3 外文翻译原文 Complex transient dynamics in periodically forced memristive Chuas circuit.pdf

NONLINEARDYN20157923332343DOI101007/S1107101418151ORIGINALPAPERCOMPLEXTRANSIENTDYNAMICSINPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITBOCHENGBAOPANJIANGHUAGANWUFENGWEIHURECEIVED25JUNE2014/ACCEPTED14NOVEMBER2014/PUBLISHEDONLINE2DECEMBER2014SPRINGERSCIENCEBUSINESSMEDIADORDRECHT2014ABSTRACTWHENASINUSOIDALVOLTAGESTIMULUSISAPPLIED,MEMRISTIVECHUASCIRCUITBECOMESANONAUTONOMOUSPERIODICALLYFORCEDNONLINEARCIRCUITBYUTILIZINGTHEORETICALFORMULATIONS,SIMULATIONSANDEXPERIMENTALVERIFICATIONS,THECOMPLEXTRANSIENTDYNAMICSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITISINVESTIGATEDINTHISPAPERITCANBEFOUNDTHATTHEEQUILIBRIUMPOINTOFTHECIRCUITSWITCHESBETWEENALINEEQUILIBRIUMANDNOEQUILIBRIUMWITHTHETIMEEVOLUTIONS,ANDTHECIRCUITEXHIBITSPERIOD,CHAOSANDALSOHYPERCHAOSINAPARAMETERRANGEOFTHESTIMULUSFREQUENCYMOREOVER,SOMEABUNDANTINTERESTINGNONLINEARPHENOMENAINCLUDINGTRANSIENTCHAOS,TRANSIENTHYPERCHAOSANDCHAOTICBEATSAREREVEALEDNUMERICALLYANDVERIFIEDEXPERIMENTALLYKEYWORDSMEMRISTIVECHUASCIRCUITSTIMULUSFREQUENCYEQUILIBRIUMCHAOTICBEATSTRANSIENTHYPERCHAOSBBAOBPJIANGFHUSCHOOLOFINFORMATIONSCIENCEANDENGINEERING,CHANGZHOUUNIVERSITY,CHANGZHOU213164,CHINAEMAILMERVINBAO126COMHWUDEPARTMENTOFELECTRONICENGINEERING,NANJINGUNIVERSITYOFSCIENCEANDTECHNOLOGY,NANJING210094,CHINA1INTRODUCTIONMEMRISTOR,CONSTITUTINGAFUNDAMENTALNONLINEARCIRCUITELEMENT,BELONGSTOTHENEWESTINNOVATIONSINTHEFIELDOFELECTRONICENGINEERINGDUETOTHENONLINEARITIES,MEMRISTORSARECONVENIENTLYUTILIZEDTOCONSTRUCTSOMENOVELCHAOTICCIRCUITS17RECENTLY,BYREPLACINGTHECHUASDIODESINVARIOUSCHUASCHAOTICCIRCUITSWITHMEMRISTORSCHARACTERIZEDBYDIFFERENTNONLINEARITIES,NEWSERIESOFMEMRISTORBASEDCHUASCHAOTICCIRCUITSAREPROPOSEDANDTHEIRCORRESPONDINGSTABILITIESANDCOMPLICATEDNONLINEARDYNAMICALBEHAVIORSAREINVESTIGATED27ITISCONCLUDEDTHAT,DIFFERENTFROMGENERALNONLINEARDYNAMICALSYSTEMS811,MEMRISTORBASEDCHUASCHAOTICCIRCUITSHAVEANEQUILIBRIUMSETLOCATEDONTHEAXIS,THEPLANEORTHEPHASESPACECONSTITUTEDBYTHEINNERSTATEVARIABLESOFMEMRISTORS,ANDTHEREFORE,THEIRSTABILITIESANDDYNAMICSDEPENDONTHEINNERINITIALCONDITIONSOFTHEMEMRISTORSOTHERTHANTHECIRCUITPARAMETERS12,13THEEQUILIBRIUMSETCORRESPONDINGTOANAXISISALSOCALLEDASALINEEQUILIBRIUM14THEUNUSUALFEATUREOFHAVINGALINEEQUILIBRIUMMAKETHEMEMRISTORBASEDCHUASCHAOTICCIRCUITSEXHIBITSOMEINTERESTINGDYNAMICALBEHAVIORSINCLUDINGTRANSIENTCHAOS4,12,TRANSIENTHYPERCHAOS13ANDMULTIPLECOMPLEXTRANSIENTTRANSITIONS4,12,13NOTABLY,WHENANEXTERNALSINUSOIDALFORCINGISADDEDINTOAMEMRISTIVECHUASCIRCUIT,THEMEMRISTIVECHUASCIRCUITISNONAUTONOMOUSANDPERIODICALLYFORCED,ANDTHEEQUILIBRIUMPOINTOFTHECIRCUITSWITCHESBETWEENALINEEQUILIBRIUMANDNO1232334BBAOETALEQUILIBRIUMWITHTHETIMEEVOLUTIONS,WHICHLEADSTOTHATTHESELFEXCITEDANDHIDDENATTRACTORSAPPEARALTERNATELYITISVERYINTERESTEDTHATASELFEXCITEDATTRACTORWITHABASINOFATTRACTIONISEXCITEDFROMUNSTABLEEQUILIBRIA,WHEREASAHIDDENATTRACTORWITHASMALLBASINOFATTRACTIONDOESNOTBEASSOCIATEDWITHANUNSTABLEEQUILIBRIUM1418HOWEVER,EXCEPTFORTHECHAOTICBEATSREPORTEDIN7,SOMECOMPLEXTRANSIENTDYNAMICSDEPENDENTONTHESTIMULUSFREQUENCYCANBEFOUNDINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUIT,ANDTHECORRESPONDINGINVESTIGATIONSAREACCORDINGLYNEEDEDTOUNDERSTANDTHECIRCUITBETTERESPECIALLY,DUETOFEWRESEARCHREPORTS5,19,ITAPPEARSVERYIMPORTANTANDNECESSARYTHATEXPERIMENTALOBSERVATIONSOFTHEPERIODICALLYFORCEDMEMRISTIVECHAOTICCIRCUITSAREPERFORMEDBYUSINGPHYSICALLYVISUALIZEDCIRCUITSTHEREFORE,BYUTILIZINGVARIOUSDYNAMICALMETHODS,WEWILLSTUDYAPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITANDSHOWITSCOMPLEXTRANSIENTDYNAMICSINTHISPAPERTHISPAPERISORGANIZEDASFOLLOWSSECTION2DEALSWITHTHEDYNAMICALMODELINGOFTHEPROPOSEDCIRCUIT,THEPHASEPORTRAITSOFCHAOTICATTRACTORINAFINITETIMEINTERVAL,THEEQUILIBRIUMPOINTANDITSCORRESPONDINGJACOBIANMATRIX,ASWELLASLYAPUNOVEXPONENTSDEPENDENTONTHESTIMULUSFREQUENCYINSECT3,SOMECOMPLEXTRANSIENTBEHAVIORS,FOREXAMPLES,TRANSIENTCHAOS,TRANSIENTHYPERCHAOSANDCHAOTICBEATS,DEPENDENTONTHESTIMULUSFREQUENCYAREDEPICTEDEXPERIMENTALVERIFICATIONSWITHDIFFERENTSINUSOIDALVOLTAGESTIMULIAREPERFORMEDTOPARTIALLYVERIFYTHENUMERICALSIMULATIONSINSECT4,INCLUDINGCHAOTICATTRACTOR,HYPERCHAOTICATTRACTORANDCHAOTICBEATSTHECONCLUSIONSARESUMMARIZEDINTHELASTSECTION2PERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITANDITSDYNAMICS21MEMRISTIVECHUASCIRCUITCONTAININGASINUSOIDALSTIMULUSANEWMEMRISTIVECHUASCIRCUITISPROPOSEDANDSHOWNINFIG1,WHICHISEXTENDEDFROMAMEMRISTIVECHUASCIRCUITIN5BYADDINGASINUSOIDALVOLTAGESTIMULUSVSINTOTHEINDUCTORBRANCHLDUETOTHEINTRODUCTIONOFTHESINUSOIDALVOLTAGESTIMULUS,THEPROPOSEDCIRCUITINFIG1ISCHANGEDINTOANONAUTONOMOUSCIRVRC1LC2V1V2II3VSMFIG1MEMRISTIVECHUASCIRCUITCONTAININGASINUSOIDALVOLTAGESTIMULUSCUITANDCANBEDEFINEDASAPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITTHENONLINEARCIRCUITINFIG1ISCOMPOSEDOFTWOCAPACITORSC1ANDC2,ANINDUCTORL,ARESISTORR,AMEMRISTORMANDASINUSOIDALVOLTAGESOURCEVSNOTETHATTHEMEMRISTORMISANUNIQUENONLINEARELEMENTLETTHATVTANDITDENOTETHEVOLTAGEACROSSANDTHECURRENTFLOWINGTHROUGHTHEMEMRISTOR,RESPECTIVELY,ANDTANDWREPRESENTTHEINNERSTATEVARIABLEANDTHEMEMDUCTANCEOFTHEMEMRISTOR,RESPECTIVELYTHUS,THEMATHEMATICALMODELOFTHEMEMRISTORMCANBEDESCRIBEDAS5ITWVTWAB|T|DT/DTVT1THEMATHEMATICALMODELOF1ACCORDSWITHTHEDEFININGEQUATIONFORTHECLASSOFIDEALFLUXCONTROLLEDMEMRISTORSANDCANEXHIBITAPINCHEDHYSTERESISLOOPINTHEVIPLANEASABIPOLARPERIODICVOLTAGESTIMULUSISAPPLIED20ITISREMARKABLETHATTHEMEMRISTORMODELINTHISPAPERISCHARACTERIZEDBYASMOOTHPIECEWISEQUADRATICNONLINEARITY5,WHEREASTHEMEMRISTORMODELINREF7ISDESCRIBEDBYAPIECEWISELINEARITYTHEREAREFOURSTATEVARIABLESOFV1,V2,I3,ANDINFIG1,WHICHREPRESENTTHEVOLTAGEOFTHECAPACITORC1,THEVOLTAGEOFTHECAPACITORC2,THECURRENTOFTHEINDUCTORLANDTHEMAGNETICFLUXOFTHEFLUXCONTROLLEDMEMRISTORM,RESPECTIVELYHERE,THEAPPLIEDSINUSOIDALVOLTAGESTIMULUSISVSFSINT,WHEREFANDARETHEAMPLITUDEANDFREQUENCYOFTHESTIMULUS,RESPECTIVELYTHUS,APPLYINGKIRCHHOFFSCIRCUITLAWSANDTHEVOLTAGECURRENTRELATIONSOFCIRCUITELEMENTS,THEDYNAMICALEQUATIONOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITINFIG1ISEXPRESSEDAS123MEMRISTIVECHUASCIRCUIT2335TABLE1CIRCUITPARAMETERSFORSIMULATIONSANDEXPERIMENTSPARAMETERSSIGNIFICATIONSVALUESC1CAPACITANCE68NFC2CAPACITANCE68NFLINDUCTANCE172MHRRESISTANCE21KOMEGA1ACOEFFICIENT06667MSBCOEFFICIENT14828S/WBFAMPLITUDE20MVRADIANFREQUENCY1,000RAD/SC1DV1DTV2V1RWV1C2DV2DTV1V2RI3LDI3DTV2FSINTDDTV12WHEREWISANONLINEARFUNCTIONEXPRESSEDBY1,V1,V2,I3ANDAREFOURCIRCUITVARIABLES,ANDC1,C2,L,A,B,FANDARESEVENCIRCUITPARAMETERSEQUATION2CORRESPONDSTOAFOURDIMENSIONALNONLINEARNONAUTONOMOUSDYNAMICALSYSTEM,UPONWHICHDYNAMICALBEHAVIORSOFTHECIRCUITINFIG1CANBEREVEALEDNUMERICALLYTOSIMPLIFYTHEDYNAMICALEQUATIONOFTHEPROPOSEDCIRCUIT,THEEQUIVALENTSERIESRESISTANCEESROFTHEINDUCTORLISIGNOREDBECAUSEOFTHATTHEPARASITICRESISTANCEDOESNOTAFFECTITSBIFURCATIONSTRUCTURESCONSEQUENTLY,THEDYNAMICALEQUATIONOF2ISSIMILARTOTHATPRESENTEDBYREF7EXCEPTFORTHATTWODIFFERENTMEMRISTORMODELSAREUSEDINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITCONSIDERINGTHATTHECIRCUITPARAMETERSINFIG1,ASLISTEDINTABLE1,ARESAMEASTHOSEINREF5ANDTHEINITIALVALUESOFFOURSTATEVARIABLESARETAKENASV101V,V200V,I300A,AND00WB,ITISDEMONSTRATEDTHATTHEMEMRISTIVECHUASCIRCUITWITHOUTTHEAPPLIEDVOLTAGESTIMULUSISCHAOTIC5WHENASINUSOIDALVOLTAGESTIMULUSWITH1,000RAD/SANDF20MVISAPPLIEDINTHECIRCUIT,THEPHASEPORTRAITSINDIFFERENTPHASEPLANESAREPLOTTEDINFIG2ACCORDINGTOTHEFOLLOWINGNUMERICALMETHOD,THELYAPUNOVEXPONENTSARECALCULATEDASLE11,2541,LE29548,LE30,LE425327,ANDLE57,1258THEREFORE,TWOLYAPUNOVEXPONENTSOFLE1ANDLE2AREGREATERTHANZERO,WHICHMEANSTHATTHECIRCUITOPERATESINHYPERCHAOTICSTATEITISNOTEDTHATTHEPHASEPORTRAITSOFTHECHAOTICATTRACTORINFIG2ANDTHECORRESPONDINGLYAPUNOVEXPONENTSAREDISPLAYEDINAFINITETIMEINTERVALFROMTHEABOVECONCLUSIONS,ITCANBEFOUNDTHATTHEFREQUENCYANDAMPLITUDEOFTHESINUSOIDALVOLTAGESTIMULUSARETWOSIGNIFICANTCONTROLPARAMETERSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUIT,WHICHCANMAKETHEDYNAMICSOFTHECIRCUITTOHAVEATRANSITIONFROMCHAOTICTOHYPERCHAOTICBEHAVIORSBECAUSETHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITISEASILYSTABILIZEDBYTHESINUSOIDALVOLTAGESTIMULUSWITHLARGERAMPLITUDE,THESINUSOIDALVOLTAGESTIMULUSWITHSMALLERAMPLITUDEISUTILIZEDANDTHEDYNAMICSOFTHECIRCUITDEPENDENTONTHEFREQUENCYOFTHESINUSOIDALVOLTAGESTIMULUSISFOCUSEDTOINVESTIGATEINTHISPAPER22EQUILIBRIUMPOINTANDJACOBIANMATRIXTHEEQUILIBRIUMPOINTSOF2AREOBTAINEDBYSETTINGTHELEFTHANDSIDETOZEROITISOBVIOUSLYTHATTHEREEXISTSTWOCASESCASE1WHENTKKISAPOSITIVEINTEGER,IE,VSFSINT0,THEEQUILIBRIUMSTATEOF2ISGIVENBYANEQUILIBRIUMSETEV1,V2,I3,|V1V2I30,C3FIG2CHAOTICATTRACTOROFPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITINAFINITETIMEINTERVALAPHASEPORTRAITINV1TV2TPLANEBPHASEPORTRAITINV1TTPLANEV1VV2VAV1VMWBB1232336BBAOETALWHERECISUNCERTAINBUTCONSTANT,CORRESPONDINGTOTHEAXIS5THUS,THEEQUILIBRIUMSETEISALSOKNOWNASALINEEQUILIBRIUM14CASE2WHENTNEGATIONSLASHKKISAPOSITIVEINTEGER,IE,VSFSINTNEGATIONSLASH0,NOEQUILIBRIUMSTATEEXISTSIN2,IMPLYINGTHATTHEPROPOSEDCIRCUITDOESNOTEXHIBITANYEQUILIBRIUMPOINTSTHECHAOTICATTRACTORSDISPLAYEDINDYNAMICALSYSTEMSWITHOUTANYEQUILIBRIUMPOINTSORWITHONLYSTABLEEQUILIBRIUMPOINTSAREHIDDENATTRACTORS1418WHENVSFSINT0,THEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITBECOMESTHEAUTONOMOUSMEMRISTIVECHUASCIRCUITTHEREFORE,THEJACOBIANMATRIXATTHEEQUILIBRIUMSETEISEXPRESSEDASJE1RC1WCC11RC1001RC21RC21C2001L0010004WHEREWCAB|C|THECHARACTERISTICEQUATIONISDERIVEDAS3A22A1A005WHEREA21RC11RC2WCC1,A1WCRC1C21LC2,A01RLC1C2WCLC1C2TAKINGADVANTAGEOFTHEVALUESOFCIRCUITPARAMETERSLISTEDINTABLE1ANDTHEROUTHHURWITZCONDITION2ISUNSTABLEWHEN|C|0ANDLE20ATFIRST,THENGOESTOPERIODICOSCILLATIONWITHLE10ANDLE20,ENTERSINTOCHAOTICSTATEWITHLE10ANDLE20AGAINAFTER160RAD/SANDFINALLYSETTLESDOWNTOHYPERCHAOTICSTATEWITHLE10,LE20,ANDLE30AFTER746RAD/SSEVERALPHASEPORTRAITSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITUNDERFOURDIFFERENTSTIMULUSFREQUENCIESAREOBTAINEDFROMNUMERICALSIMULATIONS,ASSHOWNINFIG5WHEN0RAD/S,IE,THEFIG4LYAPUNOVEXPONENTSPECTRAWITHTHEVARIATIONSOFTHESTIMULUSFREQUENCYATHEFIRSTFOURLYAPUNOVEXPONENTS,BTHEFIFTHLYAPUNOVEXPONENT,CTHEEXPANDEDPORTIONOFAFORASHORTSTRETCHOFKRAD/SLE103ALE1LE3LE2LE4LE103KRAD/SBCLE5LE4LE2LE3LE1FIG5PHASEPORTRAITSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITACHAOTICATTRACTORWITHOUTTHEAPPLIEDVOLTAGESTIMULUS,BLIMITCYCLEWITHSMALLDYNAMICRANGE100RAD/S,CCHAOTICATTRACTORWITHLARGEDYNAMICRANGE400RAD/S,DHYPERCHAOTICATTRACTOR2,000RAD/SV1VV2VAV1VV2VBV1VV2VCV1VV2VD1232338BBAOETALFIG6TRANSITIONDYNAMICSFROMTRANSIENTCHAOTICTOSTEADYPERIODICBEHAVIORSWHEN400RAD/SAGLOBALTIMEDOMAINWAVEFORMOFTHEVARIABLEV2INTHETIMEINTERVAL0,08S,BSTEADYTIMEDOMAINWAVEFORMOFTHEVARIABLEV2INTHETIMEINTERVAL07,08S,CSTEADYLIMITCYCLEINV1TV2TPLANE,DFIRSTFOURLYAPUNOVEXPONENTSAGAINSTTIMETSV2VATSV2VBV1VV2VCTSLE103DLE1LE2LE3LE4SINUSOIDALVOLTAGESTIMULUSISNOTAPPLIED,ACHAOTICATTRACTORISSHOWNINFIG5A,THECORRESPONDINGLYAPUNOVEXPONENTSARELE117964,LE20,LE30,LE435529,ANDLE57,7153,RESPECTIVELYWHEN100RAD/S,ALIMITCYCLEWITHSMALLDYNAMICRANGEISDISPLAYEDINFIG5B,THELYAPUNOVEXPONENTSARELE10,LE20,LE35212,LE452345,ANDLE513,606,RESPECTIVELYWHEN400RAD/S,ACHAOTICATTRACTORWITHLARGEDYNAMICRANGEISPLOTTEDINFIG5C,THELYAPUNOVEXPONENTSARELE1326,LE20,LE319933,LE41,1687,ANDLE510,354,RESPECTIVELYWHILEWHEN2,000RAD/S,AHYPERCHAOTICATTRACTORISDEPICTEDINFIG5D,THELYAPUNOVEXPONENTSARELE11,1069,LE27625,LE30,LE446919,ANDLE57,1842,RESPECTIVELYITSHOULDBESTATEDTHATTHEDYNAMICALBEHAVIORSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITDEPICTEDINFIGS3,4AND5ARETRANSIENTINAFINITETIMEINTERVALDUETOTHEAPPLIEDSINUSOIDALVOLTAGESTIMULUS,SOMEINTERFERENCESALWAYSEXISTINTHECIRCUIT,RESULTINGINTHATTHEINTRINSICCHAOTICOSCILLATIONMAYNOTOPERATENORMALLYAFTERALONGTIMEINTERVALESPECIALLY,WHENTHEAPPLIEDSINUSOIDALVOLTAGEFORCINGDOMINATESTHEMEMRISTIVECHUASCIRCUIT,THEOSCILLATIONSTATEWILLFOLLOWTHEAPPLIEDSINUSOIDALVOLTAGESTIMULUSWITHTHESMALLDYNAMICRANGEANDTHESAMEOSCILLATIONFREQUENCY3COMPLEXTRANSIENTDYNAMICSDEPENDENTONTHESTIMULUSFREQUENCYACCORDINGTOTHENUMERICALSIMULATIONSOF2,ITISEASYTOFINDTHATTHEREEXISTSCOMPLEXTRANSIENTDYNAMICSINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITTODEMONSTRATETHESETRANSIENTDYNAMICS,SEVERALFOLLOWINGEXAMPLESARECONSIDEREDTHECIRCUITPARAMETERSARELISTEDINTABLE1,THEABOVEINITIALVALUESAREUTILIZED,ANDONLYSEVERALDIFFERENTSTIMULUSFREQUENCIESAREPICKED31TRANSIENTCHAOSWHENTHERADIANFREQUENCYOFTHESTIMULUSISSELECTEDAS400RAD/S,APHENOMENONOFTRANSIENTCHAOSGENERATESINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITIE,THETRAJECTORIESOFTHECIRCUITHAVEATRANSITIONFROMTRANSIENTCHAOTICTOSTEADYPERIODICBEHAVIORSWITHTIMEEVOLUTIONS4,12FIGURE6ADISPLAYSTHETIMEDOMAINWAVEFORMOFVARIABLEV2,WHEREACHAOTICATTRACTORISLOCATEDINTHETIMEINTERVAL0,0505SANDTHENALIMITCYCLEWITHPERIOD1FORMSAFTERT0505SFORCLARITY,THETIMEDOMAINWAVEFORMOFVARIABLEV2INTHESTEADYSTATEOFTHETIMEINTERVAL07,08SISSHOWNINFIG6B,WHERETHEOSCILLATIONFREQUENCYISEQUALTOTHESTIMULUSFREQUENCYOF400RAD/SANDTHEOSCILLATIONAMPLITUDEEQUALSTOTHESTIMULUSAMPLI123MEMRISTIVECHUASCIRCUIT2339FIG7TRANSITIONDYNAMICSFROMTRANSIENTHYPERCHAOTICTOSTEADYCHAOTICBEHAVIORSWHEN1,000RAD/SATHEHYPERCHAOTICATTRACTORCOMBINEDWITHTHECHAOTICATTRACTORINV1TV2TPLANE,BTHEHYPERCHAOTICATTRACTORCOMBINEDWITHTHECHAOTICATTRACTORINV1TTPLANE,CFIRSTFOURLYAPUNOVEXPONENTSAGAINSTTIME,DTIMEDOMAINWAVEFORMOFTHEVARIABLEV2INTHETIMEINTERVAL0S,04SV1VV2VAV1VBMWBTSLE103CLE1LE2LE3LE4TSV2VDTUDEOFF20MVCORRESPONDINGLY,THEPHASEPORTRAITINV1TV2TPLANEISDEMONSTRATEDINFIG6C,WHICHSTANDSFORTHEFORMEDORBITOFTHELIMITCYCLEWITHPERIOD1ULTIMATELYAFTERATRANSIENTTIMEINTERVALINTERESTINGLY,THECIRCUITWITHBIPOLARPERIODICVOLTAGESTIMULUSCANEXHIBITAPINCHEDHYSTERESISLOOPINV1TV2TPLANE,WHICHISSIMILARTOTHECHARACTERISTICFINGERPRINTOFTHEMEMRISTOR20ADDITIONALLY,THEFIRSTFOURLYAPUNOVEXPONENTSLE1,LE2,LE3ANDLE4AGAINSTTIMESHOWNINFIG6DILLUSTRATESTHATTHELYAPUNOVEXPONENTLE1ISGRADUALLYDROPPINGTOZEROWITHTHETIMEEVOLUTIONS,IMPLYINGTHATTHENONLINEARBEHAVIORSINTHECIRCUITCANBESTABILIZEDFINALLYITISREMARKABLETHATWHENTHESTEADYSTATEARRIVES,THECHAOTICOSCILLATIONSWITHLARGEDYNAMICRANGESARECOMPLETELYREPLACEDBYTHEBIPOLARPERIODICOSCILLATIONSWITHSMALLDYNAMICRANGES,WHICHILLUSTRATETHATTHEAPPLIEDFORCINGCANACHIEVETHECHAOSSTABILIZATIONCONTROLOFTHEMEMRISTIVECHUASCIRCUIT32TRANSIENTHYPERCHAOSTRANSIENTCHAOSISACOMMONPHENOMENONOBSERVEDINMANYNONLINEARDYNAMICALSYSTEMS9,10,WHEREINAORBITBEHAVESCHAOTICALLYFORAFINITETIMEINTERVALBEFORESETTLINGINTOAFINALNONCHAOTICSTATETHISARISESDUETOTHEPRESENCEOFNONATTRACTINGCHAOTICSADDLESINPHASESPACEITHASALSOBEENREPORTEDINSOMEMEMRISTORBASEDCHAOTICCIRCUITBY4,12INCONTRASTTOTHETRANSIENTCHAOS,TRANSIENTHYPERCHAOS,ACOMPLEXNONLINEARPHENOMENONWITHTWOPOSITIVELYAPUNOVEXPONENTSINTHEINFINITETIMEINTERVAL,HASBEENFOUNDINTHEMEMRISTIVEHYPERCHAOTICCIRCUITS13,22WHEN1,000RAD/S,THEPHASEPORTRAITSOFTHEHYPERCHAOTICATTRACTORCOMBINEDWITHTHECHAOTICATTRACTOR,TIMEDOMAINWAVEFORMANDFIRSTFOURLYAPUNOVEXPONENTSLE1,LE2,LE3ANDLE4AGAINSTTIMEARESHOWNINFIG7,FROMWHICHITCANBEFOUNDTHATTHETRANSIENTHYPERCHAOSPHENOMENONEXISTSINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITFIGURE7A,BDEPICTSTWOPHASEPORTRAITSOFHYPERCHAOTICATTRACTORCOMBINEDWITHTHECHAOTICATTRACTORINDIFFERENTPHASEPLANESTHESECONDLYAPUNOVEXPONENTLE2INFIG7CCONTINUOUSLYDECREASESFROMAPOSITIVEVALUETOANEGATIVEVALUETHROUGHZEROWITHTHETIMEEVOLUTIONS,IMPLYINGTHATTHEDYNAMICSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITHASATRANSITIONFROMTRANSIENTHYPERCHAOTICTOSTEADYCHAOTICBEHAVIORS,ASSHOWNINFIG7DHENCE,THETIMESERIESPLOTTEDINFIG7DSHOWSTRANSIENTHYPERCHAOSANDTHELOCALLYAPUNOVSPECTRAINFIG7CCONFIRMTHESAME33CHAOTICBEATSWHEN200RAD/S,APHENOMENONOFCHAOTICBEATSOCCURSINTHEPERIODICALLYFORCEDMEMRISTIVE1232340BBAOETALFIG8COMPLEXDYNAMICSOFCHAOTICBEATSWHEN200RAD/SAPHASEPORTRAITINV1TV2TPLANE,BPHASEPORTRAITINV1TTPLANE,CANDDTIMEDOMAINWAVEFORMSOFTHEVARIABLESV2ANDINTHETIMEINTERVAL0,1S,RESPECTIVELYWHERETHEZOOMINWAVEFORMSOFC1ANDD1ARESHOWNINC2ANDD2,RESPECTIVELYV1VV2VAMWBV1VBTSV2VV2VC2C1TSD2D1MWBMWBCHUASCIRCUITTHECORRESPONDINGPHASEPORTRAITSANDTIMEDOMAINWAVEFORMSARESHOWNINFIG8,WHERETHELYAPUNOVEXPONENTSINAFINITETIMEINTERVALARE12030,018,1806,1810,13,690OBSERVEDFROMFIG8C,D,THEENVELOPEFREQUENCYSLOWSCALEFREQUENCYEQUALSTOTHESTIMULUSFREQUENCY,IE,200RAD/S,ANDTHEOSCILLATIONFREQUENCYINEACHENVELOPEDOESJUSTBETHECHAOTICFREQUENCYINOTHERWORDS,THEEXTERNALSTIMULUSAPPLIEDTOTHEMEMRISTIVECHUASCIRCUITISMODULATEDBYTHEINTRINSICCHAOTICOSCILLATIONONCETHEINTERFERENCEBETWEENDIFFERENTOSCILLATIONSEXISTS,ACOMPLEXNONLINEARPHENOMENONOFCHAOTICBEATSISPOSSIBLETOBURSTCHAOTICBEATSWEREREPORTEDANDSTUDIEDINMANYNONLINEARDYNAMICALSYSTEMS22244EXPERIMENTALVERIFICATIONSTHECIRCUITSCHEMATICTOREALIZETHEFLUXCONTROLLEDMEMRISTORCHARACTERIZEDBY1ISSHOWNINFIG93,WHERETHECIRCUITPARAMETERSARELISTEDINTABLE2,WITHAD633JNASANALOGMULTIPLIER,AD711KNASOPERATIONALAMPLIFIER,ANDOPERATIONVOLTAGESOF15VASARESULT,THEMEMRISTOREQUIVALENTPARAMETERSAREA06667MSANDB14828S/WBMOREDETAILSOFTHEMEMRISTOREQUIVALENTCIRCUITREALIZATIONAREDESCRIBEDIN5THUS,ANEXPERIMENTALCIRCUITOFTHEPERIODICALLYVC0IU1R0_U2U3R1R2R3VAVC_HM1VBAVAU4RSAT_M2VTEMPR4_VBU5BFIG9EQUIVALENTCIRCUITOFMEMRISTORAMAINCIRCUITBHCIRCUITWITHABSOLUTEVALUENONLINEARITYTABLE2CIRCUITPARAMETERSOFTHEMEMRISTOREQUIVALENTCIRCUITPARAMETERSSIGNIFICATIONSVALUESC0CAPACITANCE68NFR0RESISTANCE4KOMEGA1R1RESISTANCE15KOMEGA1R2,3RESISTANCE2KOMEGA1,2KOMEGA1R4RESISTANCE605KOMEGA1RSATRESISTANCE135KOMEGA1M1SCALEFACTOR01M2SCALEFACTOR1123MEMRISTIVECHUASCIRCUIT2341FIG10MEASUREDPHASEPORTRAITSOFHYPERCHAOTICATTRACTORWITHASINUSOIDALVOLTAGESTIMULUS1,000RAD/SAV1VERSUSV2,BV2VERSUSVAV21V/DIVV12V/DIVAVA05V/DIVV12V/DIVBFIG11MEASUREDPHASEPORTRAITSOFCHAOTICATTRACTORANDHYPERCHAOTICATTRACTORACHAOTICATTRACTORWITHOUTSTIMULUS,BHYPERCHAOTICATTRACTORWITHASINUSOIDALVOLTAGESTIMULUSHAVING2,000RAD/SV2075V/DIVV115V/DIVAV2075V/DIVV115V/DIVBFIG12CAPTUREDCHAOTICBEATSWITHAVOLTAGESTIMULUS200RAD/SANDF22MVATIMEDOMAINWAVEFORMOFTHEVARIABLESV2,BTHEZOOMINWAVEFORMOFTHEVARIABLESV2V21V/DIVT40MS/DIVAV205V/DIVT1MS/DIVBFORCEDMEMRISTIVECHUASCIRCUITISDESIGNEDANDUSEDTOINVESTIGATETHESTIMULUSFREQUENCYDEPENDEDCOMPLEXDYNAMICSWITHTHESAMECIRCUITPARAMETERSASFORNUMERICALSIMULATIONSWHENASINUSOIDALVOLTAGESTIMULUSWITH1,000RAD/SISAPPLIED,THEEXPERIMENTALOBSERVATIONSINDIFFERENTPHASEPLANESARESHOWNINFIG10WHEREVATINFIG10BSTANDSFORTHEOUTPUTVOLTAGEOFTHEINTEGRATORWITHTIMECONSTANTR0C0INFIG9A,ANDTHEREEXISTSTHERELATIONOFVATT/WHENNOSINUSOIDALVOLTAGESTIMULUSANDASINUSOIDALVOLTAGESTIMULUSWITH2,000RAD/SAREADDEDINTHECIRCUIT,RESPECTIVEL