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.
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
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)
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
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,
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
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.
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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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
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
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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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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λln P(N,)
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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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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