Vol.43(2023)No.5����J.ofMath.(PRC)DYNAMICSANALYSISFORTHEFRACTIONAL-ORDERHINDMARSH-ROSENEURONALMODELWITHTIME-DELAYFEEDBACKCONTROLWANGTian,WANGHai-xia(CollegeofMathematicsandStatistics,NanjingUniversityofScienceandTechnology,Nanjing210094,China)Abstract:Inthispaper,westudythedynamicsofthefractional-orderHindmarsh-Rose(HR)neuronalmodelwithtime-delayfeedbackcontrol.Usingstabilitytheoryandbifurcationanalysismethod,theeffectsoffractionalorderandtime-delayonequilibriumstability,bifurcationanddis-chargebehaviorareinvestigatedindetail.Atthesametime,theanalyticalconditionsforHopfbifurcationinfractional-ordermodelareobtained,andtherelevantresultsoftheinteger-orderHRneuronalmodelwithtime-delayfeedbackcontrolareextended.Keywords:fractional-order;Hopfbifurcation;time-delayfeedbackcontrol;stability;Hindmarsh-Roseneuronalmodel2010MRSubjectClassification:37J20;37J25Documentcode:AArticleID:0255-7797(2023)05-0377-121IntroductionInrecentyears,thefractionaldifferentialequationshaveattractedmoreandmoreat-tentionfromresearchersathomeandabroad.Duetothewideapplicationbackground,theyhaveplayedanimportantroleinmathematics[1],physics[2],biology[3],engineering[4]andotherfields[5].Asabranchofcalculus,fractionalcalculusistheextensionofordinaryintegerorderdifferentialandintegraltoarbitraryrealorderdifferentialandintegral.Withtheincreasingmaturityoffractionalcalculustheoryandtherapiddevelopmentofcomputerscienceandtechnology,itisfoundthatusingfractionalcalculustobuildamodelcanaccu-ratelydescribethepropertiesandmorphologyofmatter,andthefractionalcalculusmodelalsohasalargerstableregioncomparedwiththeintegermodel.Thetheoreticalstudyoffractionaldifferentialsystemnotonlyhastheoreticalguidingsignificance,butalsohasaverywideapplicationprospect.Atpresent,thestudyoffractionalneurodynamicshasbecomeanimportantbranchinthefieldofbiologicalnervoussystem.∗Receiveddate:2022-06-09Accepteddate:2022-09-13Foundationitem:SupportedbyNationalNaturalScienceFoundationofChina(12272062).Biography:WangTian(1998–),fema...
