-4172 383 12 ” " PUNCT ejpam-4172 383 13 . . PUNCT ejpam-4172 384 1 in in ADP ejpam-4172 384 2 : : PUNCT ejpam-4172 384 3 computers computer NOUN ejpam-4172 384 4 & & CCONJ ejpam-4172 384 5 mathematics mathematic NOUN ejpam-4172 384 6 with with ADP ejpam-4172 384 7 applications application NOUN ejpam-4172 384 8 61 61 NUM ejpam-4172 384 9 ( ( PUNCT ejpam-4172 384 10 2011 2011 NUM ejpam-4172 384 11 ) ) PUNCT ejpam-4172 384 12 , , PUNCT ejpam-4172 384 13 pp pp ADP ejpam-4172 384 14 . . PUNCT ejpam-4172 384 15 2237–2240 2237–2240 NUM ejpam-4172 384 16 . . PUNCT ejpam-4172 385 1 doi doi NOUN ejpam-4172 385 2 : : PUNCT ejpam-4172 385 3 10.1016 10.1016 NUM ejpam-4172 385 4 / / SYM ejpam-4172 385 5 j.camwa.2010.09.025 j.camwa.2010.09.025 PROPN ejpam-4172 385 6 . . PUNCT ejpam-4172 386 1 ( ( PUNCT ejpam-4172 386 2 visited visit VERB ejpam-4172 386 3 on on ADP ejpam-4172 386 4 11/19/2021 11/19/2021 NUM ejpam-4172 386 5 ) ) PUNCT ejpam-4172 386 6 . . PUNCT ejpam-4172 387 1 [ [ X ejpam-4172 387 2 35 35 NUM ejpam-4172 387 3 ] ] X ejpam-4172 387 4 s. s. PROPN ejpam-4172 387 5 s. s. PROPN ejpam-4172 387 6 ganji ganji PROPN ejpam-4172 387 7 et et PROPN ejpam-4172 387 8 al al PROPN ejpam-4172 387 9 . . PUNCT ejpam-4172 388 1 “ " PUNCT ejpam-4172 388 2 periodic periodic ADJ ejpam-4172 388 3 solution solution NOUN ejpam-4172 388 4 for for ADP ejpam-4172 388 5 strongly strongly ADV ejpam-4172 388 6 nonlinear nonlinear ADJ ejpam-4172 388 7 vibration vibration NOUN ejpam-4172 388 8 systems system NOUN ejpam-4172 388 9 by by ADP ejpam-4172 388 10 he he PRON ejpam-4172 388 11 ’s ’ VERB ejpam-4172 388 12 energy energy NOUN ejpam-4172 388 13 balance balance NOUN ejpam-4172 388 14 method method NOUN ejpam-4172 388 15 ” " PUNCT ejpam-4172 388 16 . . PUNCT ejpam-4172 389 1 in in ADP ejpam-4172 389 2 : : PUNCT ejpam-4172 389 3 acta acta PROPN ejpam-4172 389 4 applicandae applicandae PROPN ejpam-4172 389 5 mathematicae mathematicae PROPN ejpam-4172 389 6 106 106 NUM ejpam-4172 389 7 ( ( PUNCT ejpam-4172 389 8 2008 2008 NUM ejpam-4172 389 9 ) ) PUNCT ejpam-4172 389 10 , , PUNCT ejpam-4172 389 11 pp pp PROPN ejpam-4172 389 12 . . PUNCT ejpam-4172 389 13 79 79 NUM ejpam-4172 389 14 – – PUNCT ejpam-4172 389 15 92 92 NUM ejpam-4172 389 16 . . PUNCT ejpam-4172 389 17 doi doi NOUN ejpam-4172 389 18 : : PUNCT ejpam-4172 389 19 10.1007 10.1007 NUM ejpam-4172 389 20 / / SYM ejpam-4172 389 21 s10440 s10440 NOUN ejpam-4172 389 22 - - PUNCT ejpam-4172 389 23 008 008 NUM ejpam-4172 389 24 - - PUNCT ejpam-4172 389 25 9283 9283 NUM ejpam-4172 389 26 - - PUNCT ejpam-4172 389 27 6 6 NUM ejpam-4172 389 28 . . PUNCT ejpam-4172 389 29 ( ( PUNCT ejpam-4172 389 30 visited visit VERB ejpam-4172 389 31 on on ADP ejpam-4172 389 32 11/19/2021 11/19/2021 NUM ejpam-4172 389 33 ) ) PUNCT ejpam-4172 389 34 . . PUNCT ejpam-4172 390 1 [ [ X ejpam-4172 390 2 36 36 NUM ejpam-4172 390 3 ] ] X ejpam-4172 390 4 y. y. PROPN ejpam-4172 390 5 khan khan PROPN ejpam-4172 390 6 et et PROPN ejpam-4172 390 7 al al PROPN ejpam-4172 390 8 . . PUNCT ejpam-4172 391 1 “ " PUNCT ejpam-4172 391 2 dynamic dynamic ADJ ejpam-4172 391 3 analysis analysis NOUN ejpam-4172 391 4 of of ADP ejpam-4172 391 5 generalized generalized ADJ ejpam-4172 391 6 conservative conservative ADJ ejpam-4172 391 7 nonlinear nonlinear ADJ ejpam-4172 391 8 oscillators oscillator NOUN ejpam-4172 391 9 via via ADP ejpam-4172 391 10 frequency frequency NOUN ejpam-4172 391 11 amplitude amplitude NOUN ejpam-4172 391 12 formulation formulation NOUN ejpam-4172 391 13 ” " PUNCT ejpam-4172 391 14 . . PUNCT ejpam-4172 392 1 in in ADP ejpam-4172 392 2 : : PUNCT ejpam-4172 392 3 arabian arabian ADJ ejpam-4172 392 4 journal journal NOUN ejpam-4172 392 5 for for ADP ejpam-4172 392 6 science science NOUN ejpam-4172 392 7 and and CCONJ ejpam-4172 392 8 engineering engineering NOUN ejpam-4172 392 9 38 38 NUM ejpam-4172 392 10 ( ( PUNCT ejpam-4172 392 11 2011 2011 NUM ejpam-4172 392 12 ) ) PUNCT ejpam-4172 392 13 , , PUNCT ejpam-4172 392 14 pp pp ADJ ejpam-4172 392 15 . . PUNCT ejpam-4172 392 16 175–179 175–179 NUM ejpam-4172 392 17 . . PUNCT ejpam-4172 393 1 doi doi NOUN ejpam-4172 393 2 : : PUNCT ejpam-4172 393 3 10.1007 10.1007 NUM ejpam-4172 393 4 / / SYM ejpam-4172 393 5 s13369 s13369 PROPN ejpam-4172 393 6 - - PUNCT ejpam-4172 393 7 011 011 NUM ejpam-4172 393 8 - - PUNCT ejpam-4172 393 9 0035 0035 NUM ejpam-4172 393 10 - - PUNCT ejpam-4172 393 11 y. y. NOUN ejpam-4172 393 12 ( ( PUNCT ejpam-4172 393 13 visited visit VERB ejpam-4172 393 14 on on ADP ejpam-4172 393 15 11/19/2021 11/19/2021 NUM ejpam-4172 393 16 ) ) PUNCT ejpam-4172 393 17 . . PUNCT ejpam-4172 394 1 [ [ X ejpam-4172 394 2 37 37 NUM ejpam-4172 394 3 ] ] PUNCT ejpam-4172 394 4 m. m. NOUN ejpam-4172 394 5 k. k. PROPN ejpam-4172 394 6 yazdi yazdi PROPN ejpam-4172 394 7 . . PUNCT ejpam-4172 395 1 “ " PUNCT ejpam-4172 395 2 an an DET ejpam-4172 395 3 accurate accurate ADJ ejpam-4172 395 4 relationship relationship NOUN ejpam-4172 395 5 between between ADP ejpam-4172 395 6 frequency frequency NOUN ejpam-4172 395 7 and and CCONJ ejpam-4172 395 8 amplitude amplitude NOUN ejpam-4172 395 9 to to ADP ejpam-4172 395 10 nonlinear nonlinear ADJ ejpam-4172 395 11 oscillations oscillation NOUN ejpam-4172 395 12 ” " PUNCT ejpam-4172 395 13 . . PUNCT ejpam-4172 396 1 in in ADP ejpam-4172 396 2 : : PUNCT ejpam-4172 396 3 journal journal PROPN ejpam-4172 396 4 of of ADP ejpam-4172 396 5 taibah taibah PROPN ejpam-4172 396 6 university university PROPN ejpam-4172 396 7 for for ADP ejpam-4172 396 8 science science NOUN ejpam-4172 396 9 12 12 NUM ejpam-4172 396 10 ( ( PUNCT ejpam-4172 396 11 2018 2018 NUM ejpam-4172 396 12 ) ) PUNCT ejpam-4172 396 13 , , PUNCT ejpam-4172 396 14 pp pp ADP ejpam-4172 396 15 . . PUNCT ejpam-4172 396 16 532 532 NUM ejpam-4172 396 17 – – PUNCT ejpam-4172 396 18 535 535 NUM ejpam-4172 396 19 . . PUNCT ejpam-4172 396 20 doi doi NOUN ejpam-4172 396 21 : : PUNCT ejpam-4172 396 22 10.1080/16583655.2018.1498290 10.1080/16583655.2018.1498290 NUM ejpam-4172 396 23 . . PUNCT ejpam-4172 397 1 ( ( PUNCT ejpam-4172 397 2 visited visit VERB ejpam-4172 397 3 on on ADP ejpam-4172 397 4 11/19/2021 11/19/2021 NUM ejpam-4172 397 5 ) ) PUNCT ejpam-4172 397 6 . . PUNCT ejpam-4172 398 1 [ [ X ejpam-4172 398 2 38 38 NUM ejpam-4172 398 3 ] ] PUNCT ejpam-4172 398 4 m m VERB ejpam-4172 398 5 kalami kalami NOUN ejpam-4172 398 6 yazdi yazdi PROPN ejpam-4172 398 7 and and CCONJ ejpam-4172 398 8 p p PROPN ejpam-4172 398 9 hosseini hosseini PROPN ejpam-4172 398 10 tehrani tehrani PROPN ejpam-4172 398 11 . . PUNCT ejpam-4172 399 1 “ " PUNCT ejpam-4172 399 2 frequency frequency NOUN ejpam-4172 399 3 analysis analysis NOUN ejpam-4172 399 4 of of ADP ejpam-4172 399 5 nonlinear nonlinear ADJ ejpam-4172 399 6 oscillations oscillation NOUN ejpam-4172 399 7 via via ADP ejpam-4172 399 8 the the DET ejpam-4172 399 9 global global ADJ ejpam-4172 399 10 error error NOUN ejpam-4172 399 11 minimization minimization NOUN ejpam-4172 399 12 ” " PUNCT ejpam-4172 399 13 . . PUNCT ejpam-4172 400 1 in in ADP ejpam-4172 400 2 : : PUNCT ejpam-4172 400 3 nonlinear nonlinear ADJ ejpam-4172 400 4 engineering engineering NOUN ejpam-4172 400 5 5 5 NUM ejpam-4172 400 6 ( ( PUNCT ejpam-4172 400 7 2016 2016 NUM ejpam-4172 400 8 ) ) PUNCT ejpam-4172 400 9 , , PUNCT ejpam-4172 400 10 pp pp ADP ejpam-4172 400 11 . . PUNCT ejpam-4172 401 1 87 87 NUM ejpam-4172 401 2 – – PUNCT ejpam-4172 401 3 92 92 NUM ejpam-4172 401 4 . . PUNCT ejpam-4172 401 5 doi doi NOUN ejpam-4172 401 6 : : PUNCT ejpam-4172 401 7 10.1515 10.1515 NUM ejpam-4172 401 8 / / SYM ejpam-4172 401 9 nleng-2015 nleng-2015 ADJ ejpam-4172 401 10 - - PUNCT ejpam-4172 401 11 0036 0036 NUM ejpam-4172 401 12 . . PUNCT ejpam-4172 402 1 ( ( PUNCT ejpam-4172 402 2 visited visit VERB ejpam-4172 402 3 on on ADP ejpam-4172 402 4 01/03/2020 01/03/2020 NUM ejpam-4172 402 5 ) ) PUNCT ejpam-4172 402 6 . . PUNCT ejpam-4172 403 1 [ [ X ejpam-4172 403 2 39 39 NUM ejpam-4172 403 3 ] ] X ejpam-4172 403 4 ji ji PROPN ejpam-4172 403 5 - - PROPN ejpam-4172 403 6 huan huan PROPN ejpam-4172 403 7 he he PROPN ejpam-4172 403 8 . . PUNCT ejpam-4172 404 1 “ " PUNCT ejpam-4172 404 2 preliminary preliminary ADJ ejpam-4172 404 3 report report NOUN ejpam-4172 404 4 on on ADP ejpam-4172 404 5 the the DET ejpam-4172 404 6 energy energy NOUN ejpam-4172 404 7 balance balance NOUN ejpam-4172 404 8 for for ADP ejpam-4172 404 9 nonlinear nonlinear ADJ ejpam-4172 404 10 oscillations oscillation NOUN ejpam-4172 404 11 ” " PUNCT ejpam-4172 404 12 . . PUNCT ejpam-4172 405 1 in in ADP ejpam-4172 405 2 : : PUNCT ejpam-4172 405 3 mech mech NOUN ejpam-4172 405 4 . . PUNCT ejpam-4172 406 1 res re NOUN ejpam-4172 406 2 . . PUNCT ejpam-4172 407 1 commu commu PROPN ejpam-4172 407 2 . . PROPN ejpam-4172 408 1 29 29 NUM ejpam-4172 408 2 ( ( PUNCT ejpam-4172 408 3 2002 2002 NUM ejpam-4172 408 4 ) ) PUNCT ejpam-4172 408 5 , , PUNCT ejpam-4172 408 6 pp pp ADP ejpam-4172 408 7 . . PUNCT ejpam-4172 409 1 107–111 107–111 NUM ejpam-4172 409 2 . . PUNCT ejpam-4172 410 1 ( ( PUNCT ejpam-4172 410 2 visited visit VERB ejpam-4172 410 3 on on ADP ejpam-4172 410 4 2002 2002 NUM ejpam-4172 410 5 ) ) PUNCT ejpam-4172 410 6 . . PUNCT ejpam-4172 411 1 [ [ X ejpam-4172 411 2 40 40 NUM ejpam-4172 411 3 ] ] X ejpam-4172 411 4 ronald ronald PROPN ejpam-4172 411 5 e e PROPN ejpam-4172 411 6 mickens mickens PROPN ejpam-4172 411 7 . . PUNCT ejpam-4172 412 1 truly truly ADV ejpam-4172 412 2 nonlinear nonlinear ADJ ejpam-4172 412 3 oscillations oscillation NOUN ejpam-4172 412 4 : : PUNCT ejpam-4172 412 5 harmonic harmonic ADJ ejpam-4172 412 6 balance balance NOUN ejpam-4172 412 7 , , PUNCT ejpam-4172 412 8 parameter parameter NOUN ejpam-4172 412 9 expansions expansion NOUN ejpam-4172 412 10 , , PUNCT ejpam-4172 412 11 iteration iteration NOUN ejpam-4172 412 12 , , PUNCT ejpam-4172 412 13 and and CCONJ ejpam-4172 412 14 averaging average VERB ejpam-4172 412 15 methods method NOUN ejpam-4172 412 16 . . PUNCT ejpam-4172 413 1 world world NOUN ejpam-4172 413 2 scientific scientific ADJ ejpam-4172 413 3 , , PUNCT ejpam-4172 413 4 cop cop NOUN ejpam-4172 413 5 , , PUNCT ejpam-4172 413 6 2010 2010 NUM ejpam-4172 413 7 . . PUNCT ejpam-4172 414 1 [ [ X ejpam-4172 414 2 41 41 NUM ejpam-4172 414 3 ] ] X ejpam-4172 414 4 davood davood ADJ ejpam-4172 414 5 younesian younesian NOUN ejpam-4172 414 6 et et PROPN ejpam-4172 414 7 al al PROPN ejpam-4172 414 8 . . PUNCT ejpam-4172 415 1 “ " PUNCT ejpam-4172 415 2 frequency frequency NOUN ejpam-4172 415 3 analysis analysis NOUN ejpam-4172 415 4 of of ADP ejpam-4172 415 5 strongly strongly ADV ejpam-4172 415 6 nonlinear nonlinear ADJ ejpam-4172 415 7 generalized generalized ADJ ejpam-4172 415 8 duffing duffing NOUN ejpam-4172 415 9 oscillators oscillator NOUN ejpam-4172 415 10 using use VERB ejpam-4172 415 11 he he PRON ejpam-4172 415 12 ’s ’s PART ejpam-4172 415 13 frequency frequency NOUN ejpam-4172 415 14 – – PUNCT ejpam-4172 415 15 amplitude amplitude NOUN ejpam-4172 415 16 formulation formulation NOUN ejpam-4172 415 17 and and CCONJ ejpam-4172 415 18 he he PRON ejpam-4172 415 19 ’s ’ VERB ejpam-4172 415 20 energy energy NOUN ejpam-4172 415 21 balance balance NOUN ejpam-4172 415 22 method method NOUN ejpam-4172 415 23 ” " PUNCT ejpam-4172 415 24 . . PUNCT ejpam-4172 416 1 in in ADP ejpam-4172 416 2 : : PUNCT ejpam-4172 416 3 computers computer NOUN ejpam-4172 416 4 & & CCONJ ejpam-4172 416 5 mathematics mathematic NOUN ejpam-4172 416 6 with with ADP ejpam-4172 416 7 applications application NOUN ejpam-4172 416 8 59 59 NUM ejpam-4172 416 9 ( ( PUNCT ejpam-4172 416 10 2010 2010 NUM ejpam-4172 416 11 ) ) PUNCT ejpam-4172 416 12 , , PUNCT ejpam-4172 416 13 pp pp ADJ ejpam-4172 416 14 . . PUNCT ejpam-4172 417 1 3222–3228 3222–3228 NUM ejpam-4172 417 2 . . PUNCT ejpam-4172 418 1 doi doi NOUN ejpam-4172 418 2 : : PUNCT ejpam-4172 418 3 10.1016 10.1016 NUM ejpam-4172 418 4 / / SYM ejpam-4172 418 5 j.camwa.2010.03.013 j.camwa.2010.03.013 PROPN ejpam-4172 418 6 . . PUNCT ejpam-4172 419 1 ( ( PUNCT ejpam-4172 419 2 visited visit VERB ejpam-4172 419 3 on on ADP ejpam-4172 419 4 10/17/2021 10/17/2021 NUM ejpam-4172 419 5 ) ) PUNCT ejpam-4172 419 6 . . PUNCT ejpam-4172 420 1 [ [ X ejpam-4172 420 2 42 42 NUM ejpam-4172 420 3 ] ] X ejpam-4172 420 4 r.e r.e PROPN ejpam-4172 420 5 . . PROPN ejpam-4172 420 6 mickens mickens PROPN ejpam-4172 420 7 . . PUNCT ejpam-4172 421 1 “ " PUNCT ejpam-4172 421 2 periodic periodic ADJ ejpam-4172 421 3 solutions solution NOUN ejpam-4172 421 4 of of ADP ejpam-4172 421 5 the the DET ejpam-4172 421 6 relativistic relativistic ADJ ejpam-4172 421 7 harmonic harmonic ADJ ejpam-4172 421 8 oscillators oscillator NOUN ejpam-4172 421 9 ” " PUNCT ejpam-4172 421 10 . . PUNCT ejpam-4172 422 1 in in ADP ejpam-4172 422 2 : : PUNCT ejpam-4172 422 3 journal journal NOUN ejpam-4172 422 4 of of ADP ejpam-4172 422 5 sound sound NOUN ejpam-4172 422 6 and and CCONJ ejpam-4172 422 7 vibration vibration NOUN ejpam-4172 422 8 212 212 NUM ejpam-4172 422 9 ( ( PUNCT ejpam-4172 422 10 1998 1998 NUM ejpam-4172 422 11 ) ) PUNCT ejpam-4172 422 12 , , PUNCT ejpam-4172 422 13 pp pp ADP ejpam-4172 422 14 . . PUNCT ejpam-4172 423 1 905–908 905–908 NUM ejpam-4172 423 2 . . PUNCT ejpam-4172 424 1 doi doi NOUN ejpam-4172 424 2 : : PUNCT ejpam-4172 424 3 10.1006 10.1006 NUM ejpam-4172 424 4 / / SYM ejpam-4172 424 5 jsvi.1997.1453 jsvi.1997.1453 NOUN ejpam-4172 424 6 . . PUNCT ejpam-4172 425 1 ( ( PUNCT ejpam-4172 425 2 visited visit VERB ejpam-4172 425 3 on on ADP ejpam-4172 425 4 11/19/2021 11/19/2021 NUM ejpam-4172 425 5 ) ) PUNCT ejpam-4172 425 6 . . PUNCT ejpam-4172 426 1 references reference NOUN ejpam-4172 426 2 99 99 NUM ejpam-4172 427 1 [ [ X ejpam-4172 427 2 43 43 NUM ejpam-4172 427 3 ] ] PUNCT ejpam-4172 427 4 a. a. NOUN ejpam-4172 427 5 beléndez beléndez PROPN ejpam-4172 427 6 et et PROPN ejpam-4172 427 7 al al PROPN ejpam-4172 427 8 . . PUNCT ejpam-4172 428 1 “ " PUNCT ejpam-4172 428 2 application application NOUN ejpam-4172 428 3 of of ADP ejpam-4172 428 4 a a DET ejpam-4172 428 5 modified modify VERB ejpam-4172 428 6 he he PRON ejpam-4172 428 7 ’s ’s NOUN ejpam-4172 428 8 homotopy homotopy NOUN ejpam-4172 428 9 perturbation perturbation NOUN ejpam-4172 428 10 method method NOUN ejpam-4172 428 11 to to PART ejpam-4172 428 12 obtain obtain VERB ejpam-4172 428 13 higher high ADJ ejpam-4172 428 14 - - PUNCT ejpam-4172 428 15 order order NOUN ejpam-4172 428 16 approximations approximation NOUN ejpam-4172 428 17 of of ADP ejpam-4172 428 18 an an DET ejpam-4172 428 19 force force NOUN ejpam-4172 428 20 nonlinear nonlinear PROPN ejpam-4172 428 21 oscillator oscillator NOUN ejpam-4172 428 22 ” " PUNCT ejpam-4172 428 23 . . PUNCT ejpam-4172 429 1 in in ADP ejpam-4172 429 2 : : PUNCT ejpam-4172 429 3 physics physics NOUN ejpam-4172 429 4 letters letter NOUN ejpam-4172 429 5 a a DET ejpam-4172 429 6 371 371 NUM ejpam-4172 429 7 ( ( PUNCT ejpam-4172 429 8 2007 2007 NUM ejpam-4172 429 9 ) ) PUNCT ejpam-4172 429 10 , , PUNCT ejpam-4172 429 11 pp pp ADP ejpam-4172 429 12 . . PUNCT ejpam-4172 430 1 421–426 421–426 NUM ejpam-4172 430 2 . . PUNCT ejpam-4172 431 1 doi doi NOUN ejpam-4172 431 2 : : PUNCT ejpam-4172 431 3 10.1016 10.1016 NUM ejpam-4172 431 4 / / SYM ejpam-4172 431 5 j.physleta.2007.06.042 j.physleta.2007.06.042 NOUN ejpam-4172 431 6 . . PUNCT ejpam-4172 432 1 ( ( PUNCT ejpam-4172 432 2 visited visit VERB ejpam-4172 432 3 on on ADP ejpam-4172 432 4 11/19/2021 11/19/2021 NUM ejpam-4172 432 5 ) ) PUNCT ejpam-4172 432 6 . . PUNCT ejpam-4172 433 1 [ [ X ejpam-4172 433 2 44 44 NUM ejpam-4172 433 3 ] ] SYM ejpam-4172 433 4 w.p w.p PROPN ejpam-4172 433 5 . . PROPN ejpam-4172 433 6 sun sun PROPN ejpam-4172 433 7 , , PUNCT ejpam-4172 433 8 b.s b.s PROPN ejpam-4172 433 9 . . PROPN ejpam-4172 433 10 wu wu PROPN ejpam-4172 433 11 , , PUNCT ejpam-4172 433 12 and and CCONJ ejpam-4172 433 13 c.w c.w PROPN ejpam-4172 433 14 . . PROPN ejpam-4172 433 15 lim lim PROPN ejpam-4172 433 16 . . PUNCT ejpam-4172 434 1 “ " PUNCT ejpam-4172 434 2 approximate approximate ADJ ejpam-4172 434 3 analytical analytical ADJ ejpam-4172 434 4 solutions solution NOUN ejpam-4172 434 5 for for ADP ejpam-4172 434 6 oscillation oscillation NOUN ejpam-4172 434 7 of of ADP ejpam-4172 434 8 a a DET ejpam-4172 434 9 mass mass NOUN ejpam-4172 434 10 attached attach VERB ejpam-4172 434 11 to to ADP ejpam-4172 434 12 a a DET ejpam-4172 434 13 stretched stretch VERB ejpam-4172 434 14 elastic elastic ADJ ejpam-4172 434 15 wire wire NOUN ejpam-4172 434 16 ” " PUNCT ejpam-4172 434 17 . . PUNCT ejpam-4172 435 1 in in ADP ejpam-4172 435 2 : : PUNCT ejpam-4172 435 3 journal journal NOUN ejpam-4172 435 4 of of ADP ejpam-4172 435 5 sound sound NOUN ejpam-4172 435 6 and and CCONJ ejpam-4172 435 7 vibration vibration NOUN ejpam-4172 435 8 300 300 NUM ejpam-4172 435 9 ( ( PUNCT ejpam-4172 435 10 2007 2007 NUM ejpam-4172 435 11 ) ) PUNCT ejpam-4172 435 12 , , PUNCT ejpam-4172 435 13 pp pp ADJ ejpam-4172 435 14 . . PUNCT ejpam-4172 435 15 1042–1047 1042–1047 NUM ejpam-4172 435 16 . . PUNCT ejpam-4172 436 1 doi doi NOUN ejpam-4172 436 2 : : PUNCT ejpam-4172 436 3 10.1016 10.1016 NUM ejpam-4172 436 4 / / SYM ejpam-4172 436 5 j.jsv.2006.08.025 j.jsv.2006.08.025 PROPN ejpam-4172 436 6 . . PUNCT ejpam-4172 437 1 ( ( PUNCT ejpam-4172 437 2 visited visit VERB ejpam-4172 437 3 on on ADP ejpam-4172 437 4 11/19/2021 11/19/2021 NUM ejpam-4172 437 5 ) ) PUNCT ejpam-4172 437 6 . . PUNCT ejpam-4172 438 1 [ [ X ejpam-4172 438 2 45 45 NUM ejpam-4172 438 3 ] ] PUNCT ejpam-4172 438 4 d. d. PROPN ejpam-4172 438 5 younesian younesian PROPN ejpam-4172 438 6 et et PROPN ejpam-4172 438 7 al al PROPN ejpam-4172 438 8 . . PUNCT ejpam-4172 439 1 “ " PUNCT ejpam-4172 439 2 analytical analytical ADJ ejpam-4172 439 3 approximate approximate ADJ ejpam-4172 439 4 solutions solution NOUN ejpam-4172 439 5 for for ADP ejpam-4172 439 6 the the DET ejpam-4172 439 7 generalized generalized ADJ ejpam-4172 439 8 nonlinear nonlinear ADJ ejpam-4172 439 9 oscillator oscillator NOUN ejpam-4172 439 10 ” " PUNCT ejpam-4172 439 11 . . PUNCT ejpam-4172 440 1 in in ADP ejpam-4172 440 2 : : PUNCT ejpam-4172 440 3 applicable applicable ADJ ejpam-4172 440 4 analysis analysis NOUN ejpam-4172 440 5 91 91 NUM ejpam-4172 440 6 ( ( PUNCT ejpam-4172 440 7 2012 2012 NUM ejpam-4172 440 8 ) ) PUNCT ejpam-4172 440 9 , , PUNCT ejpam-4172 440 10 pp pp ADP ejpam-4172 440 11 . . PUNCT ejpam-4172 441 1 965–977 965–977 NUM ejpam-4172 441 2 . . PUNCT ejpam-4172 442 1 doi doi NOUN ejpam-4172 442 2 : : PUNCT ejpam-4172 442 3 10.1080/00036811 10.1080/00036811 NUM ejpam-4172 442 4 . . PUNCT ejpam-4172 443 1 2011.559464 2011.559464 NUM ejpam-4172 443 2 . . PUNCT ejpam-4172 444 1 ( ( PUNCT ejpam-4172 444 2 visited visit VERB ejpam-4172 444 3 on on ADP ejpam-4172 444 4 11/19/2021 11/19/2021 NUM ejpam-4172 444 5 ) ) PUNCT ejpam-4172 444 6 . . PUNCT