Dynamics of a class of discrete-time neural networ

MathematicsandComputersinSimulation53(2000)1–39Dynamicsofaclassofdiscrete-timeneuralnetworksandtheircontinuous-timecounterpartsS.Mohamad;1,K.GopalsamySchoolofInformaticsandEngineering,FlindersUniversityofSouthAustralia,BedfordPark,SA5042,AustraliaReceived1January2000;accepted1February2000AbstractThedynamicalcharacteristicsofcontinuous-timeadditiveHopfield-typeneuralnetworksarestudied.Sufficientconditionsareobtainedforexponentiallystableencodingoftemporallyuniformexternalstimuli.Discrete-timeanaloguesofthecorrespondingcontinuous-timemodelsareformulatedanditisshownanalyticallythatthedynamicsofthenetworksarepreservedbybothcontinuous-timeanddiscrete-timesystems.Twomajorconclusionsaredrawnfromthisstudy:firstly,itdemonstratesthesuitabilityoftheformulateddiscrete-timeanaloguesasmathematicalmodelsforstableencodingofassociativememoriesassociatedwithexternalstimuliindiscretetime,andsecondly,itillustratesthesuitabilityofourdiscrete-timeanaloguesasnumericalalgorithmsinsimulatingthecontinuous-timenetworks.©2000IMACS.PublishedbyElsevierScienceB.V.Allrightsreserved.MSC:34C35;34D99;34K20;39A10;39A11;39A12;65C20;65D30;65L20;92B20Keywords:Neuralnetworkswithtimedelays;Continuous-timemodels;Discrete-timemodels;Globalexponentialasymptoticstability;Numericalsolutions1.IntroductionInmodellingandanalysisofdynamicalsystems,varioustypesofsystemsrangingdownwardincom-plexityfrompartialdifferentialequations,functionaldifferentialequations,integrodifferentialequations,stochasticequationswithhereditaryterms,differenceequationsandalgebraicequationshavebeenused.Itiscommontoapproximatemodelsofhigherlevelsofcomplexitybymodelsoflowerlevelsofcom-plexity.OneofthemostwidelyusedtechniquesinthestudyofmodelsinvolvingordinarydifferentialequationsistoapproximatethesystembymeansofasystemofdifferenceequationswhosesolutionsareCorrespondingauthor.Tel.:C61-8-8201-2890;fax:C61-8-8201-2904.E-mailaddresses:ms-sec@ms.flinders.edu.au(S.Mohamad),gopal@ist.flinders.edu.au(K.Gopalsamy).1OnleavefromDepartmentofMathematics,UniversityofBruneiDarussalam,BandarSeriBegawanBE1410,BruneiDarussalam.0378-4754/00/$20.00©2000IMACS.PublishedbyElsevierScienceB.V.Allrightsreserved.PII:S0378-4754(00)00168-32S.Mohamad,K.Gopalsamy/MathematicsandComputersinSimulation53(2000)1–39expectedtobesamplesofthesolutionsofdifferentialequationsatdiscreteinstancesoftime.Thistypeofapproximationoccurswhennumericalintegrationalgorithmsareusedforcomputersimulationofthecontinuous-timesystems,asinthecaseofEuler-typemethodsandRunge–Kuttamethods.Ithasbeenshownbyseveralauthorsthatthedynamicsofnumericaldiscretizationsofdifferentialequationscandiffersignificantlyfromthoseoftheoriginaldifferentialequations.Forinstance,itisknown[27,28,34,43]thatdiscrete-timeversionscanpossessspurioussteady-statesolutionsandspuriousasymptoticbehaviourwhicharenotinherentintheoriginalmotherversionsofdifferentialequations.Byspuriousasymptoticbehaviour,wemeantheasymptoticbehaviourofthediscretizedsystemswhichisnotpossessedbythecontinuous-timesystems.Considerforexamplethesimplescalardifferentialequationdy.t/dtD−y.t/;t0:(1.1)AnEuler-typediscretizationof(1.1)leadstoadiscreteversionof(1.1)ofthefollowingform:y.nC1/D.1−h/y.n/;nD0;1;2;:::;(1.2)inwhichh0denotesthediscretizationstepsizeandy(n)thevalueofy(t)fortDnh.Solutionsof(1.1)and(1.2)are,respectively,givenbyy.t/De−ty.0/;t0;andy.n/D.1−h/ny.0/;nD1;2;3;::::(1.3)Itiselementarytonotethaty(t)of(1.1)satisfies|y(t)|!0ast!1andtheconvergenceismonotonic.However,y(n)of(1.2)forvariousvaluesofh0hasthefollowingbehaviouralaspects:0h1:|y(n)|!0asn!1(monotonicconvergence).hD1:y(n)D0forallnD1;2;3;:::foranychoiceofy(0).1h2:y(n)isoscillatorywith|y(n)|!0asn!1(oscillatoryconvergence).hD2:y(n)isperiodicwithperiod,pD2.h2:y(n)isoscillatoryand|y(n)|!1asn!1.ThisexampledemonstratesthecapabilityofthestandardEulermethodinproducingspuriousdynamicswhichcertainlyarenotpresentintheoriginalcontinuous-timeversion.Ifonediscretizes(1.1)byusingacentraldifferencescheme,thecorrespondingdiscrete-timesystemisgivenbyy.nC1/Dy.n−1/−2hy.n/;nD0;1;2;:::;(1.4)(wherey(n)denotesy(nh))whichhasaphasespaceofdimension2(theinitialvaluesfor(1.4)belongtoR2)withsolutionsgivenbyy.n/Dc1rn1Cc2rn2;nD0;1;2;:::;(1.5)wherec1andc2areconstantsdependingontheinitialvaluesy(0)andy(−1)andr1D−hCp1Ch2;r2D−h−p1Ch2:(1.6)Onecanverifythat0r11andr2−1forh0.Itfollowsthat,forlargen,y(n)in(1.5)oscillatesand|y(n)|!1asn!1whenc26D0.Thesetwoexamplesindicatethatdiscrete-timeversionsofcontinuous-timesystemsdonotsharetheasymptoticbehaviouroftheircontinuous-timemotherversions.S.Mohamad,K.Gopalsamy/MathematicsandComputersinSimulation53(2000)1–393Weconsideranotherwellknownelementarynonlinearordinarydifferentialequation:dy.t/dtDy.t/[1−y.t/];t0:(1.7)Theautonomoussystem(1.7)hastwoequilibriumstatesy(t)D0andy(t)D1.Itisnotdifficulttoseethatthetrivialsolutionisunstablewhilethepositiveequilibriumof(1.7)isasymptoticallystableandallsolutionsof(1.7)satisfyy.0/0)y.t/0fort0andy.t/!1ast!1;(1.8)andtheconvergenceismonotonic.Oneofthediscreteversionsof(1.7)isgivenbythequadraticl