周期性强迫忆阻蔡氏电路中的复杂瞬态动力学【中文7600字】
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周期性
强迫
逼迫
强制
忆阻蔡氏
电路
中的
复杂
繁杂
瞬态
动力学
中文
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周期性强迫忆阻蔡氏电路中的复杂瞬态动力学【中文7600字】,周期性,强迫,逼迫,强制,忆阻蔡氏,电路,中的,复杂,繁杂,瞬态,动力学,中文
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13NONLINEARDYN20157923332343DOI101007/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,ANDTHEEQUILIBRIUMPOINTOFTHECIRCUITSWITCHESBETWEENALINEEQUILIBRIUMANDNO132334BBAOETALEQUILIBRIUMWITHTHETIMEEVOLUTIONS,WHICHLEADSTOTHATTHESELFEXCITEDANDHIDDENATTRACTORSAPPEARALTERNATELYITISVERYINTERESTEDTHATASELFEXCITEDATTRACTORWITHABASINOFATTRACTIONISEXCITEDFROMUNSTABLEEQUILIBRIA,WHEREASAHIDDENATTRACTORWITHASMALLBASINOFATTRACTIONDOESNOTBEASSOCIATEDWITHANUNSTABLEEQUILIBRIUM1418HOWEVER,EXCEPTFORTHECHAOTICBEATSREPORTEDIN7,SOMECOMPLEXTRANSIENTDYNAMICSDEPENDENTONTHESTIMULUSFREQUENCYCANBEFOUNDINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUIT,ANDTHECORRESPONDINGINVESTIGATIONSAREACCORDINGLYNEEDEDTOUNDERSTANDTHECIRCUITBETTERESPECIALLY,DUETOFEWRESEARCHREPORTS5,19,ITAPPEARSVERYIMPORTANTANDNECESSARYTHATEXPERIMENTALOBSERVATIONSOFTHEPERIODICALLYFORCEDMEMRISTIVECHAOTICCIRCUITSAREPERFORMEDBYUSINGPHYSICALLYVISUALIZEDCIRCUITSTHEREFORE,BYUTILIZINGVARIOUSDYNAMICALMETHODS,WEWILLSTUDYAPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITANDSHOWITSCOMPLEXTRANSIENTDYNAMICSINTHISPAPERTHISPAPERISORGANIZEDASFOLLOWSSECTION2DEALSWITHTHEDYNAMICALMODELINGOFTHEPROPOSEDCIRCUIT,THEPHASEPORTRAITSOFCHAOTICATTRACTORINAFINITETIMEINTERVAL,THEEQUILIBRIUMPOINTANDITSCORRESPONDINGJACOBIANMATRIX,ASWELLASLYAPUNOVEXPONENTSDEPENDENTONTHESTIMULUSFREQUENCYINSECT3,SOMECOMPLEXTRANMFIG1MEMRISTIVECHUASCIRCUITCONTAININGASINUSOIDALVOLTAGESTIMULUSCUITANDCANBEDEFINEDASAPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITTHENONLINEARCIRCUITINFIG1ISCOMPOSEDOFTWOCAPACITORSC1ANDC2,ANINDUCTORL,ARESISTORR,AMEMRISTORMANDASINUSOIDALVOLTAGESOURCEVSNOTETHATTHEMEMRISTORMISANUNIQUENONLINEARELEMENTLETTHATVTANDITDENOTETHEVOLTAGEACROSSANDTHECURRENTFLOWINGTHROUGHTHEMEMRISTOR,RESPECTIVELY,ANDTANDWREPRESENTTHEINNERSTATEVARIABLEANDTHEMEMDUCTANCEOFTHEMEMRISTOR,RESPECTIVELYTHUS,THEMATHEMATICALMODELOFTHEMEMRISTORMCANBEDESCRIBEDAS5ITWVTSIENTBEHAVIORS,FOREXAMPLES,TRANSIENTCHAOS,TRANSIENTHYPERCHAOSANDCHAOTICBEATS,DEPENDENTONTHESTIMULUSFREQUENCYAREDEPICTEDEXPERIMENTALVERIFICATIONSWAB|T|DT/DTVT1WITHDIFFERENTSINUSOIDALVOLTAGESTIMULIAREPERFORMEDTOPARTIALLYVERIFYTHENUMERICALSIMULATIONSINSECT4,INCLUDINGCHAOTICATTRACTOR,HYPERCHAOTICATTRACTORANDCHAOTICBEATSTHECONCLUSIONSARESUMMARIZEDINTHELASTSECTION2PERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITANDITSDYNAMICS21MEMRISTIVECHUASCIRCUITCONTAININGASINUSOIDALSTIMULUSANEWMEMRISTIVECHUASCIRCUITISPROPOSEDANDSHOWNINFIG1,WHICHISEXTENDEDFROMAMEMRISTIVECHUASCIRCUITIN5BYADDINGASINUSOIDALVOLTAGESTIMULUSVSINTOTHEINDUCTORBRANCHLDUETOTHEINTRODUCTIONOFTHESINUSOIDALVOLTAGESTIMULUS,THEPROPOSEDCIRCUITINFIG1ISCHANGEDINTOANONAUTONOMOUSCIRTHEMATHEMATICALMODELOF1ACCORDSWITHTHEDEFININGEQUATIONFORTHECLASSOFIDEALFLUXCONTROLLEDMEMRISTORSANDCANEXHIBITAPINCHEDHYSTERESISLOOPINTHEVIPLANEASABIPOLARPERIODICVOLTAGESTIMULUSISAPPLIED20ITISREMARKABLETHATTHEMEMRISTORMODELINTHISPAPERISCHARACTERIZEDBYASMOOTHPIECEWISEQUADRATICNONLINEARITY5,WHEREASTHEMEMRISTORMODELINREF7ISDESCRIBEDBYAPIECEWISELINEARITYTHEREAREFOURSTATEVARIABLESOFV1,V2,I3,ANDINFIG1,WHICHREPRESENTTHEVOLTAGEOFTHECAPACITORC1,THEVOLTAGEOFTHECAPACITORC2,THECURRENTOFTHEINDUCTORLANDTHEMAGNETICFLUXOFTHEFLUXCONTROLLEDMEMRISTORM,RESPECTIVELYHERE,THEAPPLIEDSINUSOIDALVOLTAGESTIMULUSISVSFSINT,WHEREFANDARETHEAMPLITUDEANDFREQUENCYOFTHESTIMULUS,RESPECTIVELYTHUS,APPLYINGKIRCHHOFFSCIRCUITLAWSANDTHEVOLTAGECURRENTRELATIONSOFCIRCUITELEMENTS,THEDYNAMICALEQUATIONOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITINFIG1ISEXPRESSEDASI3RILV2C2V1VC1VS13MEMRISTIVECHUASCIRCUIT2335DTTABLE1CIRCUITPARAMETERSFORSIMULATIONSANDEXPERIMENTSPARAMETERSSIGNIFICATIONSVALUESC1CAPACITANCE68NFC2CAPACITANCE68NFLINDUCTANCE172MHRRESISTANCE21KQACOEFFICIENT06667MSBCOEFFICIENT14828S/WBFAMPLITUDE20MVRADIANFREQUENCY1,000RAD/SASV101V,V200V,I300A,AND00WB,ITISDEMONSTRATEDTHATTHEMEMRISTIVECHUASCIRCUITWITHOUTTHEAPPLIEDVOLTAGESTIMULUSISCHAOTIC5WHENASINUSOIDALVOLTAGESTIMULUSWITH1,000RAD/SANDF20MVISAPPLIEDINTHECIRCUIT,THEPHASEPORTRAITSINDIFFERENTPHASEPLANESAREPLOTTEDINFIG2ACCORDINGTOTHEFOLLOWINGNUMERICALMETHOD,THELYAPUNOVEXPONENTSARECALCULATEDASLE11,2541,LE29548,LE30,LE425327,ANDLE57,1258THEREFORE,TWOLYAPUNOVEXPONENTSOFLE1ANDLE2AREGREATERTHANZERO,WHICHMEANSTHATTHECIRCUITOPERATESINHYPERCHAOTICSTATEITISNOTEDTHATTHEPHASEPORTRAITSOFTHECHAOTICATTRACTORINFIG2ANDTHECORRESPONDINGLYAPUNOVDV1V2V1C1DTRWV1EXPONENTSAREDISPLAYEDINAFINITETIMEINTERVALFROMTHEABOVECONCLUSIONS,ITCANBEFOUNDTHATC2DV2V1V2R2THEFREQUENCYANDAMPLITUDEOFTHESINUSOIDALVOLTAGELDI3V2FSINTSTIMULUSARETWOSIGNIFICANTCONTROLPARAMETERSOFTHEDTDDTV1WHEREWISANONLINEARFUNCTIONEXPRESSEDBY1,V1,V2,I3ANDAREFOURCIRCUITVARIABLES,ANDC1,C2,L,A,B,FANDARESEVENCIRCUITPARAMETERSEQUATION2CORRESPONDSTOAFOURDIMENSIONALNONLINEARNONAUTONOMOUSDYNAMICALSYSTEM,UPONWHICHDYNAMICALBEHAVIORSOFTHECIRCUITINFIG1CANBEREVEALEDNUMERICALLYTOSIMPLIFYTHEDYNAMICALEQUATIONOFTHEPROPOSEDCIRCUIT,THEEQUIVALENTSERIESRESISTANCEESROFTHEINDUCTORLISIGNOREDBECAUSEOFTHATTHEPARASITICRESISTANCEDOESNOTAFFECTITSBIFURCATIONSTRUCTURESCONSEQUENTLY,THEDYNAMICALEQUATIONOF2ISSIMILARTOTHATPRESENTEDBYREF7EXCEPTFORTHATTWODIFFERENTMEMRISTORMODELSAREUSEDINTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITCONSIDERINGTHATTHECIRCUITPARAMETERSINFIG1,ASLISTEDINTABLE1,ARESAMEASTHOSEINREF5ANDTHEINITIALVALUESOFFOURSTATEVARIABLESARETAKENPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUIT,WHICHCANMAKETHEDYNAMICSOFTHECIRCUITTOHAVEATRANSITIONFROMCHAOTICTOHYPERCHAOTICBEHAVIORSBECAUSETHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITISEASILYSTABILIZEDBYTHESINUSOIDALVOLTAGESTIMULUSWITHLARGERAMPLITUDE,THESINUSOIDALVOLTAGESTIMULUSWITHSMALLERAMPLITUDEISUTILIZEDANDTHEDYNAMICSOFTHECIRCUITDEPENDENTONTHEFREQUENCYOFTHESINUSOIDALVOLTAGESTIMULUSISFOCUSEDTOINVESTIGATEINTHISPAPER22EQUILIBRIUMPOINTANDJACOBIANMATRIXTHEEQUILIBRIUMPOINTSOF2AREOBTAINEDBYSETTINGTHELEFTHANDSIDETOZEROITISOBVIOUSLYTHATTHEREEXISTSTWOCASESCASE1WHENTKKISAPOSITIVEINTEGER,IE,VSFSINT0,THEEQUILIBRIUMSTATEOF2ISGIVENBYANEQUILIBRIUMSETEV1,V2,I3,|V1V2I30,C3FIG2CHAOTICATTRACTOROFPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITINAFINITETIMEINTERVALAPHASEPORTRAITINV1TV2TPLANEBPHASEPORTRAITINV1TTPLANEV1VV1VABI3V2VMWB132336BBAOETALV2V1DDWLDI310LCWHERECISUNCERTAINBUTCONSTANT,CORRESPONDINGTOTHEAXIS5THUS,THEEQUILIBRIUMSETEISALSOKNOWNASALINEEQUILIBRIUM14CASE2WHENT/KKISAPOSITIVEINTEGER,IE,VSFSINT/0,NOEQUILIBRIUMSTATEEXISTSIN2,IFLETWT,THENTHENONAUTONOMOUSSYSTEMDESCRIBEDBY2ISCONVERTEDINTOAUTONOMOUSONE,ANDTHEFOURDIMENSIONALSYSTEMBECOMESAFIVEDIMENSIONALSYSTEM,WHICHCANBEREWRITTENASIMPLYINGTHATTHEPROPOSEDCIRCUITDOESNOTEXHIBITANYEQUILIBRIUMPOINTSTHECHAOTICATTRACTORSDISPLAYEDINCDV11DTRWV1DYNAMICALSYSTEMSWITHOUTANYEQUILIBRIUMPOINTSORWITHONLYSTABLEEQUILIBRIUMPOINTSAREHIDDENATTRACC2DV2DTV1V23RITORS1418WHENVSFSINT0,THEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITBECOMESTHEAUTONOMOUSMEMRISTIVECHUASCIRCUITTHEREFORE,THEJACOBIANMATRIXATTHEEQUILIBRIUMSETEISEXPRESSEDASDTV2FSINWDTV1DTTHEJACOBIANMATRIXOF8CANTHENBEYIELDEDAS81RC1WCC1RC1001111W1BV1SGNJERC2RC2C20RC1C1RC10C104111100RC2RC22001000J100FCOSWLWHEREWCAB|C|THECHARACTERISTICEQUATIONISDERIVEDAS3A22A1A00510000000009WHEREA211WC,A1WC1,BASEDON9,THELYAPUNOVEXPONENTSPECTRUMWITHTHERC1RC2C1RC1C2LC2VARIATIONSOFTHECIRCUITPARAMETERSCANBECALCULATEDA01WCRLC1C2LC1C2NUMERICALLYTAKINGADVANTAGEOFTHEVALUESOFCIRCUITPARAMETERSLISTEDINTABLE1ANDTHEROUTHHURWITZCONDITION2ISUNSTABLEWHEN|C|0ANDLE20ATFIRST,THENGOESTOPERIODICOSCILLATIONWITHLE10ANDLE20,ENTERSINTOCHAOTICSTATEWITHLE10ANDLE20AGAINAFTER160RAD/SANDFINALLYSETTLESDOWNTOHYPERCHAOTICSTATEWITHLE10,LE20,ANDLE30AFTER746RAD/SSEVERALPHASEPORTRAITSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITUNDERFOURDIFFERENTSTIMULUSFREQUENCIESAREOBTAINEDFROMNUMERICALSIMULATIONS,ASSHOWNINFIG5WHEN0RAD/S,IE,THEFIG4LYAPUNOVEXPONENTSPECTRAWITHTHEVARIATIONSOFTHESTIMULUSFREQUENCYATHEFIRSTFOURLYAPUNOVEXPONENTS,BTHEFIFTHLYAPUNOVEXPONENT,CTHEEXPANDEDPORTIONOFAFORASHORTSTRETCHOFKRAD/SKRAD/SFIG5PHASEPORTRAITSOFTHEPERIODICALLYFORCEDMEMRISTIVECHUASCIRCUITACHAOTICATTRACTORWITHOUTTHEAPPLIEDVOLTAGESTIMULUS,BLIMITCYCLEWITHSMALLDYNAMICRANGE100RAD/S,CCHAOTICATTRACTORWITHLARGEDYNAMICRANGE400RAD/S,DHYPERCHAOTICATTRACTOR2,000RAD/SV1VV1VLE5BLE1CLE2LE3LE4LE1LE2LE4LE3AABV1VV1VCDV2VV2VV2VLE103LE103VVVV22132338BBAOETALFIG6TRANSITIONDYNAMICSFROMTRANSIENTCHAOTICTOSTEADYPERIODICBEHAVIORSWHEN400RAD/SAGLOBALTIMEDOMAINWAVEFORMOFTHEVARIABLEV2INTHETIMEINTERVAL0,08S,BSTEADYTIMEDOMAINWAVEFORMOFTHEVARIABLEV2INTHETIMEINTERVAL07,08S,CSTEADYLIMITCYCLEINV1TV2TPLANE,DFIRSTFOURLYAPUNOVEXPONENTSAGAINSTTIMEV1VTSSINUSOIDALVOLTAGESTIMULUSISNOTAPPLIED,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/SANDTHEOSCILLATIONAMPLITUDEEQUALSTOTHESTIMULUSAMPLIABTSTSLE1LE2LE4LE3CDV2VV2VLE103V2V13MEMRISTIVECHUASCIRCUIT2339FIG7TRANSITIONDYNAMICSFROMTRANSIENTHYPERCHAOTICTOSTEADYCHAOTICBEHAVIORSWHEN1,000RAD/SATHEHYPERCHAOTICATTRACTORCOMBINEDWITHTHECHAOTICATTRACTORINV1TV2TPLANE,BTHEHYPERCHAOTICATTRACTORCOMBINEDWITHTHECHAOTICATTRACTORINV1TTPLANE,CFIRSTFOURLYAPUNOVEXPONENTSAGAINSTTIME,DTIMEDOMAINWAVEFORMOFTHEVARIABLEV2INTHETIMEINTERVAL0S,04STSTSTUDEOFF20MVCORRESPONDINGLY,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,APHENOMENONOFCHAOTICBEATSOCCURSINTHEPERIODICALLYFORCEDMEMRISTIVEABV1VV1VLE1LE3LE2LE4CD3LE10V2VV2VMWB132340BBAOETALIVM1VCU1R0V_C0BR2U3_HRU2VAR13FIG8COMPLEXDYNAMICSOFCHAOTICBEATSWHEN200RAD/SAPHASEPORTRAITINV1TV2TPLANE,BPHASEPORTRAITINV1TTPLANE,CANDDTIMEDOMAINWAVEFORMSOFTHEVARIABLESV2ANDINTHETIMEINTERVAL0,1S,RESPECTIVELYWHERETHEZOOMINWAVEFORMSOFC1ANDD1ARESHOWNINC2ANDD2,RESPECTIVELYTSTSCHUASCIRCUITTHECORRESPONDINGPHASEPORTRAITSANDTIMEDOMAINWAVEFORMSARESHOWNINFIG8,WHERETHELYAPUNOVEXPONENTSINAFINITETIMEINTERVALARE12030,018,1806,1810,13,690OBSERVEDFROMFIG8C,D,THEENVELOPEFREQUENCYSLOWSCALEFREQUENCYEQUALSTOTHESTIMULUSFREQUENCY,IE,200RAD/S,ANDTHEOSCILLATIONFREQUENCYINEACHENVELOPEDOESJUSTBETHECHAOTICFREQUENCYINOTHERWORDS,THEEXTERNALSTIMULUSAPPLIEDVATOTHEMEMRISTIVECHUASCIRCUITISMODULATEDBYTHEINTRINSICCHAOTICOSCILLATIONONCETHEINTERFERENCEBETWEENDIFFERENTOSCILLATIONSEXISTS,ACOMPLEXNONLINAVBBEARPHENOMENONOFCHAOTICBEATSISPOSSIBLETOBURSTCHAOTICBEATSWEREREPORTEDANDSTUDIEDINMANYNONLINEARDYNAMICALSYSTEMS22244EXPERIMENTALVERIFICATIONSTHECIRCUITSCHEMATICTOREALIZETHEFLUXCONTROLLEDMEMRISTORCHARACTERIZEDBY1ISSHOWNINFIG93,WHERETHECIRCUITPARAMETERSARELISTEDINTABLE2,WITHAD633JNASANALOGMULTIPLIER,AD711KNASOPERATIONALAMPLIFIER,ANDOPERATIONVOLTAGESOF15VASARESULT,THEMEMRISTOREQUIVALENTPARAMETERSAREA06667MSANDB14828S/WBMOREDETAILSOFTHEMEMRISTOREQUIVALENTCIRCUITREALIZATIONAREDESCRIBEDIN5THUS,ANEXPERIMENTALCIRCUITOFTHEPERIODICALLYFIG9EQUIVALENTCIRCUITOFMEMRISTORAMAINCIRCUITBHCIRCUITWITHABSOLUTEVALUENONLINEARITYTABLE2CIRCUITPARAMETERSOFTHEMEMRISTOREQUIVALENTCIRCUITPARAMETERSSIGNIFICATIONSVALUESC0CAPACITANCE68NFR0RESISTANCE4KQR1RESISTANCE15KQR2,3RESISTANCE2KQ,2KQR4RESISTANCE605KQRSATRESISTANCE135KQM1SCALEFACTOR01M2SCALEFACTOR1ABV1VV1VC1D1C2D2_U4RSAT_R4M2U5VTEMPV2VV2VV2VMWBMWBMWB13MEMRISTIVECHUASCIRCUIT2341FIG10MEASUREDPHASEPORTRAITSOFHYPERCHAOTICATTRACTORWITHASINUSOIDALVOLTAGESTIMULUS1,000RAD/SAV1VERSUSV2,BV2VERSUSVAFIG11MEASUREDPHASEPORTRAITSOFCHAOTICATTRACTORANDHYPERCHAOTICATTRACTORACHAOTICATTRACTORWITHOUTSTIMULUS,BHYPERCHAOTICATTRACTORWITHASINUSOIDALVOLTAGESTIMULUSHAVING2,000RAD/SFIG12CAPTUREDCHAOTICBEATSWITHAVOLTAGESTIMULUS200RAD/SANDF22MVATIMEDOMAINWAVEFORMOFTHEVARIABLESV2,BTHEZOOMINWAVEFORMOFTHEVARIABLESV2FORCEDMEMRISTIVECHUASCIRCUITISDESIGNEDANDUSEDTOINVESTIGATETHESTIMULUSFREQUENCYDEPENDEDCOMPLEXDYNAMICSWITHTHESAMECIRCUITPARAMETERSASFORNUMERICALSIMULATIONSWHENASINUSOIDALVOLTAGESTIMULUSWITH1,000RAD/SISAPPLIED,THEEXPERIMENTALOBSERVATION
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