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具时滞人机系统的Hopf-zero分支分析(英文).docx

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具时滞人机系统的Hopf-zero分支分析(英文) Hopf-zerobifurcationanalysisoftime-delayedsystems Introduction: Hopf-zerobifurcationanalysishasbeenwidelyusedtostudythebehaviorofnonlineardynamicalsystems.Thesesystemsoftenexhibitlimitcyclesandoscillations,whichcanbedifficulttopredictandunderstand.Inthispaper,wewillexplorethebehavioroftime-delayedsystemswiththeHopf-zerobifurcation.Wewilldefinewhattime-delayedsystemsare,anddiscussthesignificanceofHopf-zerobifurcationanalysisinstudyingtheirbehavior.Wewillthenpresentsomeexamplesoftime-delayedsystemsandtheirHopf-zerobifurcations,illustratingthedifferenttypesoflimitcyclesandoscillationsthatcanoccur. Time-delayedSystems: Time-delayedsystemsarethoseinwhichafeedbackloopcausesadelayintheresponseofthesystem.Thesesystemsarefoundinmanyfieldsofscienceandengineering,includingthestudyofbiologicalandphysiologicalsystems,aswellasmechanicalandelectricalsystems.Time-delayedsystemsareparticularlychallengingtoanalyzebecausethedelayedresponsecreatesanonlinearbehaviorthatcanleadtocomplex,unpredictablebehavior.ThestudyoftheHopf-zerobifurcationintime-delayedsystemsprovidesinsightintothebehaviorofthesesystemsandcanhelppredicttheonsetoflimitcyclesandoscillations. Hopf-zeroBifurcation: TheHopf-zerobifurcationisatypeofbifurcationthatoccurswhenasystemexperiencesachangeindynamicsduetoachangeinaparameter.Inthiscase,thesystemtransitionsfromastableequilibriumpointtoanunstableone,leadingtotheappearanceofalimitcycleoroscillation.TheHopf-zerobifurcationhasbeenstudiedextensivelyinnonlineardynamicalsystemsandprovidesausefultoolinunderstandingandpredictingtheirbehavior.Hopf-zerobifurcationanalysiscanrevealthetypeoflimitcyclesthatcanoccur,aswellastheirfrequenciesandamplitudes. Examples: Oneexampleofatime-delayedsystemisasimplependulumwithadelayinthefeedbackloop.Thesystemcanexhibitdifferenttypesofoscillations,suchasperiodic,quasiperiodic,orchaotic.BystudyingtheHopf-zerobifurcationinthissystem,wecanpredictthefrequencyandamplitudeoftheoscillation. Anotherexampleisachemicalreactionsys