Finite-time Lyapunov exponents of Strange Nonchaotic Attractors

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The probability distribution of finite-time Lyapunov exponents provides an important characterization of dynamical attractors. We study such distributions for strange nonchaotic attractors (SNAs) created through several different mechanisms in quasiperiodi

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aFinite–timeLyapunovexponentsofStrangeNonchaoticAttractorsAwadheshPrasadandRamakrishnaRamaswamySchoolofPhysicalSciencesJawaharlalNehruUniversity,NewDelhi110067.(February5,2008)Theprobabilitydistributionof nite–timeLyapunovexponentsprovidesanimportantcharac-terizationofdynamicalattractors.Westudysuchdistributionsforstrangenonchaoticattractors(SNAs)createdthroughseveraldi erentmechanismsinquasiperiodicallyforcednonlineardynam-icalsystems.StatisticalpropertiesofthedistributionssuchasthevarianceandtheskewnessalsodistinguishbetweenSNAsformedbydi erentbifurcationroutes.

I.INTRODUCTION

Thecharacterizationofattractorsinnonlineardynamicalsystemsisaproblemthathasseenconsiderableprogressinthepastdecade[1,2].Beyondclassi cationassimpleorstrange,thecalculationoffractaldimensionsandthedescriptionofthemeasureintermofamultifractalspectrumofsingularitieshasbecomeanimportantmeansofdescribingthestructureofdynamicalattractors[1].Forhyperbolicattractorstherearerigorousresultsconnectingthemultifractalstructure,throughthethermodynamicformalism,withdynamicalinformationasembodiedin nite–timeLyapunovexponents[2,3].

Grebogietal.[4] rstdescribeddynamicalsystemswhereintheattractorsthatresultarefractal,butthedynamicsisnotchaotic,inthatthelargestLyapunovexponentisnotgreaterthanzero.Thesestrangenonchaoticattractors(SNAs)aregenericinquasiperiodicallyforcedsystems.Subsequently,considerablee orthasbeendirectedtowardthecharacterizationandstudyofSNAs[5–11],whichhavealsobeenobservedexperimentally[12–14].ApotentialuseofSNAsisintheareaofsecurecommunications,andrecentapplications[15,16]exploittheeaseofsynchronizationofsuchsystems.

Thestrangenonchaoticstateisonlyoneofthepossibledynamicalstatesrealizedinquasiperiodicallydrivensystems;periodic,quasiperiodicandchaoticattractorscanalsobeobtainedasparametersarevaried.SNAsaretypicallyfoundforparametervaluesveryclosetotheboundariesofthechaoticregions,andthedi erentbifurcationmechanismsthroughwhichtheyarecreatedisaproblemofinterest.ThereareanumberofroutesorscenariosforthecreationofSNAs,someofwhichcanbecorrelatedwithbifurcations.Theseinclude

(i)theHeagy–Hammel(HH)[8]mechanisminvolvingacollisionbetweenaperiod-doubledtorusanditsunstableparent,

(ii)theblowoutbifurcationroute[11],and

(iii)intermittency[17],whenasafunctionofdrivingparameterachaoticstrangeattractordisappearsandiseventuallyreplacedbyatorusthroughananalogueofthesaddle-nodebifurcation.

ThesignatureofthesebifurcationsintermsofthebehaviourofthelargestLyapunovexponenthasbeendiscussedindetail[18].Theblowoutbifurcationmechanism[11]requiresthatthequasiperiodictorusofasystemwithaninvariantsubspacelossesitstransversestabilityasaparameterchangesacrossthetransitionandleadtothebirthofanSNA.InthisprocessthetransverseLyapunovexponentbecomespositivewhilethenontrivialLyapunovexponentforthewholesystemremainsnegative.IntheHHmechanism[8],assystemparametersarevaried,theperiod-doubledtorusgetsprogressivelymorewrinkledandcollideswithaparentunstabletorus;thisscenarioislikeanattractormergingcrisis[19].ThedistinctivesignatureoftheintermittencyroutetoSNAisasharpchangeintheLyapunovexponentwhichshowslargevarianceandscalingbehavior[17]atthebifurcation.

Ageneralmechanismthatisfrequentlyobservedbutforwhichthereisnowell-identi edbifurcationistheso–calledfractalizationroute[9],wherebyasmoothtorusgetsincreasinglywrinkledandtransformsintoaSNAwithoutanyinteractionwithanearbyunstableperiodicorbit(incontrasttoHH).ThisisprobablythemostcommonroutetoSNAinanumberofmapsand ows[6,7].

ThepresentpaperaddressestheissueofdistinguishingamongSNAsformedbydi erentroutesthroughtheuseof nite–time(orlocal)Lyapunovexponents.Weshowthatthemorphologiesofdi erentSNAsdi erincrucialways,particularlyforintermittentSNAs[17,18].ThisisseenmostdramaticallyinthecharacteristicdistributionsoflocalLyapunovexponentsandthestatisticalpropertiesofthedistributionssuchasthevarianceandtheskewness.

InSec.II,webrie yintroducethedynamicalsystemsthatarestudiedhere.ResultsarediscussedinSecIII.ThisisfollowedbyasummaryinSec.IV.

II.DYNAMICALSYSTEMS

Severalquasiperiodicallydrivensystems—bothmapsand ows[6–11]—havebeenshowntohaveSNAs.Weconsiderthequasiperiodicallyforcedlogisticmap[8]whereinthreeoftheroutestoSNAscanbeobserved.Thissystemisde nedbytheequations

xn+1=α[1+ cos(2πφn)]xn(1 xn),

φn+1=φn+ω(mod1),

√wherex∈R1,φ∈S1,ω=((1)

λN(xn)=1

5 1)/2)whichisboundedandhasnootherstableattractorsotherthanthe

invariantsubspace(

x=0,φ).Asparameterschangesacrossthecriticalvalues,thedynamicsofxleadstoastrangenonchaoticattractor[11].OurinterestisincontrastingthismechanismforSNAformationwhichisalsoaccompaniedbyon–o intermittency,withtheintermittentSNA[17].

III.RESULTS

AlthoughλNdependsoninitialconditions,theprobabilitydensity,de nedthrough

P(N,λ)dλ=ProbabilitythatλNliesbetweenλandλ+dλ,(5)

doesnot.Thisdistributioncanbeobtainedbytakingan(in nitely)long,ergodictrajectory,anddividingitinsegmentsoflengthN,fromwhichthelocalLyapunovexponentcanbecalculatedthroughEq.(2).

Forchaoticmotionithasbeenargued[2,3]thatsincethelocalLyapunovexponentscanbetreatedasindependentrandom uctuations,thecentrallimittheoremisvalid,leadingtoanormaldistributionforλN,

P(N,λ)≈1

2πNG′′(Λ)exp[ NG′′(Λ)(λ Λ)2/2](6)

withthefunctionG,thespectrumofe ectiveLyapunovexponents[3],beingappropriatelyde ned[2].

Theseexpectationsarenotalwayssatis edsincetherecanbeimportantcorrelationsinthedynamics.Wehaverecentlydescribedthecharacteristicdistributionsfor nitetimeLyapunovexponentsinlowdimensionalchaoticsystemswheretherearesigni cantdeparturesfromcentral–limitbehaviour[20].

ForSNAsthereareadditionalcomplications.AlthoughΛisbyde nitionnonpositive,P(N,λ)canhaveasigni cantcontributionfromλ>0:forsomeofthetime,thesesystemsbehavechaoticallybecauseofthefractalstructureoftheattractors[7].InthelimitoflargeN,thecontributionfrompositiveλdecreasesandthedensitycollapsestoaδ–function,limN→∞P(N,λ)→δ(Λ λ).

ShowninFigs.2and3arelocalLyapunovdistributionsforthefourroutestoSNAs(parametersarespeci edinthecaption),forshort(N=50)andlongbut nite(N=1000)times.ThefractalizedandHHSNAsbothshowagradualapproachtothenormaldistribution(agaussianis ttothedatainFig.3),whiletheblowoutSNAs,andmorespectacularly,theintermittentSNAsshowadistinctivedeparturefromthegaussiandistribution.

TheintermittentSNAismorphologicallyanddynamicallyverydi erentfromtheotherSNAs,andtheshapeofthecharacteristicdistribution,P(N,λ)isacombinationofagaussianandanexponential[20].IncontrasttotheotherSNAs,thedistributionisasymmetric,andthelargeλtaildecaysveryslowly.

ItappearsthattheintermittentSNAisinadistinctuniversalityclass[20]andanumberofquantitativemeasurescanbedevisedinordertoshowthisdistinction.Consider,forexample,thefractionofexponentslyingaboveλ=0,

F+(N)= ∞

P(N,λ)dλ,(7)

F+(N)vsNforthedi erentSNAsareshowninFig.4.ExceptfortheintermittentSNA,forwhichF+(N)～N β,thisquantitydecaysexponentially,F+(N)～exp( γN),withtheexponentsβandγdependingstronglyontheparametersofthesystem.Wefoundthatthevaluesoftheexponentsareβ=0.72fortheintermittentSNAandγ=0.02,0.007and0.042respectivelyforHH,fractalized,andblowoutSNAs.Similarly,otherstatisticalpropertiesofthethesedistributionscanbestudied.Wecalculatethe rsttwomomentsaboutthearithmeticmeanofalldistributionsandobtainthevariance,

∞

σ2=(λ Λ)2P(N,λ)dλ,(8)

∞

andthecoe cientofskewness,namely

γ1= ∞

(λ Λ)3P(N,λ)dλ/(σ3),(9)

∞

whichareshowninFig.5(a)andFig.5(b)respectively.GenerallyforalltypesofSNAs,thevarianceofP(N,λ)decreasesasapowerofN,σ2～1/Nδwheretheexponentδisdi erentforeachSNA.ThevarianceforintermittentSNAsdecreasesveryslowlycomparedtootherSNAs,andournumericalresultsfortheexponents,fortheexam-plesshownhereare=0.97,1.71,1.63,and1.7forintermittent,fractalized,HH,andblowoutSNAs.Thedegreeofasymmetryinthedistributionisquanti edbythesigni cantlylargerskewnessγ1(seeFig.5(b)).

IV.SUMMARY

Inthepresentpaperwehavestudiedthedynamicalstructureofstrangenonchaoticattractorsformedbydi erentbifurcationmechanismsinquasiperiodicallydrivensystems,byexaminingthedistributionof nite–timeLyapunovexponents.AlthoughtheLyapunovexponentisnegativeonaSNA,overshorttimes,nearbytrajectoriescanseparatefromoneanothersincetheattractorisstrange:thiscorrespondstoalocalpositiveLyapunovexponent.Themannerinwhichthisdistributionchangesasafunctionoftimeischaracteristicoftheattractor,andofthebifurcationroutesthroughwhichattractorsarecreated:intermittentdynamicsleadstoverydistinctivedistributionsoflocalLyapunovexponents[20].

Ourpresentresultsfurtherunderscoretheutilityof nite–timeLyapunovexponentsindescribingthelocalstructureofdynamicalattractors[2,3,20]ingeneral.Forthecaseofhyperbolicattractors,thetheoryconnectingthesetotheinvariantmeasureiswell–developed.Thepresentpaperispartofapreliminarysteptowardsunderstandingtheconnectionbetweenanunderlyingfractalstructureandgloballynonchaoticdynamicsonstrangenonchaoticattractors.

ACKNOWLEDGMENT

ThisresearchwassupportedbygrantNo.SP/S2/E07/96fromtheDepartmentofScienceandTechnology,India.WethankVishalMehrafordiscussions.

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Fig. 1. Phase diagram for the quasiperiodically forced logistic map (schematic) 18], corresponding to the period 3 window of the unforced case. In order to obtain this diagram, is calculated in a 100 100 grid. The shaded region (S) along the boundary (the= 0 contour) shows the region of SNAs. Intermittent SNAs are found on the edge of the C1 region marked I, while the right boundary, denoted C1 has fractalized SNAs. The left boundary between the periodic region and C1 does not show any SNA. The boundaries of the attractors can be interwoven in complicated manner (especially along C1).0 0

λln P(N,)

-5

ln F+ (N)-10-15

-20

-25024

Ln N68

)

(1 γn

ln N

ln N2

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