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1、Hopfbifurcationforinertialtwo・neuronsystemwithdistributeddelaysXiaofengLiaoSchoolofComputerScienceandEngineering,ChongqingUniversity,Chongqing400030,P.R.ChinaEmail:xfliao@cqu.edu.cnAbstractInthispaper,aninertialtwo-neuronsystemwithdistributeddelaysfortheweakkernelis
2、investigated・Byapplyingthefrequency-domainapproach,theexistenceofbifurcationparameterpointisdetermined・Somenumericalsimulationsforjustifyingthetheoreticalanalysisarealsogiven.Keywords:inertial,neuralnetwork,distributeddelay,Hopfbifurcation.LIntroductionNeuralnetwork
3、s[IJarehighlycomplexandlarge-scalenonlineardynamicalsystems.Tosimplifytheanalysisandcomputation,manyresearchershavedirectedtheirattentiontothestudyofsimplesystems[2-8].Thisisstillusefulsincethecomplexityfoundmaybeearnedovertolargenetworks.WheelerandSchieve[2]hadcons
4、ideredthefollowinginertialneuronsystem:磷=-〃]阱-KM+丿]]tanh(〃]),⑴whereU、denotesthemeansomapotentialoftheeffectiveneuron,Mand丿]]arenon-zerorealnumbers.Thestabilityofthefixedpointsofsystem(1)havebeenfoundandanalyzed,andatwo-neuronsystemwithoneortwoinertialtermsaddedhasbe
5、enshowntoexhibitchaos[2].Inordertodescribethedynamicalbehaviorofthemodelthatdependsonthepasthistoryofthesystem,itisoftennecessarytointroducetime-delaysintothemodels・Recently,somesimpleneuronmodelswithdiscreteanddistributeddelaysareproposed[9-16].Lietal.[9]studiedthe
6、followingsingleinertialneuronmodelwithsingledelay:d-a&r-bx+cf(x—hx(t一r)).(2)whereconstantsa,h,c>0,/?>0,andr>0isthetimedelay.Hopfbifurcationwasstudiedbyusingthenormalformtheoryofretardedfunctionaldifferentialequations・Whenadoptinganonmonotonicactivationfunction,chaot
7、icbehaviorwasalsoobservedin[9].Liaoetal.[10]studiedthefollowingtwo-neuronsystemwithdistributeddelays:dt如⑴dt=一(/)+0：.门兀2⑴一b2『F(Q)X2=-x2⑴+(0-b{P(/-r)dr-c2],*where%、biand(z=1,2)arenonnegativenumbers,andx((i=1,2)denotesthemeansomapotential*ofthezthneuronwhile%(z=1,2)coi
8、Tespondtoitsvaluerange,fori=1,2.Moreover,thememoryHopfbifurcationforinertialtwo・neuronsystemwithdistributeddelaysXiaofengLiaoSchoolofCompu
