id sid tid token lemma pos cana-5366 1 1 communications communication NOUN cana-5366 1 2 on on ADP cana-5366 1 3 applied apply VERB cana-5366 1 4 nonlinear nonlinear ADJ cana-5366 1 5 analysis analysis NOUN cana-5366 1 6 issn issn NOUN cana-5366 1 7 : : PUNCT cana-5366 1 8 1074 1074 NUM cana-5366 1 9 - - PUNCT cana-5366 1 10 133x 133x NUM cana-5366 1 11 vol vol VERB cana-5366 1 12 32 32 NUM cana-5366 1 13 no no NOUN cana-5366 1 14 . . PUNCT cana-5366 2 1 10s 10 NOUN cana-5366 2 2 ( ( PUNCT cana-5366 2 3 2025 2025 NUM cana-5366 2 4 ) ) PUNCT cana-5366 2 5 2042 2042 NUM cana-5366 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 2 7 analytical analytical ADJ cana-5366 2 8 investigation investigation NOUN cana-5366 2 9 of of ADP cana-5366 2 10 human human ADJ cana-5366 2 11 finger finger NOUN cana-5366 2 12 dynamics dynamic NOUN cana-5366 2 13 using use VERB cana-5366 2 14 lie lie NOUN cana-5366 2 15 symmetry symmetry NOUN cana-5366 2 16 analysis analysis NOUN cana-5366 2 17 1amara 1amara NUM cana-5366 2 18 chandoul chandoul NOUN cana-5366 2 19 , , PUNCT cana-5366 2 20 2alanod 2alanod NUM cana-5366 2 21 m. m. NOUN cana-5366 2 22 sibih sibih PROPN cana-5366 3 1 1department 1department NUM cana-5366 3 2 of of ADP cana-5366 3 3 computer computer NOUN cana-5366 3 4 science science NOUN cana-5366 3 5 multimedia multimedia NOUN cana-5366 3 6 , , PUNCT cana-5366 3 7 higher high ADJ cana-5366 3 8 institute institute NOUN cana-5366 3 9 of of ADP cana-5366 3 10 informatics informatics PROPN cana-5366 3 11 multimedia multimedia NOUN cana-5366 3 12 of of ADP cana-5366 3 13 sfax sfax NOUN cana-5366 3 14 , , PUNCT cana-5366 3 15 sfax sfax ADJ cana-5366 3 16 university university NOUN cana-5366 3 17 , , PUNCT cana-5366 3 18 sfax sfax NOUN cana-5366 3 19 , , PUNCT cana-5366 3 20 tunisia tunisia PROPN cana-5366 3 21 2department 2department NUM cana-5366 3 22 of of ADP cana-5366 3 23 mathematics mathematic NOUN cana-5366 3 24 , , PUNCT cana-5366 3 25 jamoum jamoum PROPN cana-5366 3 26 university university PROPN cana-5366 3 27 college college NOUN cana-5366 3 28 , , PUNCT cana-5366 3 29 umm umm INTJ cana-5366 3 30 al al PROPN cana-5366 3 31 - - PUNCT cana-5366 3 32 qura qura PROPN cana-5366 3 33 university university PROPN cana-5366 3 34 , , PUNCT cana-5366 3 35 holy holy PROPN cana-5366 3 36 makkah makkah PROPN cana-5366 3 37 21955 21955 NUM cana-5366 3 38 , , PUNCT cana-5366 3 39 saudi saudi PROPN cana-5366 3 40 arabia arabia PROPN cana-5366 3 41 article article NOUN cana-5366 3 42 history history NOUN cana-5366 3 43 : : PUNCT cana-5366 3 44 received receive VERB cana-5366 3 45 : : PUNCT cana-5366 3 46 12 12 NUM cana-5366 3 47 - - SYM cana-5366 3 48 01 01 NUM cana-5366 3 49 - - PUNCT cana-5366 3 50 2025 2025 NUM cana-5366 3 51 revised revise VERB cana-5366 3 52 : : PUNCT cana-5366 3 53 15 15 NUM cana-5366 3 54 - - NUM cana-5366 3 55 02 02 NUM cana-5366 3 56 - - PUNCT cana-5366 3 57 2025 2025 NUM cana-5366 3 58 accepted accept VERB cana-5366 3 59 : : PUNCT cana-5366 3 60 01 01 NUM cana-5366 3 61 - - SYM cana-5366 3 62 03 03 NUM cana-5366 3 63 - - PUNCT cana-5366 3 64 2025 2025 NUM cana-5366 3 65 abstract abstract NOUN cana-5366 3 66 : : PUNCT cana-5366 3 67 this this DET cana-5366 3 68 study study NOUN cana-5366 3 69 explores explore VERB cana-5366 3 70 the the DET cana-5366 3 71 dynamics dynamic NOUN cana-5366 3 72 of of ADP cana-5366 3 73 human human ADJ cana-5366 3 74 finger finger NOUN cana-5366 3 75 movement movement NOUN cana-5366 3 76 through through ADP cana-5366 3 77 the the DET cana-5366 3 78 lens lens NOUN cana-5366 3 79 of of ADP cana-5366 3 80 a a DET cana-5366 3 81 nonlinear nonlinear ADJ cana-5366 3 82 differential differential ADJ cana-5366 3 83 equation equation NOUN cana-5366 3 84 . . PUNCT cana-5366 4 1 by by ADP cana-5366 4 2 employing employ VERB cana-5366 4 3 lie lie NOUN cana-5366 4 4 symmetry symmetry NOUN cana-5366 4 5 analysis analysis NOUN cana-5366 4 6 , , PUNCT cana-5366 4 7 we we PRON cana-5366 4 8 derive derive VERB cana-5366 4 9 the the DET cana-5366 4 10 exact exact ADJ cana-5366 4 11 form form NOUN cana-5366 4 12 of of ADP cana-5366 4 13 the the DET cana-5366 4 14 torque torque NOUN cana-5366 4 15 parameter parameter NOUN cana-5366 4 16 Ο„(ΞΈ Ο„(ΞΈ NUM cana-5366 4 17 ) ) PUNCT cana-5366 4 18 and and CCONJ cana-5366 4 19 obtain obtain VERB cana-5366 4 20 an an DET cana-5366 4 21 analytical analytical ADJ cana-5366 4 22 solution solution NOUN cana-5366 4 23 to to ADP cana-5366 4 24 the the DET cana-5366 4 25 governing govern VERB cana-5366 4 26 equation equation NOUN cana-5366 4 27 . . PUNCT cana-5366 5 1 the the DET cana-5366 5 2 results result NOUN cana-5366 5 3 are be AUX cana-5366 5 4 visualized visualize VERB cana-5366 5 5 for for ADP cana-5366 5 6 various various ADJ cana-5366 5 7 initial initial ADJ cana-5366 5 8 conditions condition NOUN cana-5366 5 9 , , PUNCT cana-5366 5 10 showcasing showcase VERB cana-5366 5 11 the the DET cana-5366 5 12 utility utility NOUN cana-5366 5 13 of of ADP cana-5366 5 14 lie lie NOUN cana-5366 5 15 symmetry symmetry NOUN cana-5366 5 16 methods method NOUN cana-5366 5 17 in in ADP cana-5366 5 18 biomechanical biomechanical ADJ cana-5366 5 19 modelling modelling NOUN cana-5366 5 20 . . PUNCT cana-5366 6 1 this this DET cana-5366 6 2 work work NOUN cana-5366 6 3 not not PART cana-5366 6 4 only only ADV cana-5366 6 5 advances advance VERB cana-5366 6 6 the the DET cana-5366 6 7 understanding understanding NOUN cana-5366 6 8 of of ADP cana-5366 6 9 human human ADJ cana-5366 6 10 finger finger NOUN cana-5366 6 11 dynamics dynamic NOUN cana-5366 6 12 but but CCONJ cana-5366 6 13 also also ADV cana-5366 6 14 provides provide VERB cana-5366 6 15 a a DET cana-5366 6 16 mathematical mathematical ADJ cana-5366 6 17 foundation foundation NOUN cana-5366 6 18 for for ADP cana-5366 6 19 applications application NOUN cana-5366 6 20 in in ADP cana-5366 6 21 robotics robotic NOUN cana-5366 6 22 and and CCONJ cana-5366 6 23 physiology physiology NOUN cana-5366 6 24 . . PUNCT cana-5366 7 1 introduction introduction NOUN cana-5366 7 2 : : PUNCT cana-5366 7 3 the the DET cana-5366 7 4 human human ADJ cana-5366 7 5 finger finger NOUN cana-5366 7 6 is be AUX cana-5366 7 7 a a DET cana-5366 7 8 complex complex ADJ cana-5366 7 9 biomechanical biomechanical ADJ cana-5366 7 10 system system NOUN cana-5366 7 11 whose whose DET cana-5366 7 12 dynamics dynamic NOUN cana-5366 7 13 can can AUX cana-5366 7 14 be be AUX cana-5366 7 15 modeled model VERB cana-5366 7 16 using use VERB cana-5366 7 17 nonlinear nonlinear ADJ cana-5366 7 18 differential differential ADJ cana-5366 7 19 equations equation NOUN cana-5366 7 20 . . PUNCT cana-5366 8 1 understanding understand VERB cana-5366 8 2 these these DET cana-5366 8 3 dynamics dynamic NOUN cana-5366 8 4 is be AUX cana-5366 8 5 crucial crucial ADJ cana-5366 8 6 for for ADP cana-5366 8 7 applications application NOUN cana-5366 8 8 in in ADP cana-5366 8 9 robotics robotic NOUN cana-5366 8 10 , , PUNCT cana-5366 8 11 prosthetics prosthetic NOUN cana-5366 8 12 , , PUNCT cana-5366 8 13 and and CCONJ cana-5366 8 14 rehabilitation rehabilitation NOUN cana-5366 8 15 medicine medicine NOUN cana-5366 8 16 . . PUNCT cana-5366 9 1 objectives objective NOUN cana-5366 9 2 : : PUNCT cana-5366 9 3 this this DET cana-5366 9 4 study study NOUN cana-5366 9 5 aims aim VERB cana-5366 9 6 to to PART cana-5366 9 7 : : PUNCT cana-5366 9 8 ( ( PUNCT cana-5366 9 9 1 1 X cana-5366 9 10 ) ) PUNCT cana-5366 9 11 derive derive VERB cana-5366 9 12 the the DET cana-5366 9 13 exact exact ADJ cana-5366 9 14 form form NOUN cana-5366 9 15 of of ADP cana-5366 9 16 the the DET cana-5366 9 17 torque torque NOUN cana-5366 9 18 parameter parameter PROPN cana-5366 9 19 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 9 20 ) ) PUNCT cana-5366 9 21 governing govern VERB cana-5366 9 22 finger finger NOUN cana-5366 9 23 movement movement NOUN cana-5366 9 24 , , PUNCT cana-5366 9 25 ( ( PUNCT cana-5366 9 26 2 2 X cana-5366 9 27 ) ) PUNCT cana-5366 9 28 obtain obtain VERB cana-5366 9 29 an an DET cana-5366 9 30 analytical analytical ADJ cana-5366 9 31 solution solution NOUN cana-5366 9 32 to to ADP cana-5366 9 33 the the DET cana-5366 9 34 nonlinear nonlinear ADJ cana-5366 9 35 differential differential ADJ cana-5366 9 36 equation equation NOUN cana-5366 9 37 describing describe VERB cana-5366 9 38 finger finger NOUN cana-5366 9 39 dynamics dynamic NOUN cana-5366 9 40 , , PUNCT cana-5366 9 41 and and CCONJ cana-5366 9 42 ( ( PUNCT cana-5366 9 43 3 3 X cana-5366 9 44 ) ) PUNCT cana-5366 9 45 validate validate VERB cana-5366 9 46 the the DET cana-5366 9 47 solution solution NOUN cana-5366 9 48 through through ADP cana-5366 9 49 graphical graphical ADJ cana-5366 9 50 representation representation NOUN cana-5366 9 51 under under ADP cana-5366 9 52 various various ADJ cana-5366 9 53 initial initial ADJ cana-5366 9 54 conditions condition NOUN cana-5366 9 55 . . PUNCT cana-5366 10 1 methods method NOUN cana-5366 10 2 : : PUNCT cana-5366 10 3 we we PRON cana-5366 10 4 employ employ VERB cana-5366 10 5 lie lie NOUN cana-5366 10 6 symmetry symmetry NOUN cana-5366 10 7 analysis analysis NOUN cana-5366 10 8 to to PART cana-5366 10 9 systematically systematically ADV cana-5366 10 10 determine determine VERB cana-5366 10 11 the the DET cana-5366 10 12 form form NOUN cana-5366 10 13 of of ADP cana-5366 10 14 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 10 15 ) ) PUNCT cana-5366 10 16 and and CCONJ cana-5366 10 17 solve solve VERB cana-5366 10 18 the the DET cana-5366 10 19 governing govern VERB cana-5366 10 20 equation equation NOUN cana-5366 10 21 . . PUNCT cana-5366 11 1 this this DET cana-5366 11 2 mathematical mathematical ADJ cana-5366 11 3 approach approach NOUN cana-5366 11 4 identifies identify VERB cana-5366 11 5 invariant invariant ADJ cana-5366 11 6 transformations transformation NOUN cana-5366 11 7 that that PRON cana-5366 11 8 reduce reduce VERB cana-5366 11 9 the the DET cana-5366 11 10 equation equation NOUN cana-5366 11 11 's 's PART cana-5366 11 12 complexity complexity NOUN cana-5366 11 13 , , PUNCT cana-5366 11 14 enabling enable VERB cana-5366 11 15 the the DET cana-5366 11 16 derivation derivation NOUN cana-5366 11 17 of of ADP cana-5366 11 18 an an DET cana-5366 11 19 exact exact ADJ cana-5366 11 20 solution solution NOUN cana-5366 11 21 . . PUNCT cana-5366 12 1 the the DET cana-5366 12 2 methodology methodology NOUN cana-5366 12 3 includes include VERB cana-5366 12 4 symmetry symmetry NOUN cana-5366 12 5 generator generator NOUN cana-5366 12 6 identification identification NOUN cana-5366 12 7 , , PUNCT cana-5366 12 8 equation equation NOUN cana-5366 12 9 reduction reduction NOUN cana-5366 12 10 , , PUNCT cana-5366 12 11 and and CCONJ cana-5366 12 12 analytical analytical ADJ cana-5366 12 13 solution solution NOUN cana-5366 12 14 derivation derivation NOUN cana-5366 12 15 . . PUNCT cana-5366 13 1 results result NOUN cana-5366 13 2 : : PUNCT cana-5366 13 3 the the DET cana-5366 13 4 analysis analysis NOUN cana-5366 13 5 yields yield VERB cana-5366 13 6 a a DET cana-5366 13 7 quadratic quadratic ADJ cana-5366 13 8 form form NOUN cana-5366 13 9 for for ADP cana-5366 13 10 the the DET cana-5366 13 11 torque torque NOUN cana-5366 13 12 parameter,𝜏(πœƒ parameter,𝜏(πœƒ NOUN cana-5366 13 13 ) ) PUNCT cana-5366 13 14 = = PUNCT cana-5366 14 1 π‘˜ π‘˜ X cana-5366 14 2 πœƒ2 πœƒ2 PROPN cana-5366 14 3 βˆ’ βˆ’ NOUN cana-5366 14 4 6πœ‡2πœƒ2 6πœ‡2πœƒ2 NUM cana-5366 14 5 5 5 NUM cana-5366 14 6 ! ! PUNCT cana-5366 14 7 , , PUNCT cana-5366 14 8 and and CCONJ cana-5366 14 9 provides provide VERB cana-5366 14 10 the the DET cana-5366 14 11 exact exact ADJ cana-5366 14 12 solution solution NOUN cana-5366 14 13 πœƒ(𝑑 πœƒ(𝑑 PROPN cana-5366 14 14 ) ) PUNCT cana-5366 15 1 = = PUNCT cana-5366 16 1 6πœ‡2 6πœ‡2 NUM cana-5366 16 2 25π‘˜πΌ[1 25π‘˜πΌ[1 NUM cana-5366 16 3 + + PROPN cana-5366 16 4 5𝐼 5𝐼 NOUN cana-5366 16 5 exp exp NOUN cana-5366 16 6 ( ( PUNCT cana-5366 16 7 πœ‡π‘‘ πœ‡π‘‘ NOUN cana-5366 16 8 5𝐼 5𝐼 PROPN cana-5366 16 9 ) ) PUNCT cana-5366 16 10 ] ] PUNCT cana-5366 17 1 2 2 X cana-5366 17 2 . . PUNCT cana-5366 17 3 graphical graphical ADJ cana-5366 17 4 representations representation NOUN cana-5366 17 5 demonstrate demonstrate VERB cana-5366 17 6 the the DET cana-5366 17 7 solution solution NOUN cana-5366 17 8 's 's PART cana-5366 17 9 behavior behavior NOUN cana-5366 17 10 for for ADP cana-5366 17 11 different different ADJ cana-5366 17 12 initial initial ADJ cana-5366 17 13 conditions condition NOUN cana-5366 17 14 , , PUNCT cana-5366 17 15 showing show VERB cana-5366 17 16 excellent excellent ADJ cana-5366 17 17 agreement agreement NOUN cana-5366 17 18 with with ADP cana-5366 17 19 expected expect VERB cana-5366 17 20 physical physical ADJ cana-5366 17 21 behaviour behaviour NOUN cana-5366 17 22 . . PUNCT cana-5366 18 1 conclusion conclusion NOUN cana-5366 18 2 : : PUNCT cana-5366 18 3 this this DET cana-5366 18 4 work work NOUN cana-5366 18 5 demonstrates demonstrate VERB cana-5366 18 6 that that DET cana-5366 18 7 lie lie NOUN cana-5366 18 8 symmetry symmetry NOUN cana-5366 18 9 analysis analysis NOUN cana-5366 18 10 is be AUX cana-5366 18 11 a a DET cana-5366 18 12 powerful powerful ADJ cana-5366 18 13 tool tool NOUN cana-5366 18 14 for for ADP cana-5366 18 15 solving solve VERB cana-5366 18 16 nonlinear nonlinear ADJ cana-5366 18 17 biomechanical biomechanical ADJ cana-5366 18 18 systems system NOUN cana-5366 18 19 . . PUNCT cana-5366 19 1 the the DET cana-5366 19 2 derived derive VERB cana-5366 19 3 analytical analytical ADJ cana-5366 19 4 solution solution NOUN cana-5366 19 5 provides provide VERB cana-5366 19 6 a a DET cana-5366 19 7 rigorous rigorous ADJ cana-5366 19 8 benchmark benchmark NOUN cana-5366 19 9 for for ADP cana-5366 19 10 numerical numerical ADJ cana-5366 19 11 simulations simulation NOUN cana-5366 19 12 and and CCONJ cana-5366 19 13 experimental experimental ADJ cana-5366 19 14 studies study NOUN cana-5366 19 15 of of ADP cana-5366 19 16 human human ADJ cana-5366 19 17 finger finger NOUN cana-5366 19 18 dynamics dynamic NOUN cana-5366 19 19 , , PUNCT cana-5366 19 20 with with ADP cana-5366 19 21 potential potential ADJ cana-5366 19 22 applications application NOUN cana-5366 19 23 in in ADP cana-5366 19 24 robotic robotic ADJ cana-5366 19 25 hand hand NOUN cana-5366 19 26 design design NOUN cana-5366 19 27 and and CCONJ cana-5366 19 28 rehabilitation rehabilitation NOUN cana-5366 19 29 engineering engineering NOUN cana-5366 19 30 . . PUNCT cana-5366 20 1 keywords keyword NOUN cana-5366 20 2 : : PUNCT cana-5366 20 3 lie lie VERB cana-5366 20 4 symmetry symmetry NOUN cana-5366 20 5 analysis analysis NOUN cana-5366 20 6 , , PUNCT cana-5366 20 7 nonlinear nonlinear ADJ cana-5366 20 8 differential differential ADJ cana-5366 20 9 equations equation NOUN cana-5366 20 10 , , PUNCT cana-5366 20 11 biomechanical biomechanical ADJ cana-5366 20 12 modelling modelling NOUN cana-5366 20 13 , , PUNCT cana-5366 20 14 human human ADJ cana-5366 20 15 finger finger NOUN cana-5366 20 16 dynamics dynamic NOUN cana-5366 20 17 , , PUNCT cana-5366 20 18 analytical analytical ADJ cana-5366 20 19 solutions solution NOUN cana-5366 20 20 , , PUNCT cana-5366 20 21 torque torque NOUN cana-5366 20 22 parameter parameter NOUN cana-5366 20 23 , , PUNCT cana-5366 20 24 robotic robotic ADJ cana-5366 20 25 applications application NOUN cana-5366 20 26 communications communication NOUN cana-5366 20 27 on on ADP cana-5366 20 28 applied apply VERB cana-5366 20 29 nonlinear nonlinear ADJ cana-5366 20 30 analysis analysis NOUN cana-5366 20 31 issn issn NOUN cana-5366 20 32 : : PUNCT cana-5366 20 33 1074 1074 NUM cana-5366 20 34 - - PUNCT cana-5366 20 35 133x 133x NUM cana-5366 20 36 vol vol VERB cana-5366 20 37 32 32 NUM cana-5366 20 38 no no NOUN cana-5366 20 39 . . PUNCT cana-5366 21 1 10s 10 NOUN cana-5366 21 2 ( ( PUNCT cana-5366 21 3 2025 2025 NUM cana-5366 21 4 ) ) PUNCT cana-5366 21 5 2043 2043 NUM cana-5366 21 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 21 7 1 1 X cana-5366 21 8 . . PUNCT cana-5366 22 1 introduction introduction NOUN cana-5366 22 2 the the DET cana-5366 22 3 human human ADJ cana-5366 22 4 finger finger NOUN cana-5366 22 5 is be AUX cana-5366 22 6 a a DET cana-5366 22 7 marvel marvel NOUN cana-5366 22 8 of of ADP cana-5366 22 9 biological biological ADJ cana-5366 22 10 engineering engineering NOUN cana-5366 22 11 , , PUNCT cana-5366 22 12 capable capable ADJ cana-5366 22 13 of of ADP cana-5366 22 14 performing perform VERB cana-5366 22 15 intricate intricate ADJ cana-5366 22 16 and and CCONJ cana-5366 22 17 precise precise ADJ cana-5366 22 18 movements movement NOUN cana-5366 22 19 essential essential ADJ cana-5366 22 20 for for ADP cana-5366 22 21 daily daily ADJ cana-5366 22 22 activities activity NOUN cana-5366 22 23 . . PUNCT cana-5366 23 1 understanding understand VERB cana-5366 23 2 the the DET cana-5366 23 3 dynamics dynamic NOUN cana-5366 23 4 of of ADP cana-5366 23 5 finger finger NOUN cana-5366 23 6 movement movement NOUN cana-5366 23 7 is be AUX cana-5366 23 8 crucial crucial ADJ cana-5366 23 9 not not PART cana-5366 23 10 only only ADV cana-5366 23 11 for for ADP cana-5366 23 12 advancing advance VERB cana-5366 23 13 our our PRON cana-5366 23 14 knowledge knowledge NOUN cana-5366 23 15 of of ADP cana-5366 23 16 human human ADJ cana-5366 23 17 physiology physiology NOUN cana-5366 23 18 but but CCONJ cana-5366 23 19 also also ADV cana-5366 23 20 for for ADP cana-5366 23 21 developing develop VERB cana-5366 23 22 robotic robotic ADJ cana-5366 23 23 systems system NOUN cana-5366 23 24 that that PRON cana-5366 23 25 can can AUX cana-5366 23 26 mimic mimic VERB cana-5366 23 27 human human ADJ cana-5366 23 28 dexterity dexterity NOUN cana-5366 23 29 . . PUNCT cana-5366 24 1 the the DET cana-5366 24 2 movement movement NOUN cana-5366 24 3 of of ADP cana-5366 24 4 the the DET cana-5366 24 5 human human ADJ cana-5366 24 6 finger finger NOUN cana-5366 24 7 is be AUX cana-5366 24 8 governed govern VERB cana-5366 24 9 by by ADP cana-5366 24 10 complex complex ADJ cana-5366 24 11 interactions interaction NOUN cana-5366 24 12 between between ADP cana-5366 24 13 muscles muscle NOUN cana-5366 24 14 , , PUNCT cana-5366 24 15 tendons tendon NOUN cana-5366 24 16 , , PUNCT cana-5366 24 17 and and CCONJ cana-5366 24 18 joints joint NOUN cana-5366 24 19 , , PUNCT cana-5366 24 20 which which PRON cana-5366 24 21 can can AUX cana-5366 24 22 be be AUX cana-5366 24 23 modeled model VERB cana-5366 24 24 using use VERB cana-5366 24 25 nonlinear nonlinear ADJ cana-5366 24 26 differential differential ADJ cana-5366 24 27 equations equation NOUN cana-5366 24 28 ( ( PUNCT cana-5366 24 29 des des PROPN cana-5366 24 30 ) ) PUNCT cana-5366 24 31 . . PUNCT cana-5366 25 1 these these DET cana-5366 25 2 models model NOUN cana-5366 25 3 provide provide VERB cana-5366 25 4 a a DET cana-5366 25 5 mathematical mathematical ADJ cana-5366 25 6 framework framework NOUN cana-5366 25 7 for for ADP cana-5366 25 8 analyzing analyze VERB cana-5366 25 9 the the DET cana-5366 25 10 underlying underlying ADJ cana-5366 25 11 mechanics mechanic NOUN cana-5366 25 12 and and CCONJ cana-5366 25 13 predicting predict VERB cana-5366 25 14 the the DET cana-5366 25 15 behavior behavior NOUN cana-5366 25 16 of of ADP cana-5366 25 17 the the DET cana-5366 25 18 system system NOUN cana-5366 25 19 under under ADP cana-5366 25 20 various various ADJ cana-5366 25 21 conditions condition NOUN cana-5366 25 22 . . PUNCT cana-5366 26 1 nonlinear nonlinear PROPN cana-5366 26 2 des des PROPN cana-5366 26 3 are be AUX cana-5366 26 4 widely widely ADV cana-5366 26 5 used use VERB cana-5366 26 6 in in ADP cana-5366 26 7 biomechanics biomechanic NOUN cana-5366 26 8 to to PART cana-5366 26 9 describe describe VERB cana-5366 26 10 the the DET cana-5366 26 11 dynamics dynamic NOUN cana-5366 26 12 of of ADP cana-5366 26 13 biological biological ADJ cana-5366 26 14 systems system NOUN cana-5366 26 15 [ [ X cana-5366 26 16 7,8 7,8 NUM cana-5366 26 17 ] ] PUNCT cana-5366 26 18 . . PUNCT cana-5366 27 1 the the DET cana-5366 27 2 analytical analytical ADJ cana-5366 27 3 solutions solution NOUN cana-5366 27 4 of of ADP cana-5366 27 5 these these DET cana-5366 27 6 equations equation NOUN cana-5366 27 7 are be AUX cana-5366 27 8 invaluable invaluable ADJ cana-5366 27 9 for for ADP cana-5366 27 10 validating validate VERB cana-5366 27 11 numerical numerical ADJ cana-5366 27 12 simulations simulation NOUN cana-5366 27 13 and and CCONJ cana-5366 27 14 experimental experimental ADJ cana-5366 27 15 data datum NOUN cana-5366 27 16 [ [ X cana-5366 27 17 9,10 9,10 NUM cana-5366 27 18 ] ] PUNCT cana-5366 27 19 . . PUNCT cana-5366 28 1 however however ADV cana-5366 28 2 , , PUNCT cana-5366 28 3 solving solve VERB cana-5366 28 4 nonlinear nonlinear ADJ cana-5366 28 5 des des PROPN cana-5366 28 6 analytically analytically ADV cana-5366 28 7 is be AUX cana-5366 28 8 often often ADV cana-5366 28 9 challenging challenge VERB cana-5366 28 10 due due ADJ cana-5366 28 11 to to ADP cana-5366 28 12 their their PRON cana-5366 28 13 inherent inherent ADJ cana-5366 28 14 complexity complexity NOUN cana-5366 28 15 . . PUNCT cana-5366 29 1 traditional traditional ADJ cana-5366 29 2 methods method NOUN cana-5366 29 3 may may AUX cana-5366 29 4 fail fail VERB cana-5366 29 5 to to PART cana-5366 29 6 yield yield VERB cana-5366 29 7 solutions solution NOUN cana-5366 29 8 , , PUNCT cana-5366 29 9 necessitating necessitate VERB cana-5366 29 10 the the DET cana-5366 29 11 use use NOUN cana-5366 29 12 of of ADP cana-5366 29 13 advanced advanced ADJ cana-5366 29 14 mathematical mathematical ADJ cana-5366 29 15 techniques technique NOUN cana-5366 29 16 such such ADJ cana-5366 29 17 as as ADP cana-5366 29 18 lie lie NOUN cana-5366 29 19 symmetry symmetry NOUN cana-5366 29 20 analysis analysis NOUN cana-5366 29 21 [ [ X cana-5366 29 22 2,3 2,3 NUM cana-5366 29 23 ] ] PUNCT cana-5366 29 24 . . PUNCT cana-5366 30 1 lie lie PROPN cana-5366 30 2 symmetry symmetry NOUN cana-5366 30 3 analysis analysis NOUN cana-5366 30 4 is be AUX cana-5366 30 5 a a DET cana-5366 30 6 powerful powerful ADJ cana-5366 30 7 mathematical mathematical ADJ cana-5366 30 8 tool tool NOUN cana-5366 30 9 for for ADP cana-5366 30 10 solving solve VERB cana-5366 30 11 differential differential ADJ cana-5366 30 12 equations equation NOUN cana-5366 30 13 by by ADP cana-5366 30 14 identifying identify VERB cana-5366 30 15 transformations transformation NOUN cana-5366 30 16 that that PRON cana-5366 30 17 leave leave VERB cana-5366 30 18 the the DET cana-5366 30 19 equation equation NOUN cana-5366 30 20 invariant invariant ADJ cana-5366 30 21 [ [ X cana-5366 30 22 4,5 4,5 NUM cana-5366 30 23 ] ] PUNCT cana-5366 30 24 . . PUNCT cana-5366 31 1 these these DET cana-5366 31 2 transformations transformation NOUN cana-5366 31 3 form form VERB cana-5366 31 4 a a DET cana-5366 31 5 lie lie NOUN cana-5366 31 6 group group NOUN cana-5366 31 7 , , PUNCT cana-5366 31 8 and and CCONJ cana-5366 31 9 their their PRON cana-5366 31 10 generators generator NOUN cana-5366 31 11 are be AUX cana-5366 31 12 known know VERB cana-5366 31 13 as as ADP cana-5366 31 14 lie lie NOUN cana-5366 31 15 symmetries symmetry NOUN cana-5366 31 16 . . PUNCT cana-5366 32 1 the the DET cana-5366 32 2 method method NOUN cana-5366 32 3 has have AUX cana-5366 32 4 been be AUX cana-5366 32 5 successfully successfully ADV cana-5366 32 6 applied apply VERB cana-5366 32 7 to to ADP cana-5366 32 8 various various ADJ cana-5366 32 9 fields field NOUN cana-5366 32 10 , , PUNCT cana-5366 32 11 including include VERB cana-5366 32 12 fluid fluid ADJ cana-5366 32 13 dynamics dynamic NOUN cana-5366 32 14 [ [ X cana-5366 32 15 11 11 NUM cana-5366 32 16 ] ] PUNCT cana-5366 32 17 , , PUNCT cana-5366 32 18 quantum quantum ADJ cana-5366 32 19 mechanics mechanic NOUN cana-5366 32 20 [ [ X cana-5366 32 21 12 12 NUM cana-5366 32 22 ] ] PUNCT cana-5366 32 23 , , PUNCT cana-5366 32 24 and and CCONJ cana-5366 32 25 biomechanics biomechanic NOUN cana-5366 32 26 [ [ X cana-5366 32 27 13 13 NUM cana-5366 32 28 ] ] PUNCT cana-5366 32 29 . . PUNCT cana-5366 33 1 in in ADP cana-5366 33 2 the the DET cana-5366 33 3 context context NOUN cana-5366 33 4 of of ADP cana-5366 33 5 human human ADJ cana-5366 33 6 finger finger NOUN cana-5366 33 7 movement movement NOUN cana-5366 33 8 , , PUNCT cana-5366 33 9 lie lie NOUN cana-5366 33 10 symmetry symmetry NOUN cana-5366 33 11 analysis analysis NOUN cana-5366 33 12 can can AUX cana-5366 33 13 be be AUX cana-5366 33 14 used use VERB cana-5366 33 15 to to PART cana-5366 33 16 determine determine VERB cana-5366 33 17 the the DET cana-5366 33 18 form form NOUN cana-5366 33 19 of of ADP cana-5366 33 20 the the DET cana-5366 33 21 torque torque NOUN cana-5366 33 22 parameter parameter PROPN cana-5366 33 23 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 33 24 ) ) PUNCT cana-5366 33 25 and and CCONJ cana-5366 33 26 derive derive VERB cana-5366 33 27 an an DET cana-5366 33 28 exact exact ADJ cana-5366 33 29 solution solution NOUN cana-5366 33 30 to to ADP cana-5366 33 31 the the DET cana-5366 33 32 governing govern VERB cana-5366 33 33 equation equation NOUN cana-5366 33 34 . . PUNCT cana-5366 34 1 the the DET cana-5366 34 2 torque torque PROPN cana-5366 34 3 parameter parameter PROPN cana-5366 34 4 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 34 5 ) ) PUNCT cana-5366 34 6 plays play VERB cana-5366 34 7 a a DET cana-5366 34 8 crucial crucial ADJ cana-5366 34 9 role role NOUN cana-5366 34 10 in in ADP cana-5366 34 11 determining determine VERB cana-5366 34 12 the the DET cana-5366 34 13 dynamics dynamic NOUN cana-5366 34 14 of of ADP cana-5366 34 15 the the DET cana-5366 34 16 finger finger NOUN cana-5366 34 17 movement movement NOUN cana-5366 34 18 . . PUNCT cana-5366 35 1 previous previous ADJ cana-5366 35 2 studies study NOUN cana-5366 35 3 have have AUX cana-5366 35 4 employed employ VERB cana-5366 35 5 numerical numerical ADJ cana-5366 35 6 and and CCONJ cana-5366 35 7 experimental experimental ADJ cana-5366 35 8 methods method NOUN cana-5366 35 9 to to PART cana-5366 35 10 estimate estimate VERB cana-5366 35 11 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 35 12 ) ) PUNCT cana-5366 36 1 [ [ X cana-5366 36 2 1,15 1,15 NUM cana-5366 36 3 ] ] X cana-5366 36 4 . . PUNCT cana-5366 37 1 however however ADV cana-5366 37 2 , , PUNCT cana-5366 37 3 these these DET cana-5366 37 4 approaches approach NOUN cana-5366 37 5 often often ADV cana-5366 37 6 rely rely VERB cana-5366 37 7 on on ADP cana-5366 37 8 assumptions assumption NOUN cana-5366 37 9 and and CCONJ cana-5366 37 10 approximations approximation NOUN cana-5366 37 11 , , PUNCT cana-5366 37 12 which which PRON cana-5366 37 13 may may AUX cana-5366 37 14 limit limit VERB cana-5366 37 15 their their PRON cana-5366 37 16 accuracy accuracy NOUN cana-5366 37 17 . . PUNCT cana-5366 38 1 in in ADP cana-5366 38 2 contrast contrast NOUN cana-5366 38 3 , , PUNCT cana-5366 38 4 lie lie NOUN cana-5366 38 5 symmetry symmetry NOUN cana-5366 38 6 analysis analysis NOUN cana-5366 38 7 provides provide VERB cana-5366 38 8 a a DET cana-5366 38 9 systematic systematic ADJ cana-5366 38 10 and and CCONJ cana-5366 38 11 rigorous rigorous ADJ cana-5366 38 12 method method NOUN cana-5366 38 13 for for ADP cana-5366 38 14 determining determine VERB cana-5366 38 15 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 38 16 ) ) PUNCT cana-5366 38 17 and and CCONJ cana-5366 38 18 obtaining obtain VERB cana-5366 38 19 an an DET cana-5366 38 20 analytical analytical ADJ cana-5366 38 21 solution solution NOUN cana-5366 38 22 [ [ X cana-5366 38 23 16,17 16,17 NUM cana-5366 38 24 ] ] PUNCT cana-5366 38 25 . . PUNCT cana-5366 39 1 the the DET cana-5366 39 2 importance importance NOUN cana-5366 39 3 of of ADP cana-5366 39 4 understanding understand VERB cana-5366 39 5 human human ADJ cana-5366 39 6 finger finger NOUN cana-5366 39 7 dynamics dynamic NOUN cana-5366 39 8 extends extend VERB cana-5366 39 9 beyond beyond ADP cana-5366 39 10 basic basic ADJ cana-5366 39 11 science science NOUN cana-5366 39 12 . . PUNCT cana-5366 40 1 applications application NOUN cana-5366 40 2 in in ADP cana-5366 40 3 robotics robotic NOUN cana-5366 40 4 and and CCONJ cana-5366 40 5 prosthetics prosthetic NOUN cana-5366 40 6 require require VERB cana-5366 40 7 accurate accurate ADJ cana-5366 40 8 models model NOUN cana-5366 40 9 of of ADP cana-5366 40 10 finger finger NOUN cana-5366 40 11 movement movement NOUN cana-5366 40 12 to to PART cana-5366 40 13 design design VERB cana-5366 40 14 systems system NOUN cana-5366 40 15 that that PRON cana-5366 40 16 can can AUX cana-5366 40 17 replicate replicate VERB cana-5366 40 18 human human ADJ cana-5366 40 19 dexterity dexterity NOUN cana-5366 40 20 [ [ X cana-5366 40 21 18,19 18,19 NUM cana-5366 40 22 ] ] PUNCT cana-5366 40 23 . . PUNCT cana-5366 41 1 analytical analytical ADJ cana-5366 41 2 solutions solution NOUN cana-5366 41 3 derived derive VERB cana-5366 41 4 from from ADP cana-5366 41 5 these these DET cana-5366 41 6 models model NOUN cana-5366 41 7 serve serve VERB cana-5366 41 8 as as ADP cana-5366 41 9 benchmarks benchmark NOUN cana-5366 41 10 for for ADP cana-5366 41 11 validating validate VERB cana-5366 41 12 numerical numerical ADJ cana-5366 41 13 simulations simulation NOUN cana-5366 41 14 and and CCONJ cana-5366 41 15 experimental experimental ADJ cana-5366 41 16 results result NOUN cana-5366 41 17 [ [ X cana-5366 41 18 20,21 20,21 NUM cana-5366 41 19 ] ] PUNCT cana-5366 41 20 . . PUNCT cana-5366 42 1 furthermore furthermore ADV cana-5366 42 2 , , PUNCT cana-5366 42 3 insights insight NOUN cana-5366 42 4 gained gain VERB cana-5366 42 5 from from ADP cana-5366 42 6 studying study VERB cana-5366 42 7 finger finger NOUN cana-5366 42 8 dynamics dynamic NOUN cana-5366 42 9 can can AUX cana-5366 42 10 inform inform VERB cana-5366 42 11 the the DET cana-5366 42 12 development development NOUN cana-5366 42 13 of of ADP cana-5366 42 14 rehabilitation rehabilitation NOUN cana-5366 42 15 strategies strategy NOUN cana-5366 42 16 for for ADP cana-5366 42 17 individuals individual NOUN cana-5366 42 18 with with ADP cana-5366 42 19 motor motor NOUN cana-5366 42 20 impairments impairment NOUN cana-5366 42 21 [ [ X cana-5366 42 22 22,23 22,23 NOUN cana-5366 42 23 ] ] PUNCT cana-5366 42 24 . . PUNCT cana-5366 43 1 in in ADP cana-5366 43 2 this this DET cana-5366 43 3 study study NOUN cana-5366 43 4 , , PUNCT cana-5366 43 5 we we PRON cana-5366 43 6 focus focus VERB cana-5366 43 7 on on ADP cana-5366 43 8 a a DET cana-5366 43 9 specific specific ADJ cana-5366 43 10 nonlinear nonlinear ADJ cana-5366 43 11 differential differential ADJ cana-5366 43 12 equation equation NOUN cana-5366 43 13 that that PRON cana-5366 43 14 models model VERB cana-5366 43 15 the the DET cana-5366 43 16 angular angular ADJ cana-5366 43 17 displacement displacement NOUN cana-5366 43 18 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 43 19 ) ) PUNCT cana-5366 43 20 of of ADP cana-5366 43 21 a a DET cana-5366 43 22 human human ADJ cana-5366 43 23 finger finger NOUN cana-5366 43 24 : : PUNCT cana-5366 43 25 𝐼 𝐼 PROPN cana-5366 43 26 οΏ½ οΏ½ PROPN cana-5366 43 27 ̈ ̈ SYM cana-5366 43 28 οΏ½ οΏ½ NOUN cana-5366 43 29 (𝑑 (𝑑 NOUN cana-5366 43 30 ) ) PUNCT cana-5366 43 31 + + CCONJ cana-5366 43 32 πœ‡ πœ‡ DET cana-5366 43 33 οΏ½ οΏ½ PROPN cana-5366 43 34 Μ‡ Μ‡ PROPN cana-5366 43 35 οΏ½ οΏ½ PROPN cana-5366 43 36 ( ( PUNCT cana-5366 43 37 𝑑 𝑑 NOUN cana-5366 43 38 ) ) PUNCT cana-5366 43 39 βˆ’ βˆ’ PROPN cana-5366 43 40 𝜏(πœƒ(𝑑 𝜏(πœƒ(𝑑 PROPN cana-5366 43 41 ) ) PUNCT cana-5366 43 42 ) ) PUNCT cana-5366 44 1 = = SYM cana-5366 44 2 0 0 NUM cana-5366 44 3 , , PUNCT cana-5366 44 4 𝑑 𝑑 X cana-5366 44 5 > > X cana-5366 44 6 0 0 NUM cana-5366 44 7 where where SCONJ cana-5366 44 8 𝐼 𝐼 PROPN cana-5366 44 9 is be AUX cana-5366 44 10 the the DET cana-5366 44 11 moment moment NOUN cana-5366 44 12 of of ADP cana-5366 44 13 inertia inertia NOUN cana-5366 44 14 , , PUNCT cana-5366 44 15 πœ‡ πœ‡ X cana-5366 44 16 is be AUX cana-5366 44 17 the the DET cana-5366 44 18 coefficient coefficient NOUN cana-5366 44 19 of of ADP cana-5366 44 20 viscosity viscosity NOUN cana-5366 44 21 , , PUNCT cana-5366 44 22 and and CCONJ cana-5366 44 23 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 44 24 ) ) PUNCT cana-5366 44 25 represents represent VERB cana-5366 44 26 the the DET cana-5366 44 27 torque torque NOUN cana-5366 44 28 acting act VERB cana-5366 44 29 on on ADP cana-5366 44 30 the the DET cana-5366 44 31 finger finger NOUN cana-5366 44 32 . . PUNCT cana-5366 45 1 previous previous ADJ cana-5366 45 2 work work NOUN cana-5366 45 3 by by ADP cana-5366 45 4 kosugi kosugi PROPN cana-5366 45 5 et et PROPN cana-5366 45 6 al al PROPN cana-5366 45 7 . . PUNCT cana-5366 46 1 [ [ X cana-5366 46 2 1 1 NUM cana-5366 46 3 ] ] PUNCT cana-5366 46 4 utilized utilize VERB cana-5366 46 5 numerical numerical ADJ cana-5366 46 6 and and CCONJ cana-5366 46 7 experimental experimental ADJ cana-5366 46 8 methods method NOUN cana-5366 46 9 to to PART cana-5366 46 10 solve solve VERB cana-5366 46 11 this this DET cana-5366 46 12 equation equation NOUN cana-5366 46 13 , , PUNCT cana-5366 46 14 deriving derive VERB cana-5366 46 15 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 46 16 ) ) PUNCT cana-5366 46 17 using use VERB cana-5366 46 18 the the DET cana-5366 46 19 principle principle NOUN cana-5366 46 20 of of ADP cana-5366 46 21 virtual virtual ADJ cana-5366 46 22 work work NOUN cana-5366 46 23 . . PUNCT cana-5366 47 1 our our PRON cana-5366 47 2 approach approach NOUN cana-5366 47 3 , , PUNCT cana-5366 47 4 however however ADV cana-5366 47 5 , , PUNCT cana-5366 47 6 employs employ VERB cana-5366 47 7 lie lie NOUN cana-5366 47 8 symmetry symmetry NOUN cana-5366 47 9 analysis analysis NOUN cana-5366 47 10 to to PART cana-5366 47 11 determine determine VERB cana-5366 47 12 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 47 13 ) ) PUNCT cana-5366 47 14 and and CCONJ cana-5366 47 15 obtain obtain VERB cana-5366 47 16 an an DET cana-5366 47 17 exact exact ADJ cana-5366 47 18 solution solution NOUN cana-5366 47 19 , , PUNCT cana-5366 47 20 offering offer VERB cana-5366 47 21 a a DET cana-5366 47 22 complementary complementary ADJ cana-5366 47 23 perspective perspective NOUN cana-5366 47 24 to to ADP cana-5366 47 25 the the DET cana-5366 47 26 existing exist VERB cana-5366 47 27 literature literature NOUN cana-5366 47 28 . . PUNCT cana-5366 48 1 communications communication NOUN cana-5366 48 2 on on ADP cana-5366 48 3 applied apply VERB cana-5366 48 4 nonlinear nonlinear ADJ cana-5366 48 5 analysis analysis NOUN cana-5366 48 6 issn issn NOUN cana-5366 48 7 : : PUNCT cana-5366 48 8 1074 1074 NUM cana-5366 48 9 - - PUNCT cana-5366 48 10 133x 133x NUM cana-5366 48 11 vol vol VERB cana-5366 48 12 32 32 NUM cana-5366 48 13 no no NOUN cana-5366 48 14 . . PUNCT cana-5366 49 1 10s 10 NOUN cana-5366 49 2 ( ( PUNCT cana-5366 49 3 2025 2025 NUM cana-5366 49 4 ) ) PUNCT cana-5366 49 5 2044 2044 NUM cana-5366 49 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 49 7 the the DET cana-5366 49 8 remainder remainder NOUN cana-5366 49 9 of of ADP cana-5366 49 10 this this DET cana-5366 49 11 paper paper NOUN cana-5366 49 12 is be AUX cana-5366 49 13 organized organize VERB cana-5366 49 14 as as SCONJ cana-5366 49 15 follows follow VERB cana-5366 49 16 . . PUNCT cana-5366 50 1 section section NOUN cana-5366 50 2 2 2 NUM cana-5366 50 3 provides provide VERB cana-5366 50 4 a a DET cana-5366 50 5 theoretical theoretical ADJ cana-5366 50 6 background background NOUN cana-5366 50 7 on on ADP cana-5366 50 8 lie lie NOUN cana-5366 50 9 symmetry symmetry NOUN cana-5366 50 10 analysis analysis NOUN cana-5366 50 11 and and CCONJ cana-5366 50 12 its its PRON cana-5366 50 13 application application NOUN cana-5366 50 14 to to ADP cana-5366 50 15 biomechanical biomechanical ADJ cana-5366 50 16 systems system NOUN cana-5366 50 17 . . PUNCT cana-5366 51 1 section section NOUN cana-5366 51 2 3 3 NUM cana-5366 51 3 details detail NOUN cana-5366 51 4 the the DET cana-5366 51 5 methodology methodology NOUN cana-5366 51 6 used use VERB cana-5366 51 7 to to PART cana-5366 51 8 derive derive VERB cana-5366 51 9 the the DET cana-5366 51 10 exact exact ADJ cana-5366 51 11 solution solution NOUN cana-5366 51 12 to to ADP cana-5366 51 13 the the DET cana-5366 51 14 governing govern VERB cana-5366 51 15 equation equation NOUN cana-5366 51 16 . . PUNCT cana-5366 52 1 section section NOUN cana-5366 52 2 4 4 NUM cana-5366 52 3 presents present VERB cana-5366 52 4 the the DET cana-5366 52 5 results result NOUN cana-5366 52 6 and and CCONJ cana-5366 52 7 discusses discuss VERB cana-5366 52 8 their their PRON cana-5366 52 9 implications implication NOUN cana-5366 52 10 for for ADP cana-5366 52 11 understanding understand VERB cana-5366 52 12 human human ADJ cana-5366 52 13 finger finger NOUN cana-5366 52 14 dynamics dynamic NOUN cana-5366 52 15 . . PUNCT cana-5366 53 1 finally finally ADV cana-5366 53 2 , , PUNCT cana-5366 53 3 section section NOUN cana-5366 53 4 5 5 NUM cana-5366 53 5 concludes conclude VERB cana-5366 53 6 the the DET cana-5366 53 7 paper paper NOUN cana-5366 53 8 and and CCONJ cana-5366 53 9 outlines outline VERB cana-5366 53 10 future future ADJ cana-5366 53 11 research research NOUN cana-5366 53 12 directions direction NOUN cana-5366 53 13 . . PUNCT cana-5366 54 1 2 2 X cana-5366 54 2 . . X cana-5366 54 3 theoretical theoretical ADJ cana-5366 54 4 framework framework NOUN cana-5366 54 5 2.1 2.1 NUM cana-5366 54 6 . . PUNCT cana-5366 55 1 lie lie PROPN cana-5366 55 2 symmetry symmetry NOUN cana-5366 55 3 analysis analysis NOUN cana-5366 55 4 lie lie NOUN cana-5366 55 5 symmetry symmetry NOUN cana-5366 55 6 analysis analysis NOUN cana-5366 55 7 is be AUX cana-5366 55 8 a a DET cana-5366 55 9 powerful powerful ADJ cana-5366 55 10 mathematical mathematical ADJ cana-5366 55 11 framework framework NOUN cana-5366 55 12 for for ADP cana-5366 55 13 solving solve VERB cana-5366 55 14 differential differential ADJ cana-5366 55 15 equations equation NOUN cana-5366 55 16 by by ADP cana-5366 55 17 identifying identify VERB cana-5366 55 18 transformations transformation NOUN cana-5366 55 19 that that PRON cana-5366 55 20 preserve preserve VERB cana-5366 55 21 the the DET cana-5366 55 22 structure structure NOUN cana-5366 55 23 of of ADP cana-5366 55 24 the the DET cana-5366 55 25 equation equation NOUN cana-5366 55 26 . . PUNCT cana-5366 56 1 these these DET cana-5366 56 2 transformations transformation NOUN cana-5366 56 3 , , PUNCT cana-5366 56 4 which which PRON cana-5366 56 5 form form VERB cana-5366 56 6 a a DET cana-5366 56 7 lie lie NOUN cana-5366 56 8 group group NOUN cana-5366 56 9 , , PUNCT cana-5366 56 10 are be AUX cana-5366 56 11 generated generate VERB cana-5366 56 12 by by ADP cana-5366 56 13 vector vector NOUN cana-5366 56 14 fields field NOUN cana-5366 56 15 known know VERB cana-5366 56 16 as as ADP cana-5366 56 17 lie lie NOUN cana-5366 56 18 symmetries symmetry NOUN cana-5366 56 19 . . PUNCT cana-5366 57 1 the the DET cana-5366 57 2 method method NOUN cana-5366 57 3 is be AUX cana-5366 57 4 particularly particularly ADV cana-5366 57 5 advantageous advantageous ADJ cana-5366 57 6 for for ADP cana-5366 57 7 addressing address VERB cana-5366 57 8 nonlinear nonlinear ADJ cana-5366 57 9 differential differential ADJ cana-5366 57 10 equations equation NOUN cana-5366 57 11 , , PUNCT cana-5366 57 12 where where SCONJ cana-5366 57 13 conventional conventional ADJ cana-5366 57 14 analytical analytical ADJ cana-5366 57 15 techniques technique NOUN cana-5366 57 16 often often ADV cana-5366 57 17 fall fall VERB cana-5366 57 18 short short ADJ cana-5366 57 19 . . PUNCT cana-5366 58 1 by by ADP cana-5366 58 2 leveraging leverage VERB cana-5366 58 3 the the DET cana-5366 58 4 inherent inherent ADJ cana-5366 58 5 symmetries symmetry NOUN cana-5366 58 6 of of ADP cana-5366 58 7 the the DET cana-5366 58 8 system system NOUN cana-5366 58 9 , , PUNCT cana-5366 58 10 lie lie NOUN cana-5366 58 11 symmetry symmetry NOUN cana-5366 58 12 analysis analysis NOUN cana-5366 58 13 enables enable VERB cana-5366 58 14 the the DET cana-5366 58 15 reduction reduction NOUN cana-5366 58 16 of of ADP cana-5366 58 17 the the DET cana-5366 58 18 equation equation NOUN cana-5366 58 19 's 's PART cana-5366 58 20 complexity complexity NOUN cana-5366 58 21 and and CCONJ cana-5366 58 22 facilitates facilitate VERB cana-5366 58 23 the the DET cana-5366 58 24 derivation derivation NOUN cana-5366 58 25 of of ADP cana-5366 58 26 exact exact ADJ cana-5366 58 27 solutions solution NOUN cana-5366 58 28 . . PUNCT cana-5366 59 1 the the DET cana-5366 59 2 systematic systematic ADJ cana-5366 59 3 application application NOUN cana-5366 59 4 of of ADP cana-5366 59 5 lie lie NOUN cana-5366 59 6 symmetry symmetry NOUN cana-5366 59 7 analysis analysis NOUN cana-5366 59 8 involves involve VERB cana-5366 59 9 the the DET cana-5366 59 10 following follow VERB cana-5366 59 11 key key ADJ cana-5366 59 12 steps step NOUN cana-5366 59 13 : : PUNCT cana-5366 59 14 1 1 X cana-5366 59 15 . . X cana-5366 59 16 identification identification NOUN cana-5366 59 17 of of ADP cana-5366 59 18 symmetry symmetry NOUN cana-5366 59 19 generators generator NOUN cana-5366 59 20 : : PUNCT cana-5366 59 21 determine determine VERB cana-5366 59 22 the the DET cana-5366 59 23 infinitesimal infinitesimal ADJ cana-5366 59 24 generators generator NOUN cana-5366 59 25 of of ADP cana-5366 59 26 the the DET cana-5366 59 27 lie lie NOUN cana-5366 59 28 group group NOUN cana-5366 59 29 that that PRON cana-5366 59 30 leave leave VERB cana-5366 59 31 the the DET cana-5366 59 32 differential differential ADJ cana-5366 59 33 equation equation NOUN cana-5366 59 34 invariant invariant ADJ cana-5366 59 35 . . PUNCT cana-5366 60 1 these these DET cana-5366 60 2 generators generator NOUN cana-5366 60 3 are be AUX cana-5366 60 4 derived derive VERB cana-5366 60 5 by by ADP cana-5366 60 6 solving solve VERB cana-5366 60 7 a a DET cana-5366 60 8 set set NOUN cana-5366 60 9 of of ADP cana-5366 60 10 determining determine VERB cana-5366 60 11 equations equation NOUN cana-5366 60 12 obtained obtain VERB cana-5366 60 13 from from ADP cana-5366 60 14 the the DET cana-5366 60 15 invariance invariance NOUN cana-5366 60 16 condition condition NOUN cana-5366 60 17 . . PUNCT cana-5366 61 1 2 2 X cana-5366 61 2 . . X cana-5366 61 3 reduction reduction NOUN cana-5366 61 4 of of ADP cana-5366 61 5 the the DET cana-5366 61 6 equation equation NOUN cana-5366 61 7 : : PUNCT cana-5366 61 8 utilize utilize VERB cana-5366 61 9 the the DET cana-5366 61 10 identified identify VERB cana-5366 61 11 symmetries symmetry NOUN cana-5366 61 12 to to PART cana-5366 61 13 reduce reduce VERB cana-5366 61 14 the the DET cana-5366 61 15 order order NOUN cana-5366 61 16 of of ADP cana-5366 61 17 the the DET cana-5366 61 18 differential differential ADJ cana-5366 61 19 equation equation NOUN cana-5366 61 20 . . PUNCT cana-5366 62 1 this this DET cana-5366 62 2 step step NOUN cana-5366 62 3 often often ADV cana-5366 62 4 involves involve VERB cana-5366 62 5 transforming transform VERB cana-5366 62 6 the the DET cana-5366 62 7 equation equation NOUN cana-5366 62 8 into into ADP cana-5366 62 9 a a DET cana-5366 62 10 simpler simple ADJ cana-5366 62 11 form form NOUN cana-5366 62 12 , , PUNCT cana-5366 62 13 such such ADJ cana-5366 62 14 as as ADP cana-5366 62 15 an an DET cana-5366 62 16 ordinary ordinary ADJ cana-5366 62 17 differential differential ADJ cana-5366 62 18 equation equation NOUN cana-5366 62 19 ( ( PUNCT cana-5366 62 20 ode ode PROPN cana-5366 62 21 ) ) PUNCT cana-5366 62 22 of of ADP cana-5366 62 23 lower low ADJ cana-5366 62 24 order order NOUN cana-5366 62 25 or or CCONJ cana-5366 62 26 a a DET cana-5366 62 27 partial partial ADJ cana-5366 62 28 differential differential NOUN cana-5366 62 29 equation equation NOUN cana-5366 62 30 ( ( PUNCT cana-5366 62 31 pde pde NOUN cana-5366 62 32 ) ) PUNCT cana-5366 62 33 with with ADP cana-5366 62 34 fewer few ADJ cana-5366 62 35 independent independent ADJ cana-5366 62 36 variables variable NOUN cana-5366 62 37 . . PUNCT cana-5366 63 1 3 3 X cana-5366 63 2 . . X cana-5366 63 3 solution solution NOUN cana-5366 63 4 of of ADP cana-5366 63 5 the the DET cana-5366 63 6 reduced reduce VERB cana-5366 63 7 equation equation NOUN cana-5366 63 8 : : PUNCT cana-5366 63 9 solve solve VERB cana-5366 63 10 the the DET cana-5366 63 11 reduced reduce VERB cana-5366 63 12 equation equation NOUN cana-5366 63 13 to to PART cana-5366 63 14 obtain obtain VERB cana-5366 63 15 an an DET cana-5366 63 16 exact exact ADJ cana-5366 63 17 solution solution NOUN cana-5366 63 18 . . PUNCT cana-5366 64 1 the the DET cana-5366 64 2 reduced reduce VERB cana-5366 64 3 equation equation NOUN cana-5366 64 4 is be AUX cana-5366 64 5 typically typically ADV cana-5366 64 6 more more ADV cana-5366 64 7 tractable tractable ADJ cana-5366 64 8 and and CCONJ cana-5366 64 9 can can AUX cana-5366 64 10 often often ADV cana-5366 64 11 be be AUX cana-5366 64 12 integrated integrate VERB cana-5366 64 13 directly directly ADV cana-5366 64 14 or or CCONJ cana-5366 64 15 solved solve VERB cana-5366 64 16 using use VERB cana-5366 64 17 standard standard ADJ cana-5366 64 18 techniques technique NOUN cana-5366 64 19 . . PUNCT cana-5366 65 1 2.2 2.2 NUM cana-5366 65 2 . . PUNCT cana-5366 65 3 application application NOUN cana-5366 65 4 to to ADP cana-5366 65 5 finger finger NOUN cana-5366 65 6 movement movement NOUN cana-5366 65 7 the the DET cana-5366 65 8 dynamics dynamic NOUN cana-5366 65 9 of of ADP cana-5366 65 10 human human ADJ cana-5366 65 11 finger finger NOUN cana-5366 65 12 movement movement NOUN cana-5366 65 13 can can AUX cana-5366 65 14 be be AUX cana-5366 65 15 modeled model VERB cana-5366 65 16 as as ADP cana-5366 65 17 a a DET cana-5366 65 18 nonlinear nonlinear ADJ cana-5366 65 19 dynamical dynamical ADJ cana-5366 65 20 system system NOUN cana-5366 65 21 , , PUNCT cana-5366 65 22 where where SCONJ cana-5366 65 23 the the DET cana-5366 65 24 torque torque NOUN cana-5366 65 25 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 65 26 ) ) PUNCT cana-5366 65 27 plays play VERB cana-5366 65 28 a a DET cana-5366 65 29 central central ADJ cana-5366 65 30 role role NOUN cana-5366 65 31 in in ADP cana-5366 65 32 governing govern VERB cana-5366 65 33 the the DET cana-5366 65 34 system system NOUN cana-5366 65 35 's 's PART cana-5366 65 36 behavior behavior NOUN cana-5366 65 37 . . PUNCT cana-5366 66 1 applying apply VERB cana-5366 66 2 lie lie NOUN cana-5366 66 3 symmetry symmetry NOUN cana-5366 66 4 analysis analysis NOUN cana-5366 66 5 to to ADP cana-5366 66 6 this this DET cana-5366 66 7 model model NOUN cana-5366 66 8 involves involve VERB cana-5366 66 9 determining determine VERB cana-5366 66 10 the the DET cana-5366 66 11 functional functional ADJ cana-5366 66 12 form form NOUN cana-5366 66 13 of of ADP cana-5366 66 14 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 66 15 ) ) PUNCT cana-5366 66 16 and and CCONJ cana-5366 66 17 solving solve VERB cana-5366 66 18 the the DET cana-5366 66 19 resulting result VERB cana-5366 66 20 governing govern VERB cana-5366 66 21 equation equation NOUN cana-5366 66 22 . . PUNCT cana-5366 67 1 this this DET cana-5366 67 2 approach approach NOUN cana-5366 67 3 not not PART cana-5366 67 4 only only ADV cana-5366 67 5 provides provide VERB cana-5366 67 6 a a DET cana-5366 67 7 rigorous rigorous ADJ cana-5366 67 8 method method NOUN cana-5366 67 9 for for ADP cana-5366 67 10 deriving derive VERB cana-5366 67 11 an an DET cana-5366 67 12 analytical analytical ADJ cana-5366 67 13 solution solution NOUN cana-5366 67 14 but but CCONJ cana-5366 67 15 also also ADV cana-5366 67 16 offers offer VERB cana-5366 67 17 insights insight NOUN cana-5366 67 18 into into ADP cana-5366 67 19 the the DET cana-5366 67 20 underlying underlying ADJ cana-5366 67 21 physical physical ADJ cana-5366 67 22 mechanisms mechanism NOUN cana-5366 67 23 of of ADP cana-5366 67 24 finger finger NOUN cana-5366 67 25 movement movement NOUN cana-5366 67 26 . . PUNCT cana-5366 68 1 by by ADP cana-5366 68 2 identifying identify VERB cana-5366 68 3 the the DET cana-5366 68 4 symmetries symmetry NOUN cana-5366 68 5 of of ADP cana-5366 68 6 the the DET cana-5366 68 7 governing govern VERB cana-5366 68 8 equation equation NOUN cana-5366 68 9 , , PUNCT cana-5366 68 10 we we PRON cana-5366 68 11 can can AUX cana-5366 68 12 reduce reduce VERB cana-5366 68 13 its its PRON cana-5366 68 14 complexity complexity NOUN cana-5366 68 15 and and CCONJ cana-5366 68 16 derive derive VERB cana-5366 68 17 an an DET cana-5366 68 18 exact exact ADJ cana-5366 68 19 solution solution NOUN cana-5366 68 20 that that PRON cana-5366 68 21 captures capture VERB cana-5366 68 22 the the DET cana-5366 68 23 system system NOUN cana-5366 68 24 's 's PART cana-5366 68 25 dynamics dynamic NOUN cana-5366 68 26 . . PUNCT cana-5366 69 1 this this DET cana-5366 69 2 analytical analytical ADJ cana-5366 69 3 solution solution NOUN cana-5366 69 4 serves serve VERB cana-5366 69 5 as as ADP cana-5366 69 6 a a DET cana-5366 69 7 benchmark benchmark NOUN cana-5366 69 8 for for ADP cana-5366 69 9 validating validate VERB cana-5366 69 10 numerical numerical ADJ cana-5366 69 11 simulations simulation NOUN cana-5366 69 12 and and CCONJ cana-5366 69 13 experimental experimental ADJ cana-5366 69 14 observations observation NOUN cana-5366 69 15 , , PUNCT cana-5366 69 16 ensuring ensure VERB cana-5366 69 17 the the DET cana-5366 69 18 accuracy accuracy NOUN cana-5366 69 19 and and CCONJ cana-5366 69 20 reliability reliability NOUN cana-5366 69 21 of of ADP cana-5366 69 22 the the DET cana-5366 69 23 model model NOUN cana-5366 69 24 . . PUNCT cana-5366 70 1 furthermore furthermore ADV cana-5366 70 2 , , PUNCT cana-5366 70 3 the the DET cana-5366 70 4 application application NOUN cana-5366 70 5 of of ADP cana-5366 70 6 lie lie NOUN cana-5366 70 7 symmetry symmetry NOUN cana-5366 70 8 analysis analysis NOUN cana-5366 70 9 to to ADP cana-5366 70 10 biomechanical biomechanical ADJ cana-5366 70 11 systems system NOUN cana-5366 70 12 , , PUNCT cana-5366 70 13 such such ADJ cana-5366 70 14 as as ADP cana-5366 70 15 finger finger NOUN cana-5366 70 16 movement movement NOUN cana-5366 70 17 , , PUNCT cana-5366 70 18 highlights highlight VERB cana-5366 70 19 the the DET cana-5366 70 20 versatility versatility NOUN cana-5366 70 21 of of ADP cana-5366 70 22 the the DET cana-5366 70 23 method method NOUN cana-5366 70 24 and and CCONJ cana-5366 70 25 its its PRON cana-5366 70 26 potential potential NOUN cana-5366 70 27 for for ADP cana-5366 70 28 advancing advance VERB cana-5366 70 29 our our PRON cana-5366 70 30 understanding understanding NOUN cana-5366 70 31 of of ADP cana-5366 70 32 complex complex ADJ cana-5366 70 33 biological biological ADJ cana-5366 70 34 processes process NOUN cana-5366 70 35 . . PUNCT cana-5366 71 1 communications communication NOUN cana-5366 71 2 on on ADP cana-5366 71 3 applied apply VERB cana-5366 71 4 nonlinear nonlinear ADJ cana-5366 71 5 analysis analysis NOUN cana-5366 71 6 issn issn NOUN cana-5366 71 7 : : PUNCT cana-5366 71 8 1074 1074 NUM cana-5366 71 9 - - PUNCT cana-5366 71 10 133x 133x NUM cana-5366 71 11 vol vol VERB cana-5366 71 12 32 32 NUM cana-5366 71 13 no no NOUN cana-5366 71 14 . . PUNCT cana-5366 72 1 10s 10 NOUN cana-5366 72 2 ( ( PUNCT cana-5366 72 3 2025 2025 NUM cana-5366 72 4 ) ) PUNCT cana-5366 72 5 2045 2045 NUM cana-5366 72 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 72 7 3 3 X cana-5366 72 8 . . PUNCT cana-5366 72 9 methodology methodology NOUN cana-5366 72 10 3.1 3.1 NUM cana-5366 72 11 . . PUNCT cana-5366 72 12 determining determine VERB cana-5366 72 13 the the DET cana-5366 72 14 symmetry symmetry NOUN cana-5366 72 15 generators generator NOUN cana-5366 72 16 the the DET cana-5366 72 17 lie lie NOUN cana-5366 72 18 symmetry symmetry NOUN cana-5366 72 19 analysis analysis NOUN cana-5366 72 20 begins begin VERB cana-5366 72 21 with with ADP cana-5366 72 22 the the DET cana-5366 72 23 identification identification NOUN cana-5366 72 24 of of ADP cana-5366 72 25 symmetry symmetry NOUN cana-5366 72 26 generators generator NOUN cana-5366 72 27 for for ADP cana-5366 72 28 the the DET cana-5366 72 29 given give VERB cana-5366 72 30 nonlinear nonlinear ADJ cana-5366 72 31 differential differential ADJ cana-5366 72 32 equation equation NOUN cana-5366 72 33 . . PUNCT cana-5366 73 1 these these DET cana-5366 73 2 generators generator NOUN cana-5366 73 3 are be AUX cana-5366 73 4 vector vector NOUN cana-5366 73 5 fields field NOUN cana-5366 73 6 that that PRON cana-5366 73 7 leave leave VERB cana-5366 73 8 the the DET cana-5366 73 9 differential differential ADJ cana-5366 73 10 equation equation NOUN cana-5366 73 11 invariant invariant ADJ cana-5366 73 12 under under ADP cana-5366 73 13 a a DET cana-5366 73 14 continuous continuous ADJ cana-5366 73 15 group group NOUN cana-5366 73 16 of of ADP cana-5366 73 17 transformations transformation NOUN cana-5366 73 18 . . PUNCT cana-5366 74 1 for for ADP cana-5366 74 2 the the DET cana-5366 74 3 governing govern VERB cana-5366 74 4 equation equation NOUN cana-5366 74 5 of of ADP cana-5366 74 6 human human ADJ cana-5366 74 7 finger finger NOUN cana-5366 74 8 movement movement NOUN cana-5366 74 9 , , PUNCT cana-5366 74 10 𝐼 𝐼 PROPN cana-5366 74 11 οΏ½ οΏ½ PROPN cana-5366 74 12 ̈ ̈ SYM cana-5366 74 13 οΏ½ οΏ½ NOUN cana-5366 74 14 (𝑑 (𝑑 NOUN cana-5366 74 15 ) ) PUNCT cana-5366 74 16 + + CCONJ cana-5366 74 17 πœ‡ πœ‡ DET cana-5366 74 18 οΏ½ οΏ½ PROPN cana-5366 74 19 Μ‡ Μ‡ PROPN cana-5366 74 20 οΏ½ οΏ½ PROPN cana-5366 74 21 ( ( PUNCT cana-5366 74 22 𝑑 𝑑 NOUN cana-5366 74 23 ) ) PUNCT cana-5366 74 24 βˆ’ βˆ’ PROPN cana-5366 74 25 𝜏(πœƒ(𝑑 𝜏(πœƒ(𝑑 PROPN cana-5366 74 26 ) ) PUNCT cana-5366 74 27 ) ) PUNCT cana-5366 75 1 = = SYM cana-5366 75 2 0 0 NUM cana-5366 75 3 , , PUNCT cana-5366 75 4 𝑑 𝑑 X cana-5366 75 5 > > X cana-5366 75 6 0 0 NUM cana-5366 75 7 , , PUNCT cana-5366 75 8 we we PRON cana-5366 75 9 seek seek VERB cana-5366 75 10 the the DET cana-5366 75 11 generator generator NOUN cana-5366 75 12 of of ADP cana-5366 75 13 lie lie NOUN cana-5366 75 14 point point NOUN cana-5366 75 15 symmetries symmetry NOUN cana-5366 75 16 in in ADP cana-5366 75 17 the the DET cana-5366 75 18 form form NOUN cana-5366 75 19 : : PUNCT cana-5366 75 20 𝑋 𝑋 PROPN cana-5366 75 21 = = PUNCT cana-5366 75 22 πœ‰(𝑑 πœ‰(𝑑 PROPN cana-5366 75 23 , , PUNCT cana-5366 75 24 πœƒ πœƒ NOUN cana-5366 75 25 ) ) PUNCT cana-5366 75 26 πœ• πœ• PROPN cana-5366 75 27 πœ•π‘‘ πœ•π‘‘ PROPN cana-5366 75 28 + + CCONJ cana-5366 75 29 πœ‚(𝑑 πœ‚(𝑑 PROPN cana-5366 75 30 , , PUNCT cana-5366 75 31 πœƒ πœƒ NOUN cana-5366 75 32 ) ) PUNCT cana-5366 75 33 πœ• πœ• NOUN cana-5366 75 34 πœ•πœƒ πœ•πœƒ NOUN cana-5366 75 35 , , PUNCT cana-5366 75 36 where where SCONJ cana-5366 75 37 πœ‰(𝑑 πœ‰(𝑑 PROPN cana-5366 75 38 , , PUNCT cana-5366 75 39 πœƒ πœƒ NOUN cana-5366 75 40 ) ) PUNCT cana-5366 75 41 and and CCONJ cana-5366 75 42 πœ‚(𝑑 πœ‚(𝑑 NOUN cana-5366 75 43 , , PUNCT cana-5366 75 44 πœƒ πœƒ X cana-5366 75 45 ) ) PUNCT cana-5366 75 46 are be AUX cana-5366 75 47 functions function NOUN cana-5366 75 48 to to PART cana-5366 75 49 be be AUX cana-5366 75 50 determined determine VERB cana-5366 75 51 . . PUNCT cana-5366 76 1 the the DET cana-5366 76 2 operator operator NOUN cana-5366 76 3 𝑋 𝑋 PROPN cana-5366 76 4 is be AUX cana-5366 76 5 a a DET cana-5366 76 6 symmetry symmetry NOUN cana-5366 76 7 generator generator NOUN cana-5366 76 8 if if SCONJ cana-5366 76 9 it it PRON cana-5366 76 10 satisfies satisfy VERB cana-5366 76 11 the the DET cana-5366 76 12 invariance invariance NOUN cana-5366 76 13 condition condition NOUN cana-5366 76 14 : : PUNCT cana-5366 76 15 𝑋[2](𝐼 𝑋[2](𝐼 X cana-5366 76 16 οΏ½ οΏ½ PROPN cana-5366 76 17 ̈ ̈ ADJ cana-5366 76 18 οΏ½ οΏ½ NOUN cana-5366 76 19 + + CCONJ cana-5366 76 20 πœ‡ πœ‡ PRON cana-5366 76 21 οΏ½ οΏ½ PROPN cana-5366 76 22 Μ‡ Μ‡ VERB cana-5366 76 23 οΏ½ οΏ½ PROPN cana-5366 76 24 βˆ’ βˆ’ PROPN cana-5366 76 25 𝜏(πœƒ))|(1 𝜏(πœƒ))|(1 PROPN cana-5366 76 26 ) ) PUNCT cana-5366 76 27 = = PUNCT cana-5366 76 28 0 0 NUM cana-5366 76 29 , , PUNCT cana-5366 76 30 where where SCONJ cana-5366 76 31 𝑋[2 𝑋[2 PROPN cana-5366 76 32 ] ] PUNCT cana-5366 76 33 is be AUX cana-5366 76 34 the the DET cana-5366 76 35 second second ADJ cana-5366 76 36 prolongation prolongation NOUN cana-5366 76 37 of of ADP cana-5366 76 38 𝑋 𝑋 PROPN cana-5366 76 39 , , PUNCT cana-5366 76 40 given give VERB cana-5366 76 41 by by ADP cana-5366 76 42 : : PUNCT cana-5366 76 43 𝑋[2 𝑋[2 PROPN cana-5366 76 44 ] ] PUNCT cana-5366 76 45 = = SYM cana-5366 76 46 πœ‰ πœ‰ PROPN cana-5366 76 47 πœ• πœ• PROPN cana-5366 76 48 πœ•π‘‘ πœ•π‘‘ PROPN cana-5366 76 49 + + ADP cana-5366 76 50 πœ‚ πœ‚ DET cana-5366 76 51 πœ• πœ• NOUN cana-5366 76 52 πœ•πœƒ πœ•πœƒ NOUN cana-5366 76 53 + + CCONJ cana-5366 76 54 𝜁1 𝜁1 PROPN cana-5366 76 55 πœ• πœ• PROPN cana-5366 76 56 πœ• πœ• PROPN cana-5366 76 57 οΏ½ οΏ½ PROPN cana-5366 76 58 Μ‡ Μ‡ PROPN cana-5366 76 59 οΏ½ οΏ½ PROPN cana-5366 76 60 + + CCONJ cana-5366 76 61 𝜁2 𝜁2 PROPN cana-5366 76 62 πœ• πœ• PROPN cana-5366 76 63 πœ• πœ• PROPN cana-5366 76 64 οΏ½ οΏ½ PROPN cana-5366 76 65 ̈ ̈ X cana-5366 76 66 οΏ½ οΏ½ PROPN cana-5366 76 67 . . PUNCT cana-5366 77 1 the the DET cana-5366 77 2 coefficients coefficient NOUN cana-5366 77 3 𝜁1 𝜁1 PROPN cana-5366 77 4 and and CCONJ cana-5366 77 5 𝜁2 𝜁2 NOUN cana-5366 77 6 are be AUX cana-5366 77 7 defined define VERB cana-5366 77 8 as as ADP cana-5366 77 9 : : PUNCT cana-5366 77 10 ΞΆ1 ΞΆ1 PROPN cana-5366 77 11 = = PROPN cana-5366 77 12 Ξ·t Ξ·t ADP cana-5366 77 13 + + NUM cana-5366 77 14 οΏ½ οΏ½ PROPN cana-5366 77 15 Μ‡ Μ‡ PROPN cana-5366 77 16 οΏ½ οΏ½ PROPN cana-5366 77 17 (Ξ·ΞΈ (Ξ·ΞΈ PUNCT cana-5366 77 18 βˆ’ βˆ’ PROPN cana-5366 77 19 ΞΎ ΞΎ PROPN cana-5366 77 20 t t PROPN cana-5366 77 21 ) ) PUNCT cana-5366 78 1 βˆ’ βˆ’ PROPN cana-5366 78 2 οΏ½ οΏ½ PROPN cana-5366 78 3 Μ‡ Μ‡ PROPN cana-5366 78 4 οΏ½ οΏ½ PROPN cana-5366 78 5 2ΞΎΞΈ 2ΞΎΞΈ NOUN cana-5366 78 6 , , PUNCT cana-5366 78 7 ΞΆ2 ΞΆ2 NOUN cana-5366 78 8 = = SYM cana-5366 78 9 Ξ·tt Ξ·tt PROPN cana-5366 78 10 + + SYM cana-5366 78 11 οΏ½ οΏ½ PROPN cana-5366 78 12 Μ‡ Μ‡ PROPN cana-5366 78 13 οΏ½ οΏ½ PROPN cana-5366 78 14 (2Ξ·tΞΈ (2Ξ·tΞΈ NOUN cana-5366 78 15 βˆ’ βˆ’ PROPN cana-5366 78 16 ΞΎ ΞΎ PROPN cana-5366 78 17 tt tt PROPN cana-5366 78 18 ) ) PUNCT cana-5366 78 19 + + PROPN cana-5366 78 20 οΏ½ οΏ½ PROPN cana-5366 78 21 Μ‡ Μ‡ PROPN cana-5366 78 22 οΏ½ οΏ½ NOUN cana-5366 78 23 2(πœ‚πœƒπœƒ 2(πœ‚πœƒπœƒ NOUN cana-5366 78 24 βˆ’ βˆ’ PROPN cana-5366 78 25 2ΞΎtΞΈ 2ΞΎtΞΈ NUM cana-5366 78 26 ) ) PUNCT cana-5366 78 27 βˆ’ βˆ’ PROPN cana-5366 78 28 οΏ½ οΏ½ PROPN cana-5366 78 29 Μ‡ Μ‡ PROPN cana-5366 78 30 οΏ½ οΏ½ PROPN cana-5366 78 31 3πœ‰πœƒπœƒ 3πœ‰πœƒπœƒ PROPN cana-5366 78 32 + + SYM cana-5366 78 33 οΏ½ οΏ½ PROPN cana-5366 78 34 ̈ ̈ SYM cana-5366 78 35 οΏ½ οΏ½ PROPN cana-5366 78 36 (πœ‚πœƒ (πœ‚πœƒ PUNCT cana-5366 78 37 βˆ’ βˆ’ PROPN cana-5366 78 38 2πœ‰ 2πœ‰ NUM cana-5366 78 39 βˆ’ βˆ’ NOUN cana-5366 78 40 3 3 NUM cana-5366 78 41 οΏ½ οΏ½ PROPN cana-5366 78 42 Μ‡ Μ‡ PROPN cana-5366 78 43 οΏ½ οΏ½ NOUN cana-5366 78 44 πœ‰πœƒ πœ‰πœƒ NOUN cana-5366 78 45 ) ) PUNCT cana-5366 78 46 . . PUNCT cana-5366 79 1 these these DET cana-5366 79 2 coefficients coefficient NOUN cana-5366 79 3 account account VERB cana-5366 79 4 for for ADP cana-5366 79 5 the the DET cana-5366 79 6 transformation transformation NOUN cana-5366 79 7 of of ADP cana-5366 79 8 the the DET cana-5366 79 9 first first ADJ cana-5366 79 10 and and CCONJ cana-5366 79 11 second second ADJ cana-5366 79 12 derivatives derivative NOUN cana-5366 79 13 of of ADP cana-5366 79 14 πœƒ πœƒ NOUN cana-5366 79 15 under under ADP cana-5366 79 16 the the DET cana-5366 79 17 symmetry symmetry NOUN cana-5366 79 18 operation operation NOUN cana-5366 79 19 . . PUNCT cana-5366 80 1 3.2 3.2 NUM cana-5366 80 2 . . PUNCT cana-5366 81 1 classifying classify VERB cana-5366 81 2 the the DET cana-5366 81 3 symmetries symmetry NOUN cana-5366 81 4 the the DET cana-5366 81 5 invariance invariance NOUN cana-5366 81 6 condition condition NOUN cana-5366 81 7 leads lead VERB cana-5366 81 8 to to ADP cana-5366 81 9 a a DET cana-5366 81 10 system system NOUN cana-5366 81 11 of of ADP cana-5366 81 12 determining determine VERB cana-5366 81 13 equations equation NOUN cana-5366 81 14 , , PUNCT cana-5366 81 15 which which PRON cana-5366 81 16 are be AUX cana-5366 81 17 partial partial ADJ cana-5366 81 18 differential differential ADJ cana-5366 81 19 equations equation NOUN cana-5366 81 20 ( ( PUNCT cana-5366 81 21 pdes pde NOUN cana-5366 81 22 ) ) PUNCT cana-5366 81 23 for for ADP cana-5366 81 24 πœ‰ πœ‰ NOUN cana-5366 81 25 and and CCONJ cana-5366 81 26 πœ‚. πœ‚. NOUN cana-5366 81 27 these these DET cana-5366 81 28 equations equation NOUN cana-5366 81 29 are be AUX cana-5366 81 30 derived derive VERB cana-5366 81 31 by by ADP cana-5366 81 32 substituting substitute VERB cana-5366 81 33 the the DET cana-5366 81 34 prolongation prolongation NOUN cana-5366 81 35 𝑋[2 𝑋[2 PROPN cana-5366 81 36 ] ] PUNCT cana-5366 81 37 into into ADP cana-5366 81 38 the the DET cana-5366 81 39 invariance invariance NOUN cana-5366 81 40 condition condition NOUN cana-5366 81 41 and and CCONJ cana-5366 81 42 equating equate VERB cana-5366 81 43 the the DET cana-5366 81 44 coefficients coefficient NOUN cana-5366 81 45 of of ADP cana-5366 81 46 like like ADP cana-5366 81 47 terms term NOUN cana-5366 81 48 to to ADP cana-5366 81 49 zero zero NUM cana-5366 81 50 . . PUNCT cana-5366 82 1 the the DET cana-5366 82 2 resulting result VERB cana-5366 82 3 determining determine VERB cana-5366 82 4 equations equation NOUN cana-5366 82 5 are be AUX cana-5366 82 6 : : PUNCT cana-5366 82 7 πœ‰πœƒπœƒ πœ‰πœƒπœƒ X cana-5366 82 8 = = SYM cana-5366 82 9 0 0 NUM cana-5366 82 10 , , PUNCT cana-5366 83 1 2ΞΌΞΎΞΈ 2ΞΌΞΎΞΈ PROPN cana-5366 83 2 βˆ’ βˆ’ PROPN cana-5366 83 3 2iΞΎtΞΈ 2iΞΎtΞΈ NUM cana-5366 83 4 + + CCONJ cana-5366 83 5 iΞ·ΞΈΞΈ iΞ·ΞΈΞΈ ADJ cana-5366 83 6 = = SYM cana-5366 83 7 0 0 NUM cana-5366 83 8 , , PUNCT cana-5366 83 9 τηθ τηθ NOUN cana-5366 83 10 βˆ’ βˆ’ PROPN cana-5366 83 11 τθη τθη NOUN cana-5366 83 12 + + CCONJ cana-5366 83 13 iΞ·tt iΞ·tt PROPN cana-5366 84 1 + + CCONJ cana-5366 84 2 ΞΌΞ·t ΞΌΞ·t NOUN cana-5366 84 3 βˆ’ βˆ’ PROPN cana-5366 85 1 2τξt 2τξt NUM cana-5366 85 2 = = SYM cana-5366 85 3 0 0 NUM cana-5366 85 4 , , PUNCT cana-5366 85 5 2 2 NUM cana-5366 85 6 iΞ·tΞΈ iΞ·tΞΈ AUX cana-5366 85 7 βˆ’ βˆ’ PROPN cana-5366 85 8 iΞΎtt iΞΎtt ADJ cana-5366 85 9 + + CCONJ cana-5366 85 10 ΞΌΞΎt ΞΌΞΎt NOUN cana-5366 85 11 βˆ’ βˆ’ PROPN cana-5366 85 12 3τξ 3τξ NOUN cana-5366 85 13 = = NOUN cana-5366 85 14 0 0 X cana-5366 85 15 . . PUNCT cana-5366 86 1 solving solve VERB cana-5366 86 2 these these DET cana-5366 86 3 equations equation NOUN cana-5366 86 4 , , PUNCT cana-5366 86 5 we we PRON cana-5366 86 6 find find VERB cana-5366 86 7 that that SCONJ cana-5366 86 8 $ $ SYM cana-5366 86 9 \xi$ \xi$ NUM cana-5366 86 10 and and CCONJ cana-5366 86 11 $ $ SYM cana-5366 86 12 \eta$ \eta$ PROPN cana-5366 86 13 can can AUX cana-5366 86 14 be be AUX cana-5366 86 15 expressed express VERB cana-5366 86 16 as as ADP cana-5366 86 17 : : PUNCT cana-5366 86 18 πœ‰ πœ‰ NOUN cana-5366 86 19 = = PUNCT cana-5366 86 20 ΞΈa(t ΞΈa(t X cana-5366 86 21 ) ) PUNCT cana-5366 87 1 + + NUM cana-5366 87 2 b(t b(t NOUN cana-5366 87 3 ) ) PUNCT cana-5366 87 4 , , PUNCT cana-5366 87 5 Ξ· Ξ· PROPN cana-5366 87 6 = = PROPN cana-5366 87 7 ΞΈ2 ΞΈ2 PROPN cana-5366 87 8 ( ( PUNCT cana-5366 87 9 οΏ½ οΏ½ PROPN cana-5366 87 10 Μ‡ Μ‡ PROPN cana-5366 87 11 οΏ½ οΏ½ PROPN cana-5366 87 12 (𝑑 (𝑑 PROPN cana-5366 87 13 ) ) PUNCT cana-5366 87 14 βˆ’ βˆ’ NOUN cana-5366 87 15 πœ‡π‘Ž(𝑑 πœ‡π‘Ž(𝑑 NOUN cana-5366 87 16 ) ) PUNCT cana-5366 87 17 𝐼 𝐼 PROPN cana-5366 87 18 ) ) PUNCT cana-5366 87 19 + + NUM cana-5366 87 20 ΞΈc(t ΞΈc(t NUM cana-5366 87 21 ) ) PUNCT cana-5366 87 22 + + CCONJ cana-5366 87 23 d(t d(t PROPN cana-5366 87 24 ) ) PUNCT cana-5366 87 25 , , PUNCT cana-5366 87 26 communications communication NOUN cana-5366 87 27 on on ADP cana-5366 87 28 applied apply VERB cana-5366 87 29 nonlinear nonlinear ADJ cana-5366 87 30 analysis analysis NOUN cana-5366 87 31 issn issn NOUN cana-5366 87 32 : : PUNCT cana-5366 87 33 1074 1074 NUM cana-5366 87 34 - - PUNCT cana-5366 87 35 133x 133x NUM cana-5366 87 36 vol vol VERB cana-5366 87 37 32 32 NUM cana-5366 87 38 no no NOUN cana-5366 87 39 . . PUNCT cana-5366 88 1 10s 10 NOUN cana-5366 88 2 ( ( PUNCT cana-5366 88 3 2025 2025 NUM cana-5366 88 4 ) ) PUNCT cana-5366 88 5 2046 2046 NUM cana-5366 88 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 88 7 where where SCONJ cana-5366 88 8 π‘Ž(𝑑 π‘Ž(𝑑 NOUN cana-5366 88 9 ) ) PUNCT cana-5366 88 10 , , PUNCT cana-5366 88 11 𝑏(𝑑 𝑏(𝑑 PROPN cana-5366 88 12 ) ) PUNCT cana-5366 88 13 , , PUNCT cana-5366 88 14 𝑐(𝑑 𝑐(𝑑 PROPN cana-5366 88 15 ) ) PUNCT cana-5366 88 16 , , PUNCT cana-5366 88 17 and and CCONJ cana-5366 88 18 𝑑(𝑑 𝑑(𝑑 PROPN cana-5366 88 19 ) ) PUNCT cana-5366 88 20 are be AUX cana-5366 88 21 arbitrary arbitrary ADJ cana-5366 88 22 functions function NOUN cana-5366 88 23 of of ADP cana-5366 88 24 time time NOUN cana-5366 88 25 . . PUNCT cana-5366 89 1 substituting substitute VERB cana-5366 89 2 these these DET cana-5366 89 3 expressions expression NOUN cana-5366 89 4 into into ADP cana-5366 89 5 the the DET cana-5366 89 6 determining determine VERB cana-5366 89 7 equations equation NOUN cana-5366 89 8 yields yield VERB cana-5366 89 9 the the DET cana-5366 89 10 classifying classify VERB cana-5366 89 11 relations relation NOUN cana-5366 89 12 : : PUNCT cana-5366 90 1 [ [ X cana-5366 90 2 πœƒ2 πœƒ2 NOUN cana-5366 90 3 ( ( PUNCT cana-5366 90 4 πœ‡π‘Ž πœ‡π‘Ž ADP cana-5366 90 5 𝐼 𝐼 PROPN cana-5366 90 6 βˆ’ βˆ’ PROPN cana-5366 90 7 οΏ½ οΏ½ PROPN cana-5366 90 8 Μ‡ Μ‡ PROPN cana-5366 90 9 οΏ½ οΏ½ PROPN cana-5366 90 10 ) ) PUNCT cana-5366 90 11 βˆ’ βˆ’ PROPN cana-5366 90 12 πœƒπ‘ πœƒπ‘ PRON cana-5366 90 13 βˆ’ βˆ’ PROPN cana-5366 90 14 𝑑 𝑑 NOUN cana-5366 90 15 ] ] X cana-5366 90 16 π‘‘πœ π‘‘πœ NOUN cana-5366 90 17 π‘‘πœƒ π‘‘πœƒ PROPN cana-5366 90 18 + + CCONJ cana-5366 90 19 ( ( PUNCT cana-5366 90 20 𝑐 𝑐 PROPN cana-5366 90 21 βˆ’ βˆ’ PROPN cana-5366 90 22 2 2 NUM cana-5366 90 23 οΏ½ οΏ½ PROPN cana-5366 90 24 Μ‡ Μ‡ VERB cana-5366 90 25 οΏ½ οΏ½ PROPN cana-5366 90 26 βˆ’ βˆ’ PROPN cana-5366 90 27 2πœƒ 2πœƒ NUM cana-5366 90 28 πœ‡π‘Ž πœ‡π‘Ž ADP cana-5366 90 29 𝐼 𝐼 NOUN cana-5366 90 30 ) ) PUNCT cana-5366 90 31 𝜏 𝜏 X cana-5366 90 32 + + NOUN cana-5366 90 33 πœƒ2 πœƒ2 NOUN cana-5366 90 34 ( ( PUNCT cana-5366 90 35 πΌπ‘Ž πΌπ‘Ž PROPN cana-5366 90 36 βˆ’ βˆ’ PROPN cana-5366 90 37 πœ‡2 πœ‡2 PROPN cana-5366 90 38 οΏ½ οΏ½ PROPN cana-5366 90 39 Μ‡ Μ‡ NOUN cana-5366 90 40 οΏ½ οΏ½ PROPN cana-5366 90 41 𝐼 𝐼 PROPN cana-5366 90 42 ) ) PUNCT cana-5366 90 43 + + CCONJ cana-5366 90 44 πœƒ(πœ‡ πœƒ(πœ‡ NOUN cana-5366 90 45 οΏ½ οΏ½ NOUN cana-5366 90 46 Μ‡ Μ‡ VERB cana-5366 90 47 οΏ½ οΏ½ PROPN cana-5366 90 48 + + CCONJ cana-5366 90 49 𝐼 𝐼 PROPN cana-5366 90 50 οΏ½ οΏ½ PROPN cana-5366 90 51 ̈ ̈ X cana-5366 90 52 οΏ½ οΏ½ PROPN cana-5366 90 53 ) ) PUNCT cana-5366 90 54 + + CCONJ cana-5366 90 55 πœ‡ πœ‡ PRON cana-5366 90 56 οΏ½ οΏ½ PROPN cana-5366 90 57 Μ‡ Μ‡ VERB cana-5366 90 58 οΏ½ οΏ½ PROPN cana-5366 90 59 + + CCONJ cana-5366 90 60 𝐼 𝐼 PROPN cana-5366 90 61 οΏ½ οΏ½ PROPN cana-5366 90 62 ̈ ̈ X cana-5366 90 63 οΏ½ οΏ½ NOUN cana-5366 90 64 = = SYM cana-5366 90 65 0 0 NUM cana-5366 90 66 , , PUNCT cana-5366 90 67 3π‘Žπœ 3π‘Žπœ NOUN cana-5366 90 68 + + NOUN cana-5366 90 69 3πœƒ(πœ‡π‘Ž 3πœƒ(πœ‡π‘Ž NUM cana-5366 90 70 βˆ’ βˆ’ PROPN cana-5366 90 71 𝐼 𝐼 PROPN cana-5366 90 72 οΏ½ οΏ½ PROPN cana-5366 90 73 ̈ ̈ X cana-5366 90 74 οΏ½ οΏ½ NOUN cana-5366 90 75 ) ) PUNCT cana-5366 90 76 + + NUM cana-5366 90 77 𝐼( 𝐼( PROPN cana-5366 90 78 οΏ½ οΏ½ PROPN cana-5366 90 79 ̈ ̈ X cana-5366 90 80 οΏ½ οΏ½ PROPN cana-5366 90 81 βˆ’ βˆ’ PROPN cana-5366 90 82 2 2 NUM cana-5366 90 83 οΏ½ οΏ½ PROPN cana-5366 90 84 Μ‡ Μ‡ PROPN cana-5366 90 85 οΏ½ οΏ½ PROPN cana-5366 90 86 ) ) PUNCT cana-5366 90 87 βˆ’ βˆ’ ADP cana-5366 90 88 πœ‡ πœ‡ ADP cana-5366 90 89 οΏ½ οΏ½ PROPN cana-5366 90 90 Μ‡ Μ‡ NOUN cana-5366 90 91 οΏ½ οΏ½ PROPN cana-5366 90 92 Μ‡ Μ‡ VERB cana-5366 90 93 = = SYM cana-5366 90 94 0 0 NUM cana-5366 90 95 . . PUNCT cana-5366 91 1 these these DET cana-5366 91 2 relations relation NOUN cana-5366 91 3 are be AUX cana-5366 91 4 used use VERB cana-5366 91 5 to to PART cana-5366 91 6 classify classify VERB cana-5366 91 7 the the DET cana-5366 91 8 symmetries symmetry NOUN cana-5366 91 9 and and CCONJ cana-5366 91 10 determine determine VERB cana-5366 91 11 the the DET cana-5366 91 12 form form NOUN cana-5366 91 13 of of ADP cana-5366 91 14 the the DET cana-5366 91 15 torque torque NOUN cana-5366 91 16 parameter parameter NOUN cana-5366 91 17 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 91 18 ) ) PUNCT cana-5366 91 19 . . PUNCT cana-5366 92 1 if if SCONJ cana-5366 92 2 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 92 3 ) ) PUNCT cana-5366 92 4 is be AUX cana-5366 92 5 arbitrary arbitrary ADJ cana-5366 92 6 , , PUNCT cana-5366 92 7 the the DET cana-5366 92 8 principal principal ADJ cana-5366 92 9 lie lie NOUN cana-5366 92 10 algebra algebra NOUN cana-5366 92 11 consists consist VERB cana-5366 92 12 of of ADP cana-5366 92 13 the the DET cana-5366 92 14 time time NOUN cana-5366 92 15 translation translation NOUN cana-5366 92 16 symmetry symmetry NOUN cana-5366 92 17 𝑋 𝑋 PROPN cana-5366 92 18 = = SYM cana-5366 92 19 πœ• πœ• PROPN cana-5366 92 20 πœ•π‘‘ πœ•π‘‘ PROPN cana-5366 92 21 . . PUNCT cana-5366 93 1 for for ADP cana-5366 93 2 nonlinear nonlinear ADJ cana-5366 93 3 dependence dependence NOUN cana-5366 93 4 of of ADP cana-5366 93 5 𝜏 𝜏 NOUN cana-5366 93 6 on on ADP cana-5366 93 7 πœƒ πœƒ NUM cana-5366 93 8 , , PUNCT cana-5366 93 9 we we PRON cana-5366 93 10 impose impose VERB cana-5366 93 11 the the DET cana-5366 93 12 condition condition NOUN cana-5366 93 13 : : PUNCT cana-5366 93 14 𝑑2𝜏 𝑑2𝜏 X cana-5366 93 15 π‘‘πœƒ2 π‘‘πœƒ2 X cana-5366 93 16 β‰  β‰  PROPN cana-5366 93 17 0 0 NUM cana-5366 93 18 . . PUNCT cana-5366 94 1 solving solve VERB cana-5366 94 2 the the DET cana-5366 94 3 classifying classify VERB cana-5366 94 4 relations relation NOUN cana-5366 94 5 under under ADP cana-5366 94 6 this this DET cana-5366 94 7 condition condition NOUN cana-5366 94 8 , , PUNCT cana-5366 94 9 we we PRON cana-5366 94 10 obtain obtain VERB cana-5366 94 11 : : PUNCT cana-5366 94 12 Ο„(ΞΈ Ο„(ΞΈ NUM cana-5366 94 13 ) ) PUNCT cana-5366 94 14 = = PUNCT cana-5366 95 1 kπœƒ2 kπœƒ2 ADJ cana-5366 95 2 βˆ’ βˆ’ PROPN cana-5366 95 3 6πœ‡2 6πœ‡2 NUM cana-5366 95 4 ΞΈ2 ΞΈ2 PROPN cana-5366 95 5 5𝐼 5𝐼 PROPN cana-5366 95 6 , , PUNCT cana-5366 95 7 where where SCONJ cana-5366 95 8 π‘˜ π‘˜ NOUN cana-5366 95 9 is be AUX cana-5366 95 10 an an DET cana-5366 95 11 arbitrary arbitrary ADJ cana-5366 95 12 nonzero nonzero NOUN cana-5366 95 13 constant constant ADJ cana-5366 95 14 . . PUNCT cana-5366 96 1 substituting substitute VERB cana-5366 96 2 this this PRON cana-5366 96 3 into into ADP cana-5366 96 4 the the DET cana-5366 96 5 governing govern VERB cana-5366 96 6 equation equation NOUN cana-5366 96 7 yields yield VERB cana-5366 96 8 : : PUNCT cana-5366 96 9 i i PROPN cana-5366 96 10 οΏ½ οΏ½ PROPN cana-5366 96 11 ̈ ̈ X cana-5366 96 12 οΏ½ οΏ½ NOUN cana-5366 96 13 (t (t NOUN cana-5366 96 14 ) ) PUNCT cana-5366 97 1 + + CCONJ cana-5366 97 2 ΞΌ ΞΌ NUM cana-5366 97 3 οΏ½ οΏ½ PROPN cana-5366 97 4 Μ‡ Μ‡ NOUN cana-5366 97 5 οΏ½ οΏ½ PROPN cana-5366 97 6 (t (t NOUN cana-5366 97 7 ) ) PUNCT cana-5366 98 1 βˆ’ βˆ’ PROPN cana-5366 99 1 kπœƒ2 kπœƒ2 NOUN cana-5366 99 2 + + CCONJ cana-5366 99 3 6πœ‡2 6πœ‡2 NUM cana-5366 99 4 ΞΈ2 ΞΈ2 ADJ cana-5366 99 5 5𝐼 5𝐼 NOUN cana-5366 99 6 = = PROPN cana-5366 99 7 0 0 PROPN cana-5366 99 8 . . PUNCT cana-5366 100 1 the the DET cana-5366 100 2 corresponding correspond VERB cana-5366 100 3 lie lie NOUN cana-5366 100 4 point point NOUN cana-5366 100 5 symmetries symmetry NOUN cana-5366 100 6 are be AUX cana-5366 100 7 : : PUNCT cana-5366 100 8 𝑋1 𝑋1 PROPN cana-5366 100 9 = = SYM cana-5366 100 10 πœ• πœ• PROPN cana-5366 100 11 πœ•π‘‘ πœ•π‘‘ PROPN cana-5366 100 12 , , PUNCT cana-5366 100 13 𝑋2 𝑋2 VERB cana-5366 100 14 = = SYM cana-5366 101 1 5𝐼 5𝐼 NOUN cana-5366 101 2 exp exp NOUN cana-5366 101 3 ( ( PUNCT cana-5366 101 4 πœ‡π‘‘ πœ‡π‘‘ PROPN cana-5366 101 5 5𝐼 5𝐼 PROPN cana-5366 101 6 ) ) PUNCT cana-5366 101 7 πœ• πœ• PROPN cana-5366 101 8 πœ•π‘‘ πœ•π‘‘ NOUN cana-5366 101 9 βˆ’ βˆ’ PROPN cana-5366 101 10 2πœ‡ 2πœ‡ PROPN cana-5366 101 11 exp exp NOUN cana-5366 101 12 ( ( PUNCT cana-5366 101 13 πœ‡π‘‘ πœ‡π‘‘ PROPN cana-5366 101 14 5𝐼 5𝐼 PROPN cana-5366 101 15 ) ) PUNCT cana-5366 101 16 πœƒ πœƒ PROPN cana-5366 101 17 πœ• πœ• PROPN cana-5366 101 18 πœ•π‘‘ πœ•π‘‘ PROPN cana-5366 101 19 , , PUNCT cana-5366 101 20 which which PRON cana-5366 101 21 form form VERB cana-5366 101 22 a a DET cana-5366 101 23 two two NUM cana-5366 101 24 - - PUNCT cana-5366 101 25 dimensional dimensional ADJ cana-5366 101 26 symmetry symmetry NOUN cana-5366 101 27 lie lie NOUN cana-5366 101 28 algebra algebra NOUN cana-5366 101 29 . . PUNCT cana-5366 102 1 3.3 3.3 NUM cana-5366 102 2 . . PUNCT cana-5366 103 1 deriving derive VERB cana-5366 103 2 the the DET cana-5366 103 3 exact exact ADJ cana-5366 103 4 solution solution NOUN cana-5366 103 5 to to PART cana-5366 103 6 derive derive VERB cana-5366 103 7 the the DET cana-5366 103 8 exact exact ADJ cana-5366 103 9 solution solution NOUN cana-5366 103 10 , , PUNCT cana-5366 103 11 we we PRON cana-5366 103 12 use use VERB cana-5366 103 13 the the DET cana-5366 103 14 symmetry symmetry NOUN cana-5366 103 15 generators generator NOUN cana-5366 103 16 to to PART cana-5366 103 17 reduce reduce VERB cana-5366 103 18 the the DET cana-5366 103 19 order order NOUN cana-5366 103 20 of of ADP cana-5366 103 21 the the DET cana-5366 103 22 differential differential ADJ cana-5366 103 23 equation equation NOUN cana-5366 103 24 . . PUNCT cana-5366 104 1 the the DET cana-5366 104 2 characteristic characteristic ADJ cana-5366 104 3 system system NOUN cana-5366 104 4 for for ADP cana-5366 104 5 the the DET cana-5366 104 6 linear linear ADJ cana-5366 104 7 combination combination NOUN cana-5366 104 8 𝑋1 𝑋1 NOUN cana-5366 104 9 + + CCONJ cana-5366 104 10 𝑋2 𝑋2 VERB cana-5366 104 11 is be AUX cana-5366 104 12 : : PUNCT cana-5366 104 13 𝑑𝑑 𝑑𝑑 ADP cana-5366 104 14 1 1 NUM cana-5366 104 15 + + NUM cana-5366 104 16 5𝐼 5𝐼 NOUN cana-5366 104 17 exp exp NOUN cana-5366 104 18 ( ( PUNCT cana-5366 104 19 πœ‡π‘‘ πœ‡π‘‘ NOUN cana-5366 104 20 5𝐼 5𝐼 NOUN cana-5366 104 21 ) ) PUNCT cana-5366 105 1 = = PUNCT cana-5366 105 2 π‘‘πœƒ π‘‘πœƒ ADP cana-5366 105 3 βˆ’2πœ‡ βˆ’2πœ‡ PROPN cana-5366 105 4 exp exp NOUN cana-5366 105 5 ( ( PUNCT cana-5366 105 6 πœ‡π‘‘ πœ‡π‘‘ NOUN cana-5366 105 7 5𝐼 5𝐼 PROPN cana-5366 105 8 ) ) PUNCT cana-5366 105 9 πœƒ πœƒ X cana-5366 105 10 . . PUNCT cana-5366 106 1 solving solve VERB cana-5366 106 2 this this DET cana-5366 106 3 system system NOUN cana-5366 106 4 , , PUNCT cana-5366 106 5 we we PRON cana-5366 106 6 obtain obtain VERB cana-5366 106 7 : : PUNCT cana-5366 106 8 πœƒ(𝑑 πœƒ(𝑑 PROPN cana-5366 106 9 ) ) PUNCT cana-5366 107 1 = = X cana-5366 107 2 𝐾1 𝐾1 PROPN cana-5366 108 1 [ [ PUNCT cana-5366 108 2 1 1 NUM cana-5366 108 3 + + NUM cana-5366 108 4 5𝐼 5𝐼 NOUN cana-5366 108 5 exp exp NOUN cana-5366 108 6 ( ( PUNCT cana-5366 108 7 πœ‡π‘‘ πœ‡π‘‘ PROPN cana-5366 108 8 5𝐼 5𝐼 PROPN cana-5366 108 9 ) ) PUNCT cana-5366 108 10 ] ] PUNCT cana-5366 108 11 2 2 NUM cana-5366 108 12 , , PUNCT cana-5366 108 13 where where SCONJ cana-5366 108 14 𝐾1 𝐾1 NOUN cana-5366 108 15 is be AUX cana-5366 108 16 an an DET cana-5366 108 17 integration integration NOUN cana-5366 108 18 constant constant ADJ cana-5366 108 19 determined determine VERB cana-5366 108 20 by by ADP cana-5366 108 21 the the DET cana-5366 108 22 initial initial ADJ cana-5366 108 23 conditions condition NOUN cana-5366 108 24 . . PUNCT cana-5366 109 1 substituting substitute VERB cana-5366 109 2 this this PRON cana-5366 109 3 into into ADP cana-5366 109 4 the the DET cana-5366 109 5 governing govern VERB cana-5366 109 6 equation equation NOUN cana-5366 109 7 and and CCONJ cana-5366 109 8 solving solving NOUN cana-5366 109 9 for for ADP cana-5366 109 10 𝐾1 𝐾1 PROPN cana-5366 109 11 , , PUNCT cana-5366 109 12 we we PRON cana-5366 109 13 find find VERB cana-5366 109 14 : : PUNCT cana-5366 109 15 25πΌπ‘˜πΎ1 25πΌπ‘˜πΎ1 NUM cana-5366 109 16 βˆ’ βˆ’ NOUN cana-5366 109 17 6πœ‡2 6πœ‡2 NUM cana-5366 109 18 = = SYM cana-5366 109 19 0 0 NUM cana-5366 109 20 . . PUNCT cana-5366 110 1 thus thus ADV cana-5366 110 2 , , PUNCT cana-5366 110 3 the the DET cana-5366 110 4 exact exact ADJ cana-5366 110 5 solution solution NOUN cana-5366 110 6 is be AUX cana-5366 110 7 : : PUNCT cana-5366 110 8 communications communication NOUN cana-5366 110 9 on on ADP cana-5366 110 10 applied apply VERB cana-5366 110 11 nonlinear nonlinear ADJ cana-5366 110 12 analysis analysis NOUN cana-5366 110 13 issn issn NOUN cana-5366 110 14 : : PUNCT cana-5366 110 15 1074 1074 NUM cana-5366 110 16 - - PUNCT cana-5366 110 17 133x 133x NUM cana-5366 110 18 vol vol VERB cana-5366 110 19 32 32 NUM cana-5366 110 20 no no NOUN cana-5366 110 21 . . PUNCT cana-5366 111 1 10s 10 NOUN cana-5366 111 2 ( ( PUNCT cana-5366 111 3 2025 2025 NUM cana-5366 111 4 ) ) PUNCT cana-5366 111 5 2047 2047 NUM cana-5366 111 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 111 7 πœƒ(𝑑 πœƒ(𝑑 PROPN cana-5366 111 8 ) ) PUNCT cana-5366 112 1 = = PUNCT cana-5366 113 1 6πœ‡2 6πœ‡2 NUM cana-5366 113 2 25π‘˜πΌ 25π‘˜πΌ NUM cana-5366 114 1 [ [ X cana-5366 114 2 1 1 NUM cana-5366 114 3 + + NUM cana-5366 114 4 5𝐼 5𝐼 NOUN cana-5366 114 5 exp exp NOUN cana-5366 114 6 ( ( PUNCT cana-5366 114 7 πœ‡π‘‘ πœ‡π‘‘ PROPN cana-5366 114 8 5𝐼 5𝐼 PROPN cana-5366 114 9 ) ) PUNCT cana-5366 114 10 ] ] PUNCT cana-5366 114 11 2 2 X cana-5366 114 12 . . X cana-5366 114 13 for for ADP cana-5366 114 14 graphical graphical ADJ cana-5366 114 15 representation representation NOUN cana-5366 114 16 , , PUNCT cana-5366 114 17 we we PRON cana-5366 114 18 use use VERB cana-5366 114 19 the the DET cana-5366 114 20 parameter parameter NOUN cana-5366 114 21 values value NOUN cana-5366 114 22 𝐼 𝐼 ADP cana-5366 114 23 = = SYM cana-5366 114 24 4.2 4.2 NUM cana-5366 114 25 Γ— Γ— NOUN cana-5366 114 26 10βˆ’3 10βˆ’3 NUM cana-5366 114 27 kg kg NOUN cana-5366 114 28 m m NOUN cana-5366 114 29 2 2 NUM cana-5366 114 30 , , PUNCT cana-5366 114 31 and and CCONJ cana-5366 114 32 πœ‡ πœ‡ X cana-5366 114 33 = = NOUN cana-5366 114 34 0.1 0.1 NUM cana-5366 114 35 from from ADP cana-5366 114 36 [ [ X cana-5366 114 37 1 1 NUM cana-5366 114 38 ] ] PUNCT cana-5366 114 39 . . PUNCT cana-5366 115 1 the the DET cana-5366 115 2 constant constant ADJ cana-5366 115 3 π‘˜ π‘˜ X cana-5366 115 4 is be AUX cana-5366 115 5 determined determine VERB cana-5366 115 6 by by ADP cana-5366 115 7 imposing impose VERB cana-5366 115 8 specific specific ADJ cana-5366 115 9 initial initial ADJ cana-5366 115 10 conditions condition NOUN cana-5366 115 11 , , PUNCT cana-5366 115 12 as as SCONJ cana-5366 115 13 illustrated illustrate VERB cana-5366 115 14 in in ADP cana-5366 115 15 figures figure NOUN cana-5366 115 16 1 1 NUM cana-5366 115 17 and and CCONJ cana-5366 115 18 2 2 NUM cana-5366 115 19 . . NOUN cana-5366 115 20 4 4 NUM cana-5366 115 21 . . X cana-5366 115 22 plotting plot VERB cana-5366 115 23 the the DET cana-5366 115 24 exact exact ADJ cana-5366 115 25 solution solution NOUN cana-5366 115 26 𝜽(𝒕 𝜽(𝒕 NOUN cana-5366 115 27 ) ) PUNCT cana-5366 115 28 the the DET cana-5366 115 29 exact exact ADJ cana-5366 115 30 solution solution NOUN cana-5366 115 31 πœƒ(𝑑 πœƒ(𝑑 PROPN cana-5366 115 32 ) ) PUNCT cana-5366 115 33 is be AUX cana-5366 115 34 given give VERB cana-5366 115 35 by by ADP cana-5366 115 36 : : PUNCT cana-5366 115 37 πœƒ(𝑑 πœƒ(𝑑 PROPN cana-5366 115 38 ) ) PUNCT cana-5366 116 1 = = PUNCT cana-5366 117 1 6πœ‡2 6πœ‡2 NUM cana-5366 117 2 25π‘˜πΌ 25π‘˜πΌ NUM cana-5366 118 1 [ [ X cana-5366 118 2 1 1 NUM cana-5366 118 3 + + NUM cana-5366 118 4 5𝐼 5𝐼 NOUN cana-5366 118 5 exp exp NOUN cana-5366 118 6 ( ( PUNCT cana-5366 118 7 πœ‡π‘‘ πœ‡π‘‘ PROPN cana-5366 118 8 5𝐼 5𝐼 PROPN cana-5366 118 9 ) ) PUNCT cana-5366 118 10 ] ] PUNCT cana-5366 118 11 2 2 NUM cana-5366 118 12 , , PUNCT cana-5366 118 13 for for ADP cana-5366 118 14 the the DET cana-5366 118 15 two two NUM cana-5366 118 16 initial initial ADJ cana-5366 118 17 conditions condition NOUN cana-5366 118 18 πœƒ(0 πœƒ(0 VERB cana-5366 118 19 ) ) PUNCT cana-5366 118 20 = = PUNCT cana-5366 118 21 πœ‹ πœ‹ PRON cana-5366 118 22 18 18 NUM cana-5366 118 23 rad rad NOUN cana-5366 118 24 and and CCONJ cana-5366 118 25 πœƒ(0 πœƒ(0 PROPN cana-5366 118 26 ) ) PUNCT cana-5366 118 27 = = SYM cana-5366 119 1 βˆ’ βˆ’ PROPN cana-5366 119 2 5πœ‹ 5πœ‹ NOUN cana-5366 119 3 18 18 NUM cana-5366 119 4 rad rad NOUN cana-5366 119 5 . . PUNCT cana-5366 120 1 to to PART cana-5366 120 2 plot plot VERB cana-5366 120 3 this this DET cana-5366 120 4 solution solution NOUN cana-5366 120 5 , , PUNCT cana-5366 120 6 we we PRON cana-5366 120 7 first first ADV cana-5366 120 8 need need VERB cana-5366 120 9 to to PART cana-5366 120 10 determine determine VERB cana-5366 120 11 the the DET cana-5366 120 12 constant constant ADJ cana-5366 120 13 π‘˜ π‘˜ NOUN cana-5366 120 14 for for ADP cana-5366 120 15 each each DET cana-5366 120 16 initial initial ADJ cana-5366 120 17 condition condition NOUN cana-5366 120 18 . . PUNCT cana-5366 121 1 this this PRON cana-5366 121 2 is be AUX cana-5366 121 3 done do VERB cana-5366 121 4 by by ADP cana-5366 121 5 solving solve VERB cana-5366 121 6 πœƒ(0 πœƒ(0 NOUN cana-5366 121 7 ) ) PUNCT cana-5366 121 8 for for ADP cana-5366 121 9 π‘˜. π‘˜. ADJ cana-5366 121 10 step step NOUN cana-5366 121 11 1 1 NUM cana-5366 121 12 : : PUNCT cana-5366 121 13 solve solve VERB cana-5366 121 14 for for ADP cana-5366 121 15 π’Œ π’Œ PRON cana-5366 121 16 for for ADP cana-5366 121 17 each each DET cana-5366 121 18 initial initial ADJ cana-5366 121 19 condition condition NOUN cana-5366 121 20 at at ADP cana-5366 121 21 𝑑 𝑑 PROPN cana-5366 121 22 = = SYM cana-5366 121 23 0 0 NUM cana-5366 121 24 , , PUNCT cana-5366 121 25 the the DET cana-5366 121 26 equation equation NOUN cana-5366 121 27 becomes become VERB cana-5366 121 28 : : PUNCT cana-5366 121 29 πœƒ(0 πœƒ(0 X cana-5366 121 30 ) ) PUNCT cana-5366 121 31 = = PUNCT cana-5366 121 32 6πœ‡2 6πœ‡2 NUM cana-5366 121 33 25π‘˜πΌ[1 25π‘˜πΌ[1 NOUN cana-5366 122 1 + + CCONJ cana-5366 122 2 5𝐼]2 5𝐼]2 ADV cana-5366 122 3 . . PUNCT cana-5366 123 1 solving solve VERB cana-5366 123 2 for for ADP cana-5366 123 3 π‘˜ π‘˜ PRON cana-5366 123 4 : : PUNCT cana-5366 123 5 π‘˜ π‘˜ X cana-5366 123 6 = = SYM cana-5366 123 7 6πœ‡2 6πœ‡2 NUM cana-5366 123 8 25π‘˜πΌ[1 25π‘˜πΌ[1 NOUN cana-5366 123 9 + + CCONJ cana-5366 123 10 5𝐼]2πœƒ(0 5𝐼]2πœƒ(0 NUM cana-5366 123 11 ) ) PUNCT cana-5366 123 12 . . PUNCT cana-5366 124 1 substitute substitute VERB cana-5366 124 2 the the DET cana-5366 124 3 given give VERB cana-5366 124 4 values value NOUN cana-5366 124 5 𝐼 𝐼 ADP cana-5366 124 6 = = SYM cana-5366 124 7 4.2 4.2 NUM cana-5366 124 8 Γ— Γ— NOUN cana-5366 124 9 10βˆ’3 10βˆ’3 NUM cana-5366 124 10 kg kg NOUN cana-5366 124 11 m2 m2 PROPN cana-5366 124 12 , , PUNCT cana-5366 124 13 πœ‡ πœ‡ ADP cana-5366 124 14 = = NOUN cana-5366 124 15 0.1 0.1 NUM cana-5366 124 16 , , PUNCT cana-5366 124 17 and and CCONJ cana-5366 124 18 the the DET cana-5366 124 19 two two NUM cana-5366 124 20 initial initial ADJ cana-5366 124 21 conditions condition NOUN cana-5366 124 22 : : PUNCT cana-5366 124 23 1 1 X cana-5366 124 24 . . X cana-5366 124 25 for for ADP cana-5366 124 26 πœƒ(0 πœƒ(0 PROPN cana-5366 124 27 ) ) PUNCT cana-5366 124 28 = = PUNCT cana-5366 125 1 πœ‹ πœ‹ PRON cana-5366 125 2 18 18 NUM cana-5366 125 3 : : PUNCT cana-5366 125 4 π‘˜1 π‘˜1 PROPN cana-5366 125 5 = = SYM cana-5366 126 1 6(0.1)2 6(0.1)2 NUM cana-5366 126 2 25(4.2 25(4.2 NUM cana-5366 126 3 Γ— Γ— NOUN cana-5366 126 4 10βˆ’3)[1 10βˆ’3)[1 NUM cana-5366 126 5 + + SYM cana-5366 126 6 5(4.2 5(4.2 NUM cana-5366 126 7 Γ— Γ— NOUN cana-5366 126 8 10βˆ’3)]2 10βˆ’3)]2 NUM cana-5366 126 9 πœ‹ πœ‹ NOUN cana-5366 126 10 18 18 NUM cana-5366 126 11 . . PUNCT cana-5366 127 1 2 2 X cana-5366 127 2 . . X cana-5366 127 3 for for ADP cana-5366 127 4 πœƒ(0 πœƒ(0 PROPN cana-5366 127 5 ) ) PUNCT cana-5366 127 6 = = SYM cana-5366 128 1 βˆ’ βˆ’ PROPN cana-5366 128 2 5πœ‹ 5πœ‹ NOUN cana-5366 128 3 18 18 NUM cana-5366 128 4 : : PUNCT cana-5366 128 5 π‘˜2 π‘˜2 PROPN cana-5366 128 6 = = PUNCT cana-5366 129 1 6(0.1)2 6(0.1)2 NUM cana-5366 129 2 25(4.2 25(4.2 NUM cana-5366 129 3 Γ— Γ— NOUN cana-5366 129 4 10βˆ’3)[1 10βˆ’3)[1 NUM cana-5366 130 1 + + SYM cana-5366 130 2 5(4.2 5(4.2 NUM cana-5366 130 3 Γ— Γ— NOUN cana-5366 130 4 10βˆ’3)]2 10βˆ’3)]2 NUM cana-5366 130 5 ( ( PUNCT cana-5366 130 6 βˆ’ βˆ’ PROPN cana-5366 130 7 5πœ‹ 5πœ‹ NOUN cana-5366 130 8 18 18 NUM cana-5366 130 9 ) ) PUNCT cana-5366 130 10 . . PUNCT cana-5366 131 1 communications communication NOUN cana-5366 131 2 on on ADP cana-5366 131 3 applied apply VERB cana-5366 131 4 nonlinear nonlinear ADJ cana-5366 131 5 analysis analysis NOUN cana-5366 131 6 issn issn NOUN cana-5366 131 7 : : PUNCT cana-5366 131 8 1074 1074 NUM cana-5366 131 9 - - PUNCT cana-5366 131 10 133x 133x NUM cana-5366 131 11 vol vol VERB cana-5366 131 12 32 32 NUM cana-5366 131 13 no no NOUN cana-5366 131 14 . . PUNCT cana-5366 132 1 10s 10 NOUN cana-5366 132 2 ( ( PUNCT cana-5366 132 3 2025 2025 NUM cana-5366 132 4 ) ) PUNCT cana-5366 132 5 2048 2048 NUM cana-5366 132 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 132 7 step step NOUN cana-5366 132 8 2 2 NUM cana-5366 132 9 : : PUNCT cana-5366 132 10 implement implement VERB cana-5366 132 11 the the DET cana-5366 132 12 solution solution NOUN cana-5366 132 13 5 5 NUM cana-5366 132 14 . . PUNCT cana-5366 133 1 discussion discussion NOUN cana-5366 133 2 the the DET cana-5366 133 3 results result NOUN cana-5366 133 4 of of ADP cana-5366 133 5 this this DET cana-5366 133 6 study study NOUN cana-5366 133 7 demonstrate demonstrate VERB cana-5366 133 8 the the DET cana-5366 133 9 effectiveness effectiveness NOUN cana-5366 133 10 of of ADP cana-5366 133 11 lie lie NOUN cana-5366 133 12 symmetry symmetry NOUN cana-5366 133 13 analysis analysis NOUN cana-5366 133 14 in in ADP cana-5366 133 15 determining determine VERB cana-5366 133 16 the the DET cana-5366 133 17 form form NOUN cana-5366 133 18 of of ADP cana-5366 133 19 the the DET cana-5366 133 20 torque torque NOUN cana-5366 133 21 parameter parameter PROPN cana-5366 133 22 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 133 23 ) ) PUNCT cana-5366 133 24 and and CCONJ cana-5366 133 25 deriving derive VERB cana-5366 133 26 an an DET cana-5366 133 27 exact exact ADJ cana-5366 133 28 solution solution NOUN cana-5366 133 29 to to ADP cana-5366 133 30 the the DET cana-5366 133 31 nonlinear nonlinear ADJ cana-5366 133 32 differential differential ADJ cana-5366 133 33 equation equation NOUN cana-5366 133 34 governing govern VERB cana-5366 133 35 human human ADJ cana-5366 133 36 finger finger NOUN cana-5366 133 37 movement movement NOUN cana-5366 133 38 . . PUNCT cana-5366 134 1 the the DET cana-5366 134 2 quadratic quadratic ADJ cana-5366 134 3 form form NOUN cana-5366 134 4 of of ADP cana-5366 134 5 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 134 6 ) ) PUNCT cana-5366 134 7 , , PUNCT cana-5366 134 8 expressed express VERB cana-5366 134 9 as as ADP cana-5366 134 10 : : PUNCT cana-5366 134 11 πœƒ(𝑑 πœƒ(𝑑 X cana-5366 134 12 ) ) PUNCT cana-5366 135 1 = = SYM cana-5366 135 2 π‘˜πœƒ2 π‘˜πœƒ2 NOUN cana-5366 136 1 βˆ’ βˆ’ NOUN cana-5366 136 2 6πœ‡2πœƒ2 6πœ‡2πœƒ2 NUM cana-5366 136 3 5𝐼 5𝐼 NOUN cana-5366 136 4 , , PUNCT cana-5366 136 5 was be AUX cana-5366 136 6 derived derive VERB cana-5366 136 7 through through ADP cana-5366 136 8 a a DET cana-5366 136 9 systematic systematic ADJ cana-5366 136 10 application application NOUN cana-5366 136 11 of of ADP cana-5366 136 12 lie lie NOUN cana-5366 136 13 symmetry symmetry NOUN cana-5366 136 14 methods method NOUN cana-5366 136 15 . . PUNCT cana-5366 137 1 this this DET cana-5366 137 2 form form NOUN cana-5366 137 3 of of ADP cana-5366 137 4 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 137 5 ) ) PUNCT cana-5366 137 6 allowed allow VERB cana-5366 137 7 us we PRON cana-5366 137 8 to to PART cana-5366 137 9 reduce reduce VERB cana-5366 137 10 the the DET cana-5366 137 11 original original ADJ cana-5366 137 12 nonlinear nonlinear ADJ cana-5366 137 13 differential differential ADJ cana-5366 137 14 equation equation NOUN cana-5366 137 15 to to ADP cana-5366 137 16 a a DET cana-5366 137 17 solvable solvable ADJ cana-5366 137 18 form form NOUN cana-5366 137 19 , , PUNCT cana-5366 137 20 yielding yield VERB cana-5366 137 21 an an DET cana-5366 137 22 exact exact ADJ cana-5366 137 23 analytical analytical ADJ cana-5366 137 24 solution solution NOUN cana-5366 137 25 : : PUNCT cana-5366 137 26 πœƒ(𝑑 πœƒ(𝑑 PROPN cana-5366 137 27 ) ) PUNCT cana-5366 138 1 = = PUNCT cana-5366 139 1 6πœ‡2 6πœ‡2 NUM cana-5366 139 2 25π‘˜πΌ 25π‘˜πΌ NUM cana-5366 140 1 [ [ X cana-5366 140 2 1 1 NUM cana-5366 140 3 + + NUM cana-5366 140 4 5𝐼 5𝐼 NOUN cana-5366 140 5 exp exp NOUN cana-5366 140 6 ( ( PUNCT cana-5366 140 7 πœ‡π‘‘ πœ‡π‘‘ PROPN cana-5366 140 8 5𝐼 5𝐼 PROPN cana-5366 140 9 ) ) PUNCT cana-5366 140 10 ] ] PUNCT cana-5366 140 11 2 2 X cana-5366 140 12 . . X cana-5366 140 13 the the DET cana-5366 140 14 graphical graphical ADJ cana-5366 140 15 representation representation NOUN cana-5366 140 16 of of ADP cana-5366 140 17 this this DET cana-5366 140 18 solution solution NOUN cana-5366 140 19 for for ADP cana-5366 140 20 different different ADJ cana-5366 140 21 initial initial ADJ cana-5366 140 22 conditions condition NOUN cana-5366 140 23 ( ( PUNCT cana-5366 140 24 see see VERB cana-5366 140 25 figures figure NOUN cana-5366 140 26 1 1 NUM cana-5366 140 27 and and CCONJ cana-5366 140 28 2 2 NUM cana-5366 140 29 ) ) PUNCT cana-5366 140 30 highlights highlight NOUN cana-5366 140 31 the the DET cana-5366 140 32 dynamic dynamic ADJ cana-5366 140 33 behavior behavior NOUN cana-5366 140 34 of of ADP cana-5366 140 35 the the DET cana-5366 140 36 system system NOUN cana-5366 140 37 under under ADP cana-5366 140 38 varying vary VERB cana-5366 140 39 starting starting NOUN cana-5366 140 40 points point NOUN cana-5366 140 41 , , PUNCT cana-5366 140 42 providing provide VERB cana-5366 140 43 valuable valuable ADJ cana-5366 140 44 insights insight NOUN cana-5366 140 45 into into ADP cana-5366 140 46 the the DET cana-5366 140 47 transient transient ADJ cana-5366 140 48 and and CCONJ cana-5366 140 49 steady steady ADJ cana-5366 140 50 - - PUNCT cana-5366 140 51 state state NOUN cana-5366 140 52 responses response NOUN cana-5366 140 53 of of ADP cana-5366 140 54 the the DET cana-5366 140 55 finger finger PROPN cana-5366 140 56 movement movement PROPN cana-5366 140 57 model model PROPN cana-5366 140 58 . . PUNCT cana-5366 141 1 a a DET cana-5366 141 2 key key ADJ cana-5366 141 3 observation observation NOUN cana-5366 141 4 from from ADP cana-5366 141 5 this this DET cana-5366 141 6 study study NOUN cana-5366 141 7 is be AUX cana-5366 141 8 the the DET cana-5366 141 9 distinction distinction NOUN cana-5366 141 10 between between ADP cana-5366 141 11 the the DET cana-5366 141 12 form form NOUN cana-5366 141 13 of of ADP cana-5366 141 14 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 141 15 ) ) PUNCT cana-5366 141 16 derived derive VERB cana-5366 141 17 here here ADV cana-5366 141 18 and and CCONJ cana-5366 141 19 that that PRON cana-5366 141 20 proposed propose VERB cana-5366 141 21 by by ADP cana-5366 141 22 kosugi kosugi PROPN cana-5366 141 23 et et PROPN cana-5366 141 24 al al PROPN cana-5366 141 25 . . PUNCT cana-5366 142 1 [ [ X cana-5366 142 2 1 1 NUM cana-5366 142 3 ] ] PUNCT cana-5366 142 4 . . PUNCT cana-5366 143 1 while while SCONJ cana-5366 143 2 kosugi kosugi PROPN cana-5366 143 3 et et PROPN cana-5366 143 4 al al PROPN cana-5366 143 5 . . PROPN cana-5366 143 6 employed employ VERB cana-5366 143 7 numerical numerical ADJ cana-5366 143 8 and and CCONJ cana-5366 143 9 experimental experimental ADJ cana-5366 143 10 methods method NOUN cana-5366 143 11 to to PART cana-5366 143 12 approximate approximate VERB cana-5366 143 13 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 143 14 ) ) PUNCT cana-5366 143 15 , , PUNCT cana-5366 143 16 our our PRON cana-5366 143 17 approach approach NOUN cana-5366 143 18 leverages leverage NOUN cana-5366 143 19 lie lie NOUN cana-5366 143 20 symmetry symmetry NOUN cana-5366 143 21 analysis analysis NOUN cana-5366 143 22 to to PART cana-5366 143 23 derive derive VERB cana-5366 143 24 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 143 25 ) ) PUNCT cana-5366 143 26 analytically analytically ADV cana-5366 143 27 . . PUNCT cana-5366 144 1 this this DET cana-5366 144 2 communications communication NOUN cana-5366 144 3 on on ADP cana-5366 144 4 applied apply VERB cana-5366 144 5 nonlinear nonlinear ADJ cana-5366 144 6 analysis analysis NOUN cana-5366 144 7 issn issn NOUN cana-5366 144 8 : : PUNCT cana-5366 144 9 1074 1074 NUM cana-5366 144 10 - - PUNCT cana-5366 144 11 133x 133x NUM cana-5366 144 12 vol vol VERB cana-5366 144 13 32 32 NUM cana-5366 144 14 no no NOUN cana-5366 144 15 . . PUNCT cana-5366 145 1 10s 10 NOUN cana-5366 145 2 ( ( PUNCT cana-5366 145 3 2025 2025 NUM cana-5366 145 4 ) ) PUNCT cana-5366 145 5 2049 2049 NUM cana-5366 145 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 145 7 difference difference NOUN cana-5366 145 8 in in ADP cana-5366 145 9 methodology methodology NOUN cana-5366 145 10 leads lead VERB cana-5366 145 11 to to ADP cana-5366 145 12 distinct distinct ADJ cana-5366 145 13 expressions expression NOUN cana-5366 145 14 for for ADP cana-5366 145 15 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 145 16 ) ) PUNCT cana-5366 145 17 , , PUNCT cana-5366 145 18 implying imply VERB cana-5366 145 19 that that SCONJ cana-5366 145 20 the the DET cana-5366 145 21 submodel submodel PROPN cana-5366 145 22 ( ( PUNCT cana-5366 145 23 13 13 NUM cana-5366 145 24 ) ) PUNCT cana-5366 145 25 in in ADP cana-5366 145 26 this this DET cana-5366 145 27 study study NOUN cana-5366 145 28 is be AUX cana-5366 145 29 not not PART cana-5366 145 30 directly directly ADV cana-5366 145 31 comparable comparable ADJ cana-5366 145 32 to to ADP cana-5366 145 33 eq eq PROPN cana-5366 145 34 . . PUNCT cana-5366 146 1 ( ( PUNCT cana-5366 146 2 1.2 1.2 NUM cana-5366 146 3 ) ) PUNCT cana-5366 146 4 in in ADP cana-5366 146 5 [ [ X cana-5366 146 6 1 1 NUM cana-5366 146 7 ] ] PUNCT cana-5366 146 8 . . PUNCT cana-5366 147 1 consequently consequently ADV cana-5366 147 2 , , PUNCT cana-5366 147 3 a a DET cana-5366 147 4 direct direct ADJ cana-5366 147 5 comparative comparative ADJ cana-5366 147 6 analysis analysis NOUN cana-5366 147 7 between between ADP cana-5366 147 8 the the DET cana-5366 147 9 two two NUM cana-5366 147 10 models model NOUN cana-5366 147 11 may may AUX cana-5366 147 12 not not PART cana-5366 147 13 be be AUX cana-5366 147 14 appropriate appropriate ADJ cana-5366 147 15 . . PUNCT cana-5366 148 1 instead instead ADV cana-5366 148 2 , , PUNCT cana-5366 148 3 this this DET cana-5366 148 4 discrepancy discrepancy NOUN cana-5366 148 5 underscores underscore VERB cana-5366 148 6 the the DET cana-5366 148 7 need need NOUN cana-5366 148 8 for for SCONJ cana-5366 148 9 future future ADJ cana-5366 148 10 work work NOUN cana-5366 148 11 to to PART cana-5366 148 12 apply apply VERB cana-5366 148 13 lie lie NOUN cana-5366 148 14 symmetry symmetry NOUN cana-5366 148 15 analysis analysis NOUN cana-5366 148 16 to to ADP cana-5366 148 17 the the DET cana-5366 148 18 specific specific ADJ cana-5366 148 19 form form NOUN cana-5366 148 20 of of ADP cana-5366 148 21 eq eq PROPN cana-5366 148 22 . . PUNCT cana-5366 149 1 ( ( PUNCT cana-5366 149 2 1.2 1.2 NUM cana-5366 149 3 ) ) PUNCT cana-5366 149 4 in in ADP cana-5366 149 5 [ [ X cana-5366 149 6 1 1 NUM cana-5366 149 7 ] ] PUNCT cana-5366 149 8 , , PUNCT cana-5366 149 9 enabling enable VERB cana-5366 149 10 a a DET cana-5366 149 11 more more ADV cana-5366 149 12 comprehensive comprehensive ADJ cana-5366 149 13 comparison comparison NOUN cana-5366 149 14 and and CCONJ cana-5366 149 15 validation validation NOUN cana-5366 149 16 of of ADP cana-5366 149 17 the the DET cana-5366 149 18 results result NOUN cana-5366 149 19 . . PUNCT cana-5366 150 1 while while SCONJ cana-5366 150 2 the the DET cana-5366 150 3 current current ADJ cana-5366 150 4 work work NOUN cana-5366 150 5 is be AUX cana-5366 150 6 primarily primarily ADV cana-5366 150 7 motivated motivate VERB cana-5366 150 8 by by ADP cana-5366 150 9 mathematical mathematical ADJ cana-5366 150 10 interest interest NOUN cana-5366 150 11 , , PUNCT cana-5366 150 12 its its PRON cana-5366 150 13 implications implication NOUN cana-5366 150 14 extend extend VERB cana-5366 150 15 to to ADP cana-5366 150 16 practical practical ADJ cana-5366 150 17 applications application NOUN cana-5366 150 18 in in ADP cana-5366 150 19 robotics robotic NOUN cana-5366 150 20 and and CCONJ cana-5366 150 21 physiology physiology NOUN cana-5366 150 22 . . PUNCT cana-5366 151 1 analytical analytical ADJ cana-5366 151 2 solutions solution NOUN cana-5366 151 3 , , PUNCT cana-5366 151 4 such such ADJ cana-5366 151 5 as as ADP cana-5366 151 6 the the DET cana-5366 151 7 one one NUM cana-5366 151 8 derived derive VERB cana-5366 151 9 here here ADV cana-5366 151 10 , , PUNCT cana-5366 151 11 serve serve VERB cana-5366 151 12 as as ADP cana-5366 151 13 critical critical ADJ cana-5366 151 14 benchmarks benchmark NOUN cana-5366 151 15 for for ADP cana-5366 151 16 validating validate VERB cana-5366 151 17 numerical numerical ADJ cana-5366 151 18 simulations simulation NOUN cana-5366 151 19 and and CCONJ cana-5366 151 20 experimental experimental ADJ cana-5366 151 21 data datum NOUN cana-5366 151 22 . . PUNCT cana-5366 152 1 in in ADP cana-5366 152 2 robotics robotic NOUN cana-5366 152 3 , , PUNCT cana-5366 152 4 accurate accurate ADJ cana-5366 152 5 models model NOUN cana-5366 152 6 of of ADP cana-5366 152 7 human human ADJ cana-5366 152 8 finger finger NOUN cana-5366 152 9 dynamics dynamic NOUN cana-5366 152 10 are be AUX cana-5366 152 11 essential essential ADJ cana-5366 152 12 for for ADP cana-5366 152 13 designing design VERB cana-5366 152 14 robotic robotic ADJ cana-5366 152 15 hands hand NOUN cana-5366 152 16 and and CCONJ cana-5366 152 17 prosthetics prosthetic NOUN cana-5366 152 18 that that PRON cana-5366 152 19 replicate replicate VERB cana-5366 152 20 human human ADJ cana-5366 152 21 dexterity dexterity NOUN cana-5366 152 22 [ [ X cana-5366 152 23 18,19 18,19 NUM cana-5366 152 24 ] ] X cana-5366 152 25 . . PUNCT cana-5366 153 1 similarly similarly ADV cana-5366 153 2 , , PUNCT cana-5366 153 3 in in ADP cana-5366 153 4 physiology physiology NOUN cana-5366 153 5 , , PUNCT cana-5366 153 6 understanding understand VERB cana-5366 153 7 the the DET cana-5366 153 8 mechanics mechanic NOUN cana-5366 153 9 of of ADP cana-5366 153 10 finger finger NOUN cana-5366 153 11 movement movement NOUN cana-5366 153 12 can can AUX cana-5366 153 13 inform inform VERB cana-5366 153 14 the the DET cana-5366 153 15 development development NOUN cana-5366 153 16 of of ADP cana-5366 153 17 rehabilitation rehabilitation NOUN cana-5366 153 18 strategies strategy NOUN cana-5366 153 19 for for ADP cana-5366 153 20 individuals individual NOUN cana-5366 153 21 with with ADP cana-5366 153 22 motor motor NOUN cana-5366 153 23 impairments impairment NOUN cana-5366 153 24 [ [ X cana-5366 153 25 22,23 22,23 NOUN cana-5366 153 26 ] ] PUNCT cana-5366 153 27 . . PUNCT cana-5366 154 1 the the DET cana-5366 154 2 exact exact ADJ cana-5366 154 3 solution solution NOUN cana-5366 154 4 provided provide VERB cana-5366 154 5 in in ADP cana-5366 154 6 this this DET cana-5366 154 7 study study NOUN cana-5366 154 8 offers offer VERB cana-5366 154 9 a a DET cana-5366 154 10 foundation foundation NOUN cana-5366 154 11 for for ADP cana-5366 154 12 such such ADJ cana-5366 154 13 applications application NOUN cana-5366 154 14 , , PUNCT cana-5366 154 15 enabling enable VERB cana-5366 154 16 researchers researcher NOUN cana-5366 154 17 to to PART cana-5366 154 18 test test VERB cana-5366 154 19 and and CCONJ cana-5366 154 20 refine refine VERB cana-5366 154 21 their their PRON cana-5366 154 22 models model NOUN cana-5366 154 23 against against ADP cana-5366 154 24 a a DET cana-5366 154 25 mathematically mathematically ADV cana-5366 154 26 rigorous rigorous ADJ cana-5366 154 27 standard standard NOUN cana-5366 154 28 . . PUNCT cana-5366 155 1 furthermore furthermore ADV cana-5366 155 2 , , PUNCT cana-5366 155 3 the the DET cana-5366 155 4 lie lie NOUN cana-5366 155 5 symmetry symmetry NOUN cana-5366 155 6 approach approach NOUN cana-5366 155 7 used use VERB cana-5366 155 8 in in ADP cana-5366 155 9 this this DET cana-5366 155 10 study study NOUN cana-5366 155 11 is be AUX cana-5366 155 12 not not PART cana-5366 155 13 limited limit VERB cana-5366 155 14 to to ADP cana-5366 155 15 the the DET cana-5366 155 16 specific specific ADJ cana-5366 155 17 problem problem NOUN cana-5366 155 18 of of ADP cana-5366 155 19 finger finger NOUN cana-5366 155 20 movement movement NOUN cana-5366 155 21 . . PUNCT cana-5366 156 1 it it PRON cana-5366 156 2 can can AUX cana-5366 156 3 be be AUX cana-5366 156 4 extended extend VERB cana-5366 156 5 to to ADP cana-5366 156 6 other other ADJ cana-5366 156 7 biomechanical biomechanical ADJ cana-5366 156 8 systems system NOUN cana-5366 156 9 governed govern VERB cana-5366 156 10 by by ADP cana-5366 156 11 nonlinear nonlinear ADJ cana-5366 156 12 differential differential ADJ cana-5366 156 13 equations equation NOUN cana-5366 156 14 , , PUNCT cana-5366 156 15 such such ADJ cana-5366 156 16 as as ADP cana-5366 156 17 joint joint ADJ cana-5366 156 18 dynamics dynamic NOUN cana-5366 156 19 , , PUNCT cana-5366 156 20 muscle muscle NOUN cana-5366 156 21 activation activation NOUN cana-5366 156 22 models model NOUN cana-5366 156 23 , , PUNCT cana-5366 156 24 and and CCONJ cana-5366 156 25 gait gait VERB cana-5366 156 26 analysis analysis NOUN cana-5366 156 27 . . PUNCT cana-5366 157 1 the the DET cana-5366 157 2 ability ability NOUN cana-5366 157 3 to to PART cana-5366 157 4 derive derive VERB cana-5366 157 5 exact exact ADJ cana-5366 157 6 solutions solution NOUN cana-5366 157 7 for for ADP cana-5366 157 8 such such ADJ cana-5366 157 9 systems system NOUN cana-5366 157 10 can can AUX cana-5366 157 11 significantly significantly ADV cana-5366 157 12 enhance enhance VERB cana-5366 157 13 our our PRON cana-5366 157 14 understanding understanding NOUN cana-5366 157 15 of of ADP cana-5366 157 16 their their PRON cana-5366 157 17 underlying underlie VERB cana-5366 157 18 mechanics mechanic NOUN cana-5366 157 19 and and CCONJ cana-5366 157 20 improve improve VERB cana-5366 157 21 the the DET cana-5366 157 22 accuracy accuracy NOUN cana-5366 157 23 of of ADP cana-5366 157 24 predictive predictive ADJ cana-5366 157 25 models model NOUN cana-5366 157 26 . . PUNCT cana-5366 158 1 in in ADP cana-5366 158 2 conclusion conclusion NOUN cana-5366 158 3 , , PUNCT cana-5366 158 4 this this DET cana-5366 158 5 study study NOUN cana-5366 158 6 highlights highlight VERB cana-5366 158 7 the the DET cana-5366 158 8 power power NOUN cana-5366 158 9 of of ADP cana-5366 158 10 lie lie NOUN cana-5366 158 11 symmetry symmetry NOUN cana-5366 158 12 analysis analysis NOUN cana-5366 158 13 in in ADP cana-5366 158 14 solving solve VERB cana-5366 158 15 complex complex ADJ cana-5366 158 16 nonlinear nonlinear ADJ cana-5366 158 17 differential differential ADJ cana-5366 158 18 equations equation NOUN cana-5366 158 19 and and CCONJ cana-5366 158 20 provides provide VERB cana-5366 158 21 a a DET cana-5366 158 22 robust robust ADJ cana-5366 158 23 mathematical mathematical ADJ cana-5366 158 24 framework framework NOUN cana-5366 158 25 for for ADP cana-5366 158 26 understanding understand VERB cana-5366 158 27 human human ADJ cana-5366 158 28 finger finger NOUN cana-5366 158 29 dynamics dynamic NOUN cana-5366 158 30 . . PUNCT cana-5366 159 1 the the DET cana-5366 159 2 results result NOUN cana-5366 159 3 not not PART cana-5366 159 4 only only ADV cana-5366 159 5 advance advance VERB cana-5366 159 6 the the DET cana-5366 159 7 theoretical theoretical ADJ cana-5366 159 8 understanding understanding NOUN cana-5366 159 9 of of ADP cana-5366 159 10 biomechanical biomechanical ADJ cana-5366 159 11 systems system NOUN cana-5366 159 12 but but CCONJ cana-5366 159 13 also also ADV cana-5366 159 14 have have VERB cana-5366 159 15 practical practical ADJ cana-5366 159 16 implications implication NOUN cana-5366 159 17 for for ADP cana-5366 159 18 robotics robotic NOUN cana-5366 159 19 , , PUNCT cana-5366 159 20 physiology physiology NOUN cana-5366 159 21 , , PUNCT cana-5366 159 22 and and CCONJ cana-5366 159 23 rehabilitation rehabilitation NOUN cana-5366 159 24 . . PUNCT cana-5366 160 1 future future ADJ cana-5366 160 2 work work NOUN cana-5366 160 3 will will AUX cana-5366 160 4 focus focus VERB cana-5366 160 5 on on ADP cana-5366 160 6 extending extend VERB cana-5366 160 7 this this DET cana-5366 160 8 methodology methodology NOUN cana-5366 160 9 to to ADP cana-5366 160 10 other other ADJ cana-5366 160 11 models model NOUN cana-5366 160 12 and and CCONJ cana-5366 160 13 exploring explore VERB cana-5366 160 14 its its PRON cana-5366 160 15 applications application NOUN cana-5366 160 16 in in ADP cana-5366 160 17 experimental experimental ADJ cana-5366 160 18 and and CCONJ cana-5366 160 19 clinical clinical ADJ cana-5366 160 20 settings setting NOUN cana-5366 160 21 . . PUNCT cana-5366 161 1 6 6 X cana-5366 161 2 . . X cana-5366 161 3 conclusion conclusion NOUN cana-5366 161 4 in in ADP cana-5366 161 5 this this DET cana-5366 161 6 paper paper NOUN cana-5366 161 7 , , PUNCT cana-5366 161 8 we we PRON cana-5366 161 9 applied apply VERB cana-5366 161 10 lie lie NOUN cana-5366 161 11 symmetry symmetry NOUN cana-5366 161 12 analysis analysis NOUN cana-5366 161 13 to to ADP cana-5366 161 14 a a DET cana-5366 161 15 nonlinear nonlinear ADJ cana-5366 161 16 differential differential ADJ cana-5366 161 17 equation equation NOUN cana-5366 161 18 modeling model VERB cana-5366 161 19 human human ADJ cana-5366 161 20 finger finger NOUN cana-5366 161 21 movement movement NOUN cana-5366 161 22 . . PUNCT cana-5366 162 1 the the DET cana-5366 162 2 model model PROPN cana-5366 162 3 parameter parameter PROPN cana-5366 162 4 𝜏(πœƒ 𝜏(πœƒ PROPN cana-5366 162 5 ) ) PUNCT cana-5366 162 6 was be AUX cana-5366 162 7 determined determine VERB cana-5366 162 8 using use VERB cana-5366 162 9 the the DET cana-5366 162 10 lie lie NOUN cana-5366 162 11 symmetry symmetry NOUN cana-5366 162 12 method method NOUN cana-5366 162 13 , , PUNCT cana-5366 162 14 and and CCONJ cana-5366 162 15 an an DET cana-5366 162 16 exact exact ADJ cana-5366 162 17 solution solution NOUN cana-5366 162 18 was be AUX cana-5366 162 19 derived derive VERB cana-5366 162 20 . . PUNCT cana-5366 163 1 the the DET cana-5366 163 2 results result NOUN cana-5366 163 3 were be AUX cana-5366 163 4 graphically graphically ADV cana-5366 163 5 represented represent VERB cana-5366 163 6 for for ADP cana-5366 163 7 different different ADJ cana-5366 163 8 initial initial ADJ cana-5366 163 9 conditions condition NOUN cana-5366 163 10 , , PUNCT cana-5366 163 11 demonstrating demonstrate VERB cana-5366 163 12 the the DET cana-5366 163 13 effectiveness effectiveness NOUN cana-5366 163 14 of of ADP cana-5366 163 15 the the DET cana-5366 163 16 method method NOUN cana-5366 163 17 . . PUNCT cana-5366 164 1 this this DET cana-5366 164 2 study study NOUN cana-5366 164 3 provides provide VERB cana-5366 164 4 a a DET cana-5366 164 5 mathematical mathematical ADJ cana-5366 164 6 framework framework NOUN cana-5366 164 7 for for ADP cana-5366 164 8 understanding understand VERB cana-5366 164 9 human human ADJ cana-5366 164 10 finger finger NOUN cana-5366 164 11 dynamics dynamic NOUN cana-5366 164 12 and and CCONJ cana-5366 164 13 can can AUX cana-5366 164 14 be be AUX cana-5366 164 15 extended extend VERB cana-5366 164 16 to to ADP cana-5366 164 17 other other ADJ cana-5366 164 18 applications application NOUN cana-5366 164 19 in in ADP cana-5366 164 20 robotics robotic NOUN cana-5366 164 21 and and CCONJ cana-5366 164 22 biomechanics biomechanic NOUN cana-5366 164 23 . . PUNCT cana-5366 165 1 declaration declaration NOUN cana-5366 165 2 of of ADP cana-5366 165 3 competing compete VERB cana-5366 165 4 interest interest NOUN cana-5366 165 5 the the DET cana-5366 165 6 authors author NOUN cana-5366 165 7 declare declare VERB cana-5366 165 8 that that SCONJ cana-5366 165 9 they they PRON cana-5366 165 10 have have VERB cana-5366 165 11 no no DET cana-5366 165 12 known know VERB cana-5366 165 13 competing compete VERB cana-5366 165 14 financial financial ADJ cana-5366 165 15 interests interest NOUN cana-5366 165 16 or or CCONJ cana-5366 165 17 personal personal ADJ cana-5366 165 18 relationships relationship NOUN cana-5366 165 19 that that PRON cana-5366 165 20 could could AUX cana-5366 165 21 have have AUX cana-5366 165 22 appeared appear VERB cana-5366 165 23 to to PART cana-5366 165 24 influence influence VERB cana-5366 165 25 the the DET cana-5366 165 26 work work NOUN cana-5366 165 27 reported report VERB cana-5366 165 28 in in ADP cana-5366 165 29 this this DET cana-5366 165 30 paper paper NOUN cana-5366 165 31 . . PUNCT cana-5366 166 1 communications communication NOUN cana-5366 166 2 on on ADP cana-5366 166 3 applied apply VERB cana-5366 166 4 nonlinear nonlinear ADJ cana-5366 166 5 analysis analysis NOUN cana-5366 166 6 issn issn NOUN cana-5366 166 7 : : PUNCT cana-5366 166 8 1074 1074 NUM cana-5366 166 9 - - PUNCT cana-5366 166 10 133x 133x NUM cana-5366 166 11 vol vol VERB cana-5366 166 12 32 32 NUM cana-5366 166 13 no no NOUN cana-5366 166 14 . . PUNCT cana-5366 167 1 10s 10 NOUN cana-5366 167 2 ( ( PUNCT cana-5366 167 3 2025 2025 NUM cana-5366 167 4 ) ) PUNCT cana-5366 167 5 2050 2050 NUM cana-5366 168 1 https://internationalpubls.com https://internationalpubls.com X cana-5366 168 2 acknowledgement acknowledgement NOUN cana-5366 168 3 the the DET cana-5366 168 4 authors author NOUN cana-5366 168 5 declare declare VERB cana-5366 168 6 that that SCONJ cana-5366 168 7 they they PRON cana-5366 168 8 have have VERB cana-5366 168 9 no no DET cana-5366 168 10 known know VERB cana-5366 168 11 competing compete VERB cana-5366 168 12 financial financial ADJ cana-5366 168 13 interests interest NOUN cana-5366 168 14 or or CCONJ cana-5366 168 15 personal personal ADJ cana-5366 168 16 relationships relationship NOUN cana-5366 168 17 that that PRON cana-5366 168 18 could could AUX cana-5366 168 19 have have AUX cana-5366 168 20 appeared appear VERB cana-5366 168 21 to to PART cana-5366 168 22 influence influence VERB cana-5366 168 23 the the DET cana-5366 168 24 work work NOUN cana-5366 168 25 reported report VERB cana-5366 168 26 in in ADP cana-5366 168 27 this this DET cana-5366 168 28 paper paper NOUN cana-5366 168 29 . . PUNCT cana-5366 169 1 data datum NOUN cana-5366 169 2 availability availability NOUN cana-5366 169 3 no no DET cana-5366 169 4 data datum NOUN cana-5366 169 5 was be AUX cana-5366 169 6 used use VERB cana-5366 169 7 for for ADP cana-5366 169 8 the the DET cana-5366 169 9 research research NOUN cana-5366 169 10 described describe VERB cana-5366 169 11 in in ADP cana-5366 169 12 the the DET cana-5366 169 13 article article NOUN cana-5366 169 14 . . PUNCT cana-5366 170 1 bibliography bibliography NOUN cana-5366 170 2 [ [ X cana-5366 170 3 1 1 NUM cana-5366 170 4 ] ] PUNCT cana-5366 170 5 kosugi kosugi NOUN cana-5366 170 6 , , PUNCT cana-5366 170 7 t. t. PROPN cana-5366 170 8 , , PUNCT cana-5366 170 9 kino kino PROPN cana-5366 170 10 , , PUNCT cana-5366 170 11 h. h. PROPN cana-5366 170 12 , , PUNCT cana-5366 170 13 goto goto NOUN cana-5366 170 14 , , PUNCT cana-5366 170 15 m. m. NOUN cana-5366 170 16 , , PUNCT cana-5366 170 17 \ \ PROPN cana-5366 170 18 & & CCONJ cana-5366 170 19 matsutani matsutani PROPN cana-5366 170 20 , , PUNCT cana-5366 170 21 y. y. PROPN cana-5366 170 22 ( ( PUNCT cana-5366 170 23 2019 2019 NUM cana-5366 170 24 ) ) PUNCT cana-5366 170 25 . . PUNCT cana-5366 171 1 stability stability NOUN cana-5366 171 2 conditions condition NOUN cana-5366 171 3 of of ADP cana-5366 171 4 an an DET cana-5366 171 5 ode ode NOUN cana-5366 171 6 arising arising NOUN cana-5366 171 7 in in ADP cana-5366 171 8 human human ADJ cana-5366 171 9 motion motion NOUN cana-5366 171 10 and and CCONJ cana-5366 171 11 its its PRON cana-5366 171 12 numerical numerical ADJ cana-5366 171 13 simulation simulation NOUN cana-5366 171 14 . . PUNCT cana-5366 172 1 results result NOUN cana-5366 172 2 in in ADP cana-5366 172 3 applied applied ADJ cana-5366 172 4 mathematics mathematic NOUN cana-5366 172 5 , , PUNCT cana-5366 172 6 3 3 NUM cana-5366 172 7 , , PUNCT cana-5366 172 8 100063 100063 NUM cana-5366 172 9 . . PUNCT cana-5366 173 1 [ [ X cana-5366 173 2 2 2 NUM cana-5366 173 3 ] ] X cana-5366 173 4 bluman bluman NOUN cana-5366 173 5 , , PUNCT cana-5366 173 6 g. g. PROPN cana-5366 173 7 w. w. PROPN cana-5366 173 8 , , PUNCT cana-5366 173 9 \ \ PROPN cana-5366 173 10 & & CCONJ cana-5366 173 11 kumei kumei PROPN cana-5366 173 12 , , PUNCT cana-5366 173 13 s. s. PROPN cana-5366 173 14 ( ( PUNCT cana-5366 173 15 1989 1989 NUM cana-5366 173 16 ) ) PUNCT cana-5366 173 17 . . PUNCT cana-5366 174 1 symmetries symmetry NOUN cana-5366 174 2 and and CCONJ cana-5366 174 3 differential differential ADJ cana-5366 174 4 equations equation NOUN cana-5366 174 5 . . PUNCT cana-5366 175 1 springer springer NOUN cana-5366 175 2 , , PUNCT cana-5366 175 3 new new PROPN cana-5366 175 4 york york PROPN cana-5366 175 5 . . PUNCT cana-5366 176 1 [ [ X cana-5366 176 2 3 3 X cana-5366 176 3 ] ] X cana-5366 176 4 olver olver ADV cana-5366 176 5 , , PUNCT cana-5366 176 6 p. p. NOUN cana-5366 176 7 j. j. PROPN cana-5366 176 8 ( ( PUNCT cana-5366 176 9 1986 1986 NUM cana-5366 176 10 ) ) PUNCT cana-5366 176 11 . . PUNCT cana-5366 177 1 applications application NOUN cana-5366 177 2 of of ADP cana-5366 177 3 lie lie NOUN cana-5366 177 4 groups group NOUN cana-5366 177 5 to to PART cana-5366 177 6 differential differential VERB cana-5366 177 7 equations equation NOUN cana-5366 177 8 . . PUNCT cana-5366 178 1 springer springer NOUN cana-5366 178 2 , , PUNCT cana-5366 178 3 new new PROPN cana-5366 178 4 york york PROPN cana-5366 178 5 . . PUNCT cana-5366 179 1 [ [ X cana-5366 179 2 4 4 NUM cana-5366 179 3 ] ] PUNCT cana-5366 179 4 ovsiannikov ovsiannikov PROPN cana-5366 179 5 , , PUNCT cana-5366 179 6 l. l. PROPN cana-5366 179 7 v. v. PROPN cana-5366 179 8 ( ( PUNCT cana-5366 179 9 1982 1982 NUM cana-5366 179 10 ) ) PUNCT cana-5366 179 11 . . PUNCT cana-5366 180 1 group group NOUN cana-5366 180 2 analysis analysis NOUN cana-5366 180 3 of of ADP cana-5366 180 4 differential differential ADJ cana-5366 180 5 equations equation NOUN cana-5366 180 6 . . PUNCT cana-5366 181 1 academic academic ADJ cana-5366 181 2 press press NOUN cana-5366 181 3 , , PUNCT cana-5366 181 4 new new PROPN cana-5366 181 5 york york PROPN cana-5366 181 6 . . PUNCT cana-5366 182 1 [ [ X cana-5366 182 2 5 5 NUM cana-5366 182 3 ] ] PUNCT cana-5366 182 4 stephani stephani PROPN cana-5366 182 5 , , PUNCT cana-5366 182 6 h. h. PROPN cana-5366 182 7 ( ( PUNCT cana-5366 182 8 1989 1989 NUM cana-5366 182 9 ) ) PUNCT cana-5366 182 10 . . PUNCT cana-5366 183 1 differential differential ADJ cana-5366 183 2 equations equation NOUN cana-5366 183 3 : : PUNCT cana-5366 183 4 their their PRON cana-5366 183 5 solutions solution NOUN cana-5366 183 6 using use VERB cana-5366 183 7 symmetries symmetry NOUN cana-5366 183 8 . . PUNCT cana-5366 184 1 cambridge cambridge PROPN cana-5366 184 2 university university PROPN cana-5366 184 3 press press PROPN cana-5366 184 4 , , PUNCT cana-5366 184 5 cambridge cambridge PROPN cana-5366 184 6 . . PUNCT cana-5366 185 1 [ [ X cana-5366 185 2 6 6 NUM cana-5366 185 3 ] ] PUNCT cana-5366 185 4 kosugi kosugi NOUN cana-5366 185 5 , , PUNCT cana-5366 185 6 s. s. PROPN cana-5366 185 7 , , PUNCT cana-5366 185 8 tanaka tanaka PROPN cana-5366 185 9 , , PUNCT cana-5366 185 10 y. y. PROPN cana-5366 185 11 , , PUNCT cana-5366 185 12 \ \ PROPN cana-5366 185 13 & & CCONJ cana-5366 185 14 nakamura nakamura PROPN cana-5366 185 15 , , PUNCT cana-5366 185 16 h. h. PROPN cana-5366 185 17 ( ( PUNCT cana-5366 185 18 2019 2019 NUM cana-5366 185 19 ) ) PUNCT cana-5366 185 20 . . PUNCT cana-5366 186 1 modeling modeling NOUN cana-5366 186 2 and and CCONJ cana-5366 186 3 analysis analysis NOUN cana-5366 186 4 of of ADP cana-5366 186 5 human human ADJ cana-5366 186 6 finger finger NOUN cana-5366 186 7 movement movement NOUN cana-5366 186 8 . . PUNCT cana-5366 187 1 journal journal PROPN cana-5366 187 2 of of ADP cana-5366 187 3 biomechanics biomechanic NOUN cana-5366 187 4 , , PUNCT cana-5366 187 5 45(3 45(3 NUM cana-5366 187 6 ) ) PUNCT cana-5366 187 7 , , PUNCT cana-5366 187 8 123 123 NUM cana-5366 187 9 - - SYM cana-5366 187 10 -130 -130 NOUN cana-5366 187 11 . . PUNCT cana-5366 188 1 [ [ X cana-5366 188 2 7 7 NUM cana-5366 188 3 ] ] X cana-5366 188 4 fung fung PROPN cana-5366 188 5 , , PUNCT cana-5366 188 6 y. y. PROPN cana-5366 188 7 c. c. PROPN cana-5366 188 8 ( ( PUNCT cana-5366 188 9 1993 1993 NUM cana-5366 188 10 ) ) PUNCT cana-5366 188 11 . . PUNCT cana-5366 189 1 biomechanics biomechanic NOUN cana-5366 189 2 : : PUNCT cana-5366 189 3 mechanical mechanical ADJ cana-5366 189 4 properties property NOUN cana-5366 189 5 of of ADP cana-5366 189 6 living live VERB cana-5366 189 7 tissues tissue NOUN cana-5366 189 8 . . PUNCT cana-5366 190 1 springer springer NOUN cana-5366 190 2 . . PUNCT cana-5366 191 1 [ [ X cana-5366 191 2 8 8 NUM cana-5366 191 3 ] ] X cana-5366 191 4 winter winter NOUN cana-5366 191 5 , , PUNCT cana-5366 191 6 d. d. PROPN cana-5366 191 7 a. a. PROPN cana-5366 191 8 ( ( PUNCT cana-5366 191 9 2009 2009 NUM cana-5366 191 10 ) ) PUNCT cana-5366 191 11 . . PUNCT cana-5366 192 1 biomechanics biomechanic NOUN cana-5366 192 2 and and CCONJ cana-5366 192 3 motor motor NOUN cana-5366 192 4 control control NOUN cana-5366 192 5 of of ADP cana-5366 192 6 human human ADJ cana-5366 192 7 movement movement NOUN cana-5366 192 8 . . PUNCT cana-5366 193 1 wiley wiley PROPN cana-5366 193 2 . . PUNCT cana-5366 194 1 [ [ X cana-5366 194 2 9 9 NUM cana-5366 194 3 ] ] SYM cana-5366 194 4 ΓΆzkaya ΓΆzkaya NOUN cana-5366 194 5 , , PUNCT cana-5366 194 6 n. n. NOUN cana-5366 194 7 , , PUNCT cana-5366 194 8 \ \ PROPN cana-5366 194 9 & & CCONJ cana-5366 194 10 nordin nordin PROPN cana-5366 194 11 , , PUNCT cana-5366 194 12 m. m. NOUN cana-5366 194 13 ( ( PUNCT cana-5366 194 14 2012 2012 NUM cana-5366 194 15 ) ) PUNCT cana-5366 194 16 . . PUNCT cana-5366 195 1 fundamentals fundamental NOUN cana-5366 195 2 of of ADP cana-5366 195 3 biomechanics biomechanic NOUN cana-5366 195 4 : : PUNCT cana-5366 195 5 equilibrium equilibrium NOUN cana-5366 195 6 , , PUNCT cana-5366 195 7 motion motion NOUN cana-5366 195 8 , , PUNCT cana-5366 195 9 and and CCONJ cana-5366 195 10 deformation deformation NOUN cana-5366 195 11 . . PUNCT cana-5366 196 1 springer springer NOUN cana-5366 196 2 . . PUNCT cana-5366 197 1 [ [ X cana-5366 197 2 10 10 NUM cana-5366 197 3 ] ] X cana-5366 197 4 zatsiorsky zatsiorsky NOUN cana-5366 197 5 , , PUNCT cana-5366 197 6 v. v. ADP cana-5366 197 7 m. m. NOUN cana-5366 197 8 ( ( PUNCT cana-5366 197 9 2002 2002 NUM cana-5366 197 10 ) ) PUNCT cana-5366 197 11 . . PUNCT cana-5366 198 1 kinetics kinetic NOUN cana-5366 198 2 of of ADP cana-5366 198 3 human human ADJ cana-5366 198 4 motion motion NOUN cana-5366 198 5 . . PUNCT cana-5366 199 1 human human ADJ cana-5366 199 2 kinetics kinetic NOUN cana-5366 199 3 . . PUNCT cana-5366 200 1 [ [ X cana-5366 200 2 11 11 NUM cana-5366 200 3 ] ] X cana-5366 200 4 cantwell cantwell PROPN cana-5366 200 5 , , PUNCT cana-5366 200 6 b. b. PROPN cana-5366 200 7 j. j. PROPN cana-5366 200 8 ( ( PUNCT cana-5366 200 9 2002 2002 NUM cana-5366 200 10 ) ) PUNCT cana-5366 200 11 . . PUNCT cana-5366 201 1 introduction introduction NOUN cana-5366 201 2 to to PART cana-5366 201 3 symmetry symmetry VERB cana-5366 201 4 analysis analysis NOUN cana-5366 201 5 . . PUNCT cana-5366 202 1 cambridge cambridge PROPN cana-5366 202 2 university university PROPN cana-5366 202 3 press press NOUN cana-5366 202 4 . . PUNCT cana-5366 203 1 [ [ X cana-5366 203 2 12 12 NUM cana-5366 203 3 ] ] PUNCT cana-5366 203 4 wybourne wybourne PROPN cana-5366 203 5 , , PUNCT cana-5366 203 6 b. b. PROPN cana-5366 203 7 g. g. PROPN cana-5366 203 8 ( ( PUNCT cana-5366 203 9 1974 1974 NUM cana-5366 203 10 ) ) PUNCT cana-5366 203 11 . . PUNCT cana-5366 204 1 classical classical ADJ cana-5366 204 2 groups group NOUN cana-5366 204 3 for for ADP cana-5366 204 4 physicists physicist NOUN cana-5366 204 5 . . PUNCT cana-5366 205 1 wiley wiley PROPN cana-5366 205 2 . . PUNCT cana-5366 206 1 [ [ X cana-5366 206 2 13 13 NUM cana-5366 206 3 ] ] SYM cana-5366 206 4 ma ma PROPN cana-5366 206 5 , , PUNCT cana-5366 206 6 w. w. PROPN cana-5366 206 7 x. x. PROPN cana-5366 206 8 ( ( PUNCT cana-5366 206 9 2013 2013 NUM cana-5366 206 10 ) ) PUNCT cana-5366 206 11 . . PUNCT cana-5366 207 1 lie lie NOUN cana-5366 207 2 symmetry symmetry NOUN cana-5366 207 3 analysis analysis NOUN cana-5366 207 4 of of ADP cana-5366 207 5 nonlinear nonlinear ADJ cana-5366 207 6 differential differential ADJ cana-5366 207 7 equations equation NOUN cana-5366 207 8 in in ADP cana-5366 207 9 biomechanics biomechanic NOUN cana-5366 207 10 . . PUNCT cana-5366 208 1 journal journal PROPN cana-5366 208 2 of of ADP cana-5366 208 3 nonlinear nonlinear PROPN cana-5366 208 4 mathematical mathematical ADJ cana-5366 208 5 physics physics NOUN cana-5366 208 6 . . PUNCT cana-5366 209 1 [ [ X cana-5366 209 2 14 14 NUM cana-5366 209 3 ] ] PUNCT cana-5366 209 4 kosugi kosugi NOUN cana-5366 209 5 , , PUNCT cana-5366 209 6 s. s. PROPN cana-5366 209 7 , , PUNCT cana-5366 209 8 et et PROPN cana-5366 209 9 al al PROPN cana-5366 209 10 . . PROPN cana-5366 209 11 ( ( PUNCT cana-5366 209 12 2019 2019 NUM cana-5366 209 13 ) ) PUNCT cana-5366 209 14 . . PUNCT cana-5366 210 1 numerical numerical ADJ cana-5366 210 2 and and CCONJ cana-5366 210 3 experimental experimental ADJ cana-5366 210 4 analysis analysis NOUN cana-5366 210 5 of of ADP cana-5366 210 6 human human ADJ cana-5366 210 7 finger finger NOUN cana-5366 210 8 movement movement NOUN cana-5366 210 9 . . PUNCT cana-5366 211 1 journal journal PROPN cana-5366 211 2 of of ADP cana-5366 211 3 biomechanics biomechanic NOUN cana-5366 211 4 . . PUNCT cana-5366 212 1 [ [ X cana-5366 212 2 15 15 NUM cana-5366 212 3 ] ] X cana-5366 212 4 valero valero PROPN cana-5366 212 5 - - PUNCT cana-5366 212 6 cuevas cuevas PROPN cana-5366 212 7 , , PUNCT cana-5366 212 8 f. f. PROPN cana-5366 212 9 j. j. PROPN cana-5366 212 10 ( ( PUNCT cana-5366 212 11 2009 2009 NUM cana-5366 212 12 ) ) PUNCT cana-5366 212 13 . . PUNCT cana-5366 213 1 fundamentals fundamental NOUN cana-5366 213 2 of of ADP cana-5366 213 3 neuromechanics neuromechanic NOUN cana-5366 213 4 . . PUNCT cana-5366 214 1 springer springer NOUN cana-5366 214 2 . . PUNCT cana-5366 215 1 [ [ X cana-5366 215 2 16 16 NUM cana-5366 215 3 ] ] X cana-5366 215 4 hydon hydon PROPN cana-5366 215 5 , , PUNCT cana-5366 215 6 p. p. PROPN cana-5366 215 7 e. e. PROPN cana-5366 215 8 ( ( PUNCT cana-5366 215 9 2000 2000 NUM cana-5366 215 10 ) ) PUNCT cana-5366 215 11 . . PUNCT cana-5366 216 1 symmetry symmetry NOUN cana-5366 216 2 methods method NOUN cana-5366 216 3 for for ADP cana-5366 216 4 differential differential ADJ cana-5366 216 5 equations equation NOUN cana-5366 216 6 : : PUNCT cana-5366 216 7 a a DET cana-5366 216 8 beginner beginner NOUN cana-5366 216 9 's 's PART cana-5366 216 10 guide guide NOUN cana-5366 216 11 . . PUNCT cana-5366 217 1 cambridge cambridge PROPN cana-5366 217 2 university university PROPN cana-5366 217 3 press press NOUN cana-5366 217 4 . . PUNCT cana-5366 218 1 [ [ X cana-5366 218 2 17 17 NUM cana-5366 218 3 ] ] PUNCT cana-5366 218 4 ibragimov ibragimov NOUN cana-5366 218 5 , , PUNCT cana-5366 218 6 n. n. PROPN cana-5366 218 7 h. h. PROPN cana-5366 218 8 ( ( PUNCT cana-5366 218 9 1999 1999 NUM cana-5366 218 10 ) ) PUNCT cana-5366 218 11 . . PUNCT cana-5366 219 1 elementary elementary PROPN cana-5366 219 2 lie lie PROPN cana-5366 219 3 group group NOUN cana-5366 219 4 analysis analysis NOUN cana-5366 219 5 and and CCONJ cana-5366 219 6 ordinary ordinary ADJ cana-5366 219 7 differential differential ADJ cana-5366 219 8 equations equation NOUN cana-5366 219 9 . . PUNCT cana-5366 220 1 wiley wiley PROPN cana-5366 220 2 . . PUNCT cana-5366 221 1 [ [ X cana-5366 221 2 18 18 NUM cana-5366 221 3 ] ] X cana-5366 221 4 salisbury salisbury PROPN cana-5366 221 5 , , PUNCT cana-5366 221 6 j. j. PROPN cana-5366 221 7 k. k. PROPN cana-5366 221 8 , , PUNCT cana-5366 221 9 \ \ PROPN cana-5366 221 10 & & CCONJ cana-5366 221 11 craig craig PROPN cana-5366 221 12 , , PUNCT cana-5366 221 13 j. j. PROPN cana-5366 221 14 j. j. PROPN cana-5366 221 15 ( ( PUNCT cana-5366 221 16 1982 1982 NUM cana-5366 221 17 ) ) PUNCT cana-5366 221 18 . . PUNCT cana-5366 222 1 articulated articulated ADJ cana-5366 222 2 hands hand NOUN cana-5366 222 3 : : PUNCT cana-5366 222 4 force force NOUN cana-5366 222 5 control control NOUN cana-5366 222 6 and and CCONJ cana-5366 222 7 kinematic kinematic ADJ cana-5366 222 8 issues issue NOUN cana-5366 222 9 . . PUNCT cana-5366 223 1 the the DET cana-5366 223 2 international international ADJ cana-5366 223 3 journal journal NOUN cana-5366 223 4 of of ADP cana-5366 223 5 robotics robotic NOUN cana-5366 223 6 research research NOUN cana-5366 223 7 . . PUNCT cana-5366 224 1 communications communication NOUN cana-5366 224 2 on on ADP cana-5366 224 3 applied apply VERB cana-5366 224 4 nonlinear nonlinear ADJ cana-5366 224 5 analysis analysis NOUN cana-5366 224 6 issn issn NOUN cana-5366 224 7 : : PUNCT cana-5366 224 8 1074 1074 NUM cana-5366 224 9 - - PUNCT cana-5366 224 10 133x 133x NUM cana-5366 224 11 vol vol VERB cana-5366 224 12 32 32 NUM cana-5366 224 13 no no NOUN cana-5366 224 14 . . PUNCT cana-5366 225 1 10s 10 NOUN cana-5366 225 2 ( ( PUNCT cana-5366 225 3 2025 2025 NUM cana-5366 225 4 ) ) PUNCT cana-5366 225 5 2051 2051 NUM cana-5366 225 6 https://internationalpubls.com https://internationalpubls.com X cana-5366 226 1 [ [ X cana-5366 226 2 19 19 NUM cana-5366 226 3 ] ] X cana-5366 226 4 cutkosky cutkosky NOUN cana-5366 226 5 , , PUNCT cana-5366 226 6 m. m. PROPN cana-5366 226 7 r. r. PROPN cana-5366 226 8 ( ( PUNCT cana-5366 226 9 1989 1989 NUM cana-5366 226 10 ) ) PUNCT cana-5366 226 11 . . PUNCT cana-5366 227 1 on on ADP cana-5366 227 2 grasp grasp NOUN cana-5366 227 3 choice choice NOUN cana-5366 227 4 , , PUNCT cana-5366 227 5 grasp grasp NOUN cana-5366 227 6 models model NOUN cana-5366 227 7 , , PUNCT cana-5366 227 8 and and CCONJ cana-5366 227 9 the the DET cana-5366 227 10 design design NOUN cana-5366 227 11 of of ADP cana-5366 227 12 hands hand NOUN cana-5366 227 13 for for ADP cana-5366 227 14 manufacturing manufacturing NOUN cana-5366 227 15 tasks task NOUN cana-5366 227 16 . . PUNCT cana-5366 228 1 ieee ieee NOUN cana-5366 228 2 transactions transaction NOUN cana-5366 228 3 on on ADP cana-5366 228 4 robotics robotic NOUN cana-5366 228 5 and and CCONJ cana-5366 228 6 automation automation NOUN cana-5366 228 7 . . PUNCT cana-5366 229 1 [ [ X cana-5366 229 2 20 20 NUM cana-5366 229 3 ] ] X cana-5366 229 4 pratt pratt PROPN cana-5366 229 5 , , PUNCT cana-5366 229 6 g. g. PROPN cana-5366 229 7 a. a. PROPN cana-5366 229 8 , , PUNCT cana-5366 229 9 \ \ PROPN cana-5366 229 10 & & CCONJ cana-5366 229 11 williamson williamson PROPN cana-5366 229 12 , , PUNCT cana-5366 229 13 m. m. NOUN cana-5366 229 14 m. m. NOUN cana-5366 229 15 ( ( PUNCT cana-5366 229 16 2001 2001 NUM cana-5366 229 17 ) ) PUNCT cana-5366 229 18 . . PUNCT cana-5366 230 1 series series PROPN cana-5366 230 2 elastic elastic ADJ cana-5366 230 3 actuators actuator NOUN cana-5366 230 4 . . PUNCT cana-5366 231 1 ieee ieee PROPN cana-5366 231 2 / / SYM cana-5366 231 3 rsj rsj NOUN cana-5366 231 4 international international ADJ cana-5366 231 5 conference conference NOUN cana-5366 231 6 on on ADP cana-5366 231 7 intelligent intelligent ADJ cana-5366 231 8 robots robot NOUN cana-5366 231 9 and and CCONJ cana-5366 231 10 systems system NOUN cana-5366 231 11 . . PUNCT cana-5366 232 1 [ [ X cana-5366 232 2 21 21 NUM cana-5366 232 3 ] ] PUNCT cana-5366 232 4 hollerbach hollerbach NOUN cana-5366 232 5 , , PUNCT cana-5366 232 6 j. j. PROPN cana-5366 232 7 m. m. PROPN cana-5366 232 8 ( ( PUNCT cana-5366 232 9 1982 1982 NUM cana-5366 232 10 ) ) PUNCT cana-5366 232 11 . . PUNCT cana-5366 233 1 a a DET cana-5366 233 2 recursive recursive ADJ cana-5366 233 3 lagrangian lagrangian ADJ cana-5366 233 4 formulation formulation NOUN cana-5366 233 5 of of ADP cana-5366 233 6 manipulator manipulator NOUN cana-5366 233 7 dynamics dynamic NOUN cana-5366 233 8 and and CCONJ cana-5366 233 9 a a DET cana-5366 233 10 comparative comparative ADJ cana-5366 233 11 study study NOUN cana-5366 233 12 of of ADP cana-5366 233 13 dynamics dynamic NOUN cana-5366 233 14 formulation formulation NOUN cana-5366 233 15 complexity complexity NOUN cana-5366 233 16 . . PUNCT cana-5366 234 1 ieee ieee NOUN cana-5366 234 2 transactions transaction NOUN cana-5366 234 3 on on ADP cana-5366 234 4 systems system NOUN cana-5366 234 5 , , PUNCT cana-5366 234 6 man man NOUN cana-5366 234 7 , , PUNCT cana-5366 234 8 and and CCONJ cana-5366 234 9 cybernetics cybernetic NOUN cana-5366 234 10 . . PUNCT cana-5366 235 1 [ [ X cana-5366 235 2 22 22 NUM cana-5366 235 3 ] ] X cana-5366 235 4 krebs krebs NOUN cana-5366 235 5 , , PUNCT cana-5366 235 6 h. h. PROPN cana-5366 235 7 i. i. PROPN cana-5366 235 8 , , PUNCT cana-5366 235 9 \ \ PROPN cana-5366 235 10 & & CCONJ cana-5366 235 11 hogan hogan PROPN cana-5366 235 12 , , PUNCT cana-5366 235 13 n. n. PROPN cana-5366 235 14 ( ( PUNCT cana-5366 235 15 1998 1998 NUM cana-5366 235 16 ) ) PUNCT cana-5366 235 17 . . PUNCT cana-5366 236 1 robot robot NOUN cana-5366 236 2 - - PUNCT cana-5366 236 3 aided aid VERB cana-5366 236 4 neurorehabilitation neurorehabilitation NOUN cana-5366 236 5 . . PUNCT cana-5366 237 1 ieee ieee NOUN cana-5366 237 2 transactions transaction NOUN cana-5366 237 3 on on ADP cana-5366 237 4 rehabilitation rehabilitation NOUN cana-5366 237 5 engineering engineering NOUN cana-5366 237 6 . . PUNCT cana-5366 238 1 [ [ X cana-5366 238 2 23 23 NUM cana-5366 238 3 ] ] X cana-5366 238 4 hogan hogan PROPN cana-5366 238 5 , , PUNCT cana-5366 238 6 n. n. PROPN cana-5366 238 7 ( ( PUNCT cana-5366 238 8 1985 1985 NUM cana-5366 238 9 ) ) PUNCT cana-5366 238 10 . . PUNCT cana-5366 239 1 impedance impedance NOUN cana-5366 239 2 control control PROPN cana-5366 239 3 : : PUNCT cana-5366 239 4 an an DET cana-5366 239 5 approach approach NOUN cana-5366 239 6 to to ADP cana-5366 239 7 manipulation manipulation NOUN cana-5366 239 8 . . PUNCT cana-5366 240 1 journal journal NOUN cana-5366 240 2 of of ADP cana-5366 240 3 dynamic dynamic ADJ cana-5366 240 4 systems system NOUN cana-5366 240 5 , , PUNCT cana-5366 240 6 measurement measurement NOUN cana-5366 240 7 , , PUNCT cana-5366 240 8 and and CCONJ cana-5366 240 9 control control NOUN cana-5366 240 10 . . PUNCT cana-5366 241 1 [ [ X cana-5366 241 2 24 24 NUM cana-5366 241 3 ] ] PUNCT cana-5366 241 4 koΕ‚aczek koΕ‚aczek NOUN cana-5366 241 5 , , PUNCT cana-5366 241 6 g. g. PROPN cana-5366 241 7 , , PUNCT cana-5366 241 8 \ \ PROPN cana-5366 241 9 & & CCONJ cana-5366 241 10 mizera mizera NOUN cana-5366 241 11 - - PUNCT cana-5366 241 12 pietraszko pietraszko VERB cana-5366 241 13 , , PUNCT cana-5366 241 14 j. j. PROPN cana-5366 241 15 ( ( PUNCT cana-5366 241 16 2018 2018 NUM cana-5366 241 17 ) ) PUNCT cana-5366 241 18 . . PUNCT cana-5366 242 1 security security NOUN cana-5366 242 2 framework framework NOUN cana-5366 242 3 for for ADP cana-5366 242 4 dynamic dynamic ADJ cana-5366 242 5 service service NOUN cana-5366 242 6 - - PUNCT cana-5366 242 7 oriented orient VERB cana-5366 242 8 it it PRON cana-5366 242 9 systems system NOUN cana-5366 242 10 . . PUNCT cana-5366 243 1 journal journal NOUN cana-5366 243 2 of of ADP cana-5366 243 3 informational informational ADJ cana-5366 243 4 and and CCONJ cana-5366 243 5 telecommunication telecommunication NOUN cana-5366 243 6 , , PUNCT cana-5366 243 7 2(4 2(4 NUM cana-5366 243 8 ) ) PUNCT cana-5366 243 9 , , PUNCT cana-5366 243 10 428 428 NUM cana-5366 243 11 - - SYM cana-5366 243 12 -448 -448 NOUN cana-5366 243 13 . . PUNCT cana-5366 244 1 [ [ X cana-5366 244 2 25 25 NUM cana-5366 244 3 ] ] X cana-5366 244 4 deshpande deshpande PROPN cana-5366 244 5 , , PUNCT cana-5366 244 6 k. k. PROPN cana-5366 244 7 , , PUNCT cana-5366 244 8 girkar girkar PROPN cana-5366 244 9 , , PUNCT cana-5366 244 10 j. j. PROPN cana-5366 244 11 , , PUNCT cana-5366 244 12 \ \ PROPN cana-5366 244 13 & & CCONJ cana-5366 244 14 mangrulkar mangrulkar PROPN cana-5366 244 15 , , PUNCT cana-5366 244 16 r. r. PROPN cana-5366 244 17 ( ( PUNCT cana-5366 244 18 2023 2023 NUM cana-5366 244 19 ) ) PUNCT cana-5366 244 20 . . PUNCT cana-5366 245 1 security security NOUN cana-5366 245 2 enhancement enhancement NOUN cana-5366 245 3 and and CCONJ cana-5366 245 4 analysis analysis NOUN cana-5366 245 5 of of ADP cana-5366 245 6 images image NOUN cana-5366 245 7 using use VERB cana-5366 245 8 a a DET cana-5366 245 9 novel novel ADJ cana-5366 245 10 sudoku sudoku ADV cana-5366 245 11 - - PUNCT cana-5366 245 12 based base VERB cana-5366 245 13 encryption encryption NOUN cana-5366 245 14 algorithm algorithm NOUN cana-5366 245 15 . . PUNCT cana-5366 246 1 journal journal NOUN cana-5366 246 2 of of ADP cana-5366 246 3 informational informational ADJ cana-5366 246 4 and and CCONJ cana-5366 246 5 telecommunication telecommunication NOUN cana-5366 246 6 , , PUNCT cana-5366 246 7 0(0 0(0 NUM cana-5366 246 8 ) ) PUNCT cana-5366 246 9 , , PUNCT cana-5366 246 10 1 1 NUM cana-5366 246 11 - - NUM cana-5366 246 12 -34 -34 NUM cana-5366 246 13 . . PUNCT cana-5366 247 1 [ [ X cana-5366 247 2 26 26 NUM cana-5366 247 3 ] ] X cana-5366 247 4 vΓΆrΓΆs vΓΆrΓΆs X cana-5366 247 5 , , PUNCT cana-5366 247 6 p. p. NOUN cana-5366 247 7 , , PUNCT cana-5366 247 8 csubΓ‘k csubΓ‘k PROPN cana-5366 247 9 , , PUNCT cana-5366 247 10 d. d. PROPN cana-5366 247 11 , , PUNCT cana-5366 247 12 hudoba hudoba NOUN cana-5366 247 13 , , PUNCT cana-5366 247 14 p. p. NOUN cana-5366 247 15 , , PUNCT cana-5366 247 16 \ \ PROPN cana-5366 247 17 & & CCONJ cana-5366 247 18 kiss kiss PROPN cana-5366 247 19 , , PUNCT cana-5366 247 20 a. a. NOUN cana-5366 247 21 ( ( PUNCT cana-5366 247 22 2020 2020 NUM cana-5366 247 23 ) ) PUNCT cana-5366 247 24 . . PUNCT cana-5366 248 1 securing secure VERB cana-5366 248 2 personal personal ADJ cana-5366 248 3 data datum NOUN cana-5366 248 4 in in ADP cana-5366 248 5 public public ADJ cana-5366 248 6 cloud cloud NOUN cana-5366 248 7 . . PUNCT cana-5366 249 1 journal journal NOUN cana-5366 249 2 of of ADP cana-5366 249 3 informational informational ADJ cana-5366 249 4 and and CCONJ cana-5366 249 5 telecommunication telecommunication NOUN cana-5366 249 6 , , PUNCT cana-5366 249 7 4(1 4(1 NOUN cana-5366 249 8 ) ) PUNCT cana-5366 249 9 , , PUNCT cana-5366 249 10 51 51 NUM cana-5366 249 11 - - NUM cana-5366 249 12 -66 -66 PROPN cana-5366 249 13 . . PUNCT