毕业论文-微分动力系统平衡点的分类

10532S050640112008042110532S05064011:::::20080421:20080518:QualitativeAnalysisofEquilibriainaRingNeuralNetworkModelbyLUXuwenB.S.(HenanUniversityofScienceandTechnology)2005AthesissubmittedinpartialsatisfactionoftheRequirementsforthedegreeofMasterofScienceinAppliedMathematicsintheGraduateSchoolofHunanUniversitySupervisorProfessorGUOShangjiangApril,20081¤2¤(X)I,D4©Z2()Hopf:;;;;HopfAbstractThefocusofthisdissertationistostudyissuesrelatedtoexistence,stabilityandbifurcationofequilibriainaringoffouridenticalneuronswithself-feedbackanddelays,whichhasanon-centero®-surroundcharacteristicforsomesuitableconnectionweights.Suchanetworkhasbeenfoundinavarietyofneuralstruc-tures,suchasneocortex,cerebellum,hippocampus,andeveninchemistryandelectricalengineering,andcanbestudiedtogaininsightintothemechanismsunderlyingthebehaviorofrecurrentnetwork.Thisdissertationisorganizedasfollows:Firstly,thebackgroundandthemotivationforthestudyofarti¯cialneuralnetworksarepresented.Then,someknownresultsofneuralnetworkmodelsareintroduced.Occurrenceofbifurcationandsometraditionalmethodsusedtostudybifurcationareintroducedsimply.Besides,thecentermanifoldreductionandnormalformcalculationina¯nite-dimensionspaceareintroduced.Secondly,basedontheisotropicsubgroupofD4©Z2,weclassifyallthepossibleequilibriaofthisneuralnetwork.Then,weobtainthesu±cientconditionsensuringtheexistenceoftheequilibriaofdi®erentpatterns.Thirdly,wecalculatethecharacteristicequationoftheequilibriaviathelin-earizedequation.Bymeansofthecharacteristicequation,wediscussthestabilityoftheequilibria.Fourthly,byregardingthetwoconnectionweightsasbifurcationparameters,whichisdi®erentfromtraditionalresearchusingthetimedelayasbifurcationparameters,withthehelpofcentermanifoldreductionandnormalformtheory,weinvestigatetheequivariantHopfbifurcationnearthetrivialequilibrium,andobtainsomecriteriaaboutthebifurcationdirectionandstabilityofbifurcatingperiodicsolutions.Finally,wediscussthebifurcationphenomenanearasynchronousequilibriaaswell.KeyWords:Neuralnetwork;Equilibrium;Characteristicequation;Stability;Hopfbifurcation......................................................................................................Abstract.........................................................................................................................................V1..................................................................11.1................................................11.2............................................................22..............................................................52.1............................................52.2..........................................................62.3................................................83............................................103.1.........................................................103.2.........................................................113.3......................................................123.4.............................................................174......................................................204.1......................................................204.2......................................................225........................................................265.1(0;0;0;0)Hopf.............................................265.2(0;0;0;0)Hopf........................................295.3....................................................32.........................................................................37.....................................................................39.................................43.........................................................................441.14......................................................43.1(x¤;x¤;x¤;x¤)........................................183.2(x1;x2;x2;x1)........................................183.3(x1;x2;x1;x2)........................................195.1®¡¯................................................37V11.1(Arti¯cialNeuralNetworksANNs)40McCullochPittsF.RosenblattWidrowJ.J.Hop¯eld()1010-1011([1])()()()([2-8])([9])([6,7,10{17]){1{(),B¶elair[10;11],Campbell[8],Pakdaman[6;7]J.Wu[13;17;19¡21]Cao[22]Chen[19;23]Ncube,WuHuang[20]on-centero®sur-roundDnn([24{26])([17,27])GuoHuang([28,29])GuoHuang[27][30][29][31,32]Campbell(Hopf)OroszSt¶ep¶an[33][34]1.2Hop¯eldHop¯eldMarcusWestervelt[9]Wu[35]Hop¯eld_ui(t)=¡¹iui(t)+nXj=1Tjifj(uj(t¡¿ji))+Ii(1.1)uii;¹i0¹i;¿ji0ji;Tjij{2{iTji0Tji0,fj:R!RjsigmoidHop¯eld([36,37]):_x(t)=¡x(t)+f(x(t¡¿));(1.2),_x(t)=ddtx(t),f2C(R;R)(1.2)Campbell[18]f(x)=¹tanh(x)(1.2),¡1¹1(1.2)¹12:¿f,(1.2)([38])([39][40][41])AlexanderAuchumty[42]Pakdaman[7]([43,44])OthmerScriven[45]Smale[46]Howard[47]([7,13,17,21,42])4(1.1):_ui(t)=¡ui(t)+®f(ui(t¡¿))+¯[f(ui¡1(t¡¿))+f(ui+1(t¡¿))];(1.3)i(mod4),f2C(R;R)f(C1)f(0)=f00(0)=0,f0(0)=1x2Rf0(x)0{3{1.14(C2)xf00(x)0x6=0(C3)f(x)(1.3)ii+1i¡1f®0;¯0i()i+1i¡1()(1.3)on-centero®surround(1.3)[48]Wu3®¡¯(thesetofsyn-chronousstatesinthephasespace)BungayCampbell[49]Hopf[50]Guo(1