Volume123, number9 PHYSICSLETTERSA 7 September1987

CHAOS IN LOW-DIMENSIONAL HAMILTONIAN MAPS

HolgerKANTZ andPeterGRASSBERGERPhysicsDepartment,Universityof Wuppertal,Gauss-Strasse20, D-5600Wupperlal1, FRG

Received28 May 1987; acceptedfor publication 1 July 1987Communicatedby A.P. Fordy

Symplecticmapswith morethantwo degreesof freedomconstructedby couplingN area-preservingChiricov—Taylorstandardmapsareinvestigatedby numericalmethods.We find theasymptotic(for N—*ix) distributionof theNpositiveLyapunovexpo-nentswhichis attainedalreadyfor surprisinglysmallN. To testtheerrorsin calculatingLyapunovexponentsfrom finite partsoftrajectorieswecalculatethefluctuationsoftheeffectiveLyapunovexponentsasa functionofthenumberofiterationsandfind anontrivialdecayon timescalesdecreasingwith increasingdegreeof freedom.Thesefluctuationsaredueto clinging oftrajectoriesto regularorbits.

I. IntroductionThischoicehastheadvantagethattheuncoupledcase

A field of considerableinterestis chaosin non- (fi = 0) andthe caseN= i are quite familiar: this isintegrablehamiltoniansystems.Whereasthis is well just thenormal standardmap [2]. Themapdefinedinvestigatedfor two degreesof freedom[i —3], there in eq. (1) is not only volume preservingbut alsoareonly fewresultsknownabouthigher-dimensional symplectic,whichcanbe provenby simply writing

down the generatingfunction:systemssuchas e.g. many-particlesystems[5—11].Themostnaturalapproachto thisproblemconsists S(q,p’) = q’p’ +~pi2

inperformingnumericalsimulationsof somemodelsystem. But numerical integration of differential N

— ~ [kcos(q1)—flcos(q,+~—q)] , (2)equationsas done in refs. [5,6,10] requires corn-parativelymuchcomputationtime,preventingthese

where eq. (1) follows fromauthorsfrom studyingveryhigh statisticsandverylongtimecorrelations.Thuswedecidedto usea sim- p.= c9S/8q1 and q = 8S/0pplesymplecticmap,which is themodelof a Poincarémapof a realsystemreducedto theenergysheli.As Thusthismapreallydescribesahamiltoniansystem.For N= 2 this map wasalsousedin ref. [7] andiswe want to study the influence of the numberof

very similarto the Froeschlémap Eli].degreesof freedom,it is reasonableto usea systemof coupled2-d maps.Wechosea linearchainof Nnonlinearlycoupledconservativestandardmapswith 2. Asymptoticdistributionof the Lyapunovperiodicboundary:

exponentsp~=p1—ksin(x1)

Thestrengthof nonlinearityin our mapdepends—fl[sin(x,÷1—x1)+sin(x1_~—x1] on both k and fi. Whereasin the caseN=i phase

spaceisdivided into differentstochasticregionssep-x =x~+p mod2x, aratedby stabletori, the mapshouldbe ergodicfor

N> 1 in thesensethatnearlyeverychaotictrajectory1= 1 ,...,N, X1+N=x1. (i) shouldfinally explorethe whole phasespace.Thus

0375-9601/87/s03.50© ElsevierSciencePublishersB.V. 437(North-HollandPhysicsPublishingDivision)

Volume 123,number9 PHYSICSLETTERSA 7 September1987

it is not surprisingthat alreadyfor N= 2 we cannot vex curve. The curves cannotbe fitted by simpleresolveany structurein thevaluesof the Lyapunov powerlaws.exponentsasa function of kor /3. ThemaximalLya- Fork~ 0 thescaling is fastestwith a scalingfactorpunov exponentgrows roughly linear with both 1/(N+ 1), as the Nth Lyapunov exponentis stillparameters.Thisis in contrastto what we found for strictly positive,whereasfork= 0 we haveAN= 0 andthe singlestandardmap [8]. thusscalingwith i/N. In fig. 1 we show curvesf(x)

An interestingquestionis how the nonlinearityis for someparametervaluesandin fig. 2 the scalingdistributedto the different unstabledirections,i.e. behaviourasa functionof N. It canbe seenthat thedegreesof freedom.Sowe computedall N positive dataforN= 4 are alreadyin perfectagreementwithLyapunov exponentsby the following algorithm: the continuouscurve which representsthe dataforparallelto the iteration inphasespaceweiteratethe N= 25.tangentmapby applying the jacobianto Northog- Scalingdistributionsof Lyapunovexponentssuchonally chosenvectorsof thetangentspace.After every as eq. (3) are alreadyknown from other systems,16 iterationsthey are orthononnalizedagain.Their dissipative[4,12] as well as conservative[13,15]increasein lengthleadsto the Lyapunovexponents. ones.Thustheexistenceof a “thermodynamic”limitBy this proceedurethe Lyapunov exponentsare seemsto be agenericresult.Thequestioniswhetheralreadyorderedby size.TheremainingNLyapunov thereexistsa universalscalingfunctionf(x) of eq.exponents are just their negatives. For N= 15, (3) for symplecticdynamics,at leastin the limit ofk= 1.48and/i= 0.25wetestedthisalgorithmby cal- strongnonlinearity.culatingall 2NLyapunovexponents.Indeedwefound In ref. [9] theLyapunovspectrumof 2Nx2Nran-A, ~ — — ,within 10—i. dom matrix productswas calculated.The authors

For N largewe find a scalingdistribution simulateda systemconsistingof harmonicoscilla-A — 7 ‘N+ 1 \\ (3 torswith nearest-neighbourcoupling.Sotheir matri-

‘ ‘ ceshaveexactlythe samestructureas ourjacobian

with a smoothcurvefix). It is concavefor k and/1 for k= 0 with zeroesandonesat the samesites.Theless than 1, convex for k or /3 greaterthan 2, and coupling elementsarerandomlychosenfrom a con-roughlya straightline inbetween.In thecaseof large stantdistributionandareconstrainedsuchasto makeparameterswe find two possibleshapes:if/I is small the matricessympletic.They find a straight line forcomparedto k, thecurvetendsto a trivial stepfunc- the functionf(x) with scalingfactor 1/N. Newmantion: all Lyapunovexponentsareequalin the limit [14] calculatedthedistributionfor randommatrices/1 —p0. For/3 largeandk smallwefind asmooth,con- of a different type analytically,and foundexponen-

~ °O~8 10 12 14 16 ~4~25

i/N+1

Fig. 1. A, versusi/(N+ 1) fordifferentk, fi, N= 15.

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1.

N=2

0.8 - ‘ N=3N=4

£ N=5

N=6

0.6- °N=9N=1O

i/(N+1)

Fig. 2. A, versusi/(N+ 1) for differentN, k= 1.48,fi = 0.25fixed.

tial aswell as lineardistributionsdependingon the we have different stochasticregionswith differentstructureof thematrices.A result of morephysical properties,especiallydifferent maximal Lyapunovrelevancewasobtainedin ref. [131: the scalingfunc- exponents,dependingon the choseninitial condi-tion of eq. (3) is found to be a straightline in the tions. But evenif thereis only one connectedsto-Fermi—Pasta—Ulam/1 model, if the energydensityis chasticregion, thetypical time a trajectoryneedstohigh enough, explore it may be very high and the above-men-

A priori, onemight haveexpectedthat the time tioned separationof phasespacemight occurin theevolutionin thetangentspaceof chaotichamilton- limit N—p~. Apart from questionsof principlesthisian systemscanbe describedby meansof random raisesat leastthe questionof the errorsin the Lya-matricesandf(x) is linear, provided the nonline- punov exponentscomputedabovefrom trajectoriesarity is strongenough.Thus,the existenceof acon- of finite length.vex curve in a regimebeyondthe straightline (i.e. Wecaiculatethe fluctuationsof the effectiveLya-k,/3<2) in oursystemis a surprise.Soat leastinour punov exponentsA1, 7- normalizedwith the time T,systemthelimit of strongnonlinearitydoesnotlead that isto a linear Lyapunovspectrum. (7~\ — ( ‘~2 p

A’S J—<’,’.IT”i) >.I

where3. Fluctuations of the effectiveLyapunov exponents — .~ T~

‘~,~— ~og it, , (4)As alreadysaid,all chaotictrajectoriesshouldlead with J(T) thejacobianof the7thiterate,it, a unitvec-

to the samemotionin phasespacebecauseof ergod- tor in the ith unstabledirectionand < > averagingidity, which is dueto Arnold diffusion. Nevertheless over all partsof length T. Furthermore,we havethereare conjecturesthat Arnold diffusion is sup- A — lim Apressedin somesystems[6]. Thiswouid meanthat ‘ T-. “~

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Volume 123,number9 PHYSICSLETTERSA 7 September1987

By this normalizationA.(T) would be a constantif to decay (fig. 3). Also the magnitudeof the fluctua-the JacobimatricesJ(T) were uncorrelated,i.e. if a tionsof correspondingLyapunovexponentsdecreasestrajectoryvisited all partsof thestochasticregionat with increasing N. This reflects the fact that therandom. dimensionof phasespaceincreasesfasterthan the

In the caseof N= 1 we found nontrivialdecayof dimensionofstabletoriwith increasingN. Sotheflowthesefluctuationsup to the maximal time scalewe shouldbecomemorehomogeneousandlessrestrictedregarded,T= 215 [8]. We could fit thesedata by a by canton,although thereis a conjecture[7] thatpowerlaw, the roleplayedby canton whenN= 1 maybeplayed

hereby familesof N-tori, formingpartialbarriersinA (T) Ta, (5) the 2N-dimensionalphasespace.

To comparethe magnitudeof the fluctuationsofwherethepowera 0.5seemstobeuniversalfor two- the different effectiveLyapunovexponentsit seemsdimensionalarea-preservingmappings.Thesefluc- to be moreappropriateto normalizethem by 1/A ~.

tuationsaredueto theexistenceofcanton surround- Theserelativefluctuationsaremaximalfor i = Nanding stable islands [1—3]. A typical trajectory will minimal for i = 1 (fig. 4), whereasthe correlationscomearbitrarily close to every cantorusand may decayslowest (i.e.A,(T) is the lastto becomecon-penetrateit. Onceinside,it will takea longtime to stant)for i~N/2.Thefirst meansthatthe directionsgetoutagain,asthesecanton arepartialbarriers.So with minimal instability see most of the regularthe trajectorystaysin a smallbandsurroundingan structurein phasespace.Thisis whatonemighthaveisland.Due to the relativeregularityof thismotion expected.the Lyapunov exponentis nearly zero within the It seemsvery likely that the weight of stabletoriband,whereasoutside it is much larger. So the decreasesvery fastwith increasingN. Theprobabil-motionis intermittentandcorrelatedoverlongtimes. ity of hitting a torus with arbitrarily choseninitial

In the caseN> 1 wealsofind anincreasingA,(T), conditionsandstayingon it duringiterationin spitei.e. a nontrivial decayof the fluctuations.Most of ofround-offerrorsshouldbe zero,astorihaveatmostour numericalresultsare obtainedfor /1 = 0.25 and dimension N in a 2N-dimensionalphase space.k= 1.02 respectivelyk= 1.48. For theseparameters Neverthelesswe find peaksatA= 0, if weplotthedis-we find abehaviourverysimilar to the caseN= 1, tributions of the Lyapunovexponentsof ±50000if N is small, butfor Nabout5, A.(T) staysroughly trajectoriesof length T= 6400 with ramdomlycho-constantfor T—~i000, for all I. So correlationsseem senstartingpoints (k= 1.48,/3 = 0.25), for all N. As

seenfrom fig. 5, the weightsof thesepeaksaretyp-

14 - N=2, ~1,2 0 100

12 - N4, =1 1=4• N=~

CN 0

10 - 0 0 ~ 80 -

: N=9, =5 0 0 60 - 1=10

2 468101214 16 _______

0. 0.2 0.4 0.6 0.8 1.

Fig. 3. A,(T) versus1’ for comparablei, differentN (k= 1.48,/3=0.25). Fig. 4.A,(T)/A~fork= l.48,fl=0.25,T=8192,differentN.

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= space average‘.5-.-— 1. .i

(1) - ‘ 3200

- 2:?~

3 :

4~~

time average

- 5tA6400

io_2 1

2

34

io~ 5

I I I I10 ~ 0. 0.1 0.2 0.3 0.4 0.5 0.6

Fig. 5. IntegrateddistributionS(A)of effectiveLyapunovexponentsA1, T obtainedfromtimeaveragerespectivelyspaceaverage(k= 1.48,

/3=0.25,N=3)

icaily —~2 x 10—s, so theyare statisticallysignificant long-timecorrelationsdescribedabovewe searchedbut do not contributesignificantly to the average for parts of trajectorieswhose effective maximalLyapunovexponent. . Lyapunovexponentis nearlyzero. Fortheseparts

On theotherhand,wefind only atail butnopeak, we plot the coordinatesand Lyapunov exponentsif we makethe sameplot with the effectiveLyapu- versustime (fig. 6). The result is stronglyreminis-nov exponentsA~,6400of i000 trajectoriesof iength cent of intermittency: embeddedin long parts of4 x 10~startednearthehyperbolicfixed point (it 0)N. chaoticmotion,we find regularbehaviouron long(fig. 5). Theabove-mentionedpeaksatA ~‘ 0 do not but finite times. During thesetimesall coordinatesdecreasesignificantly with trajectorylength. So we show quasiperiodicityandall Lyapunovexponentsconjecturethat therearerelativelydensefamiliesof are almostzero.tori in phasespace,which effectively do seperate Sowe concludethat fluctuationsof effectiveLya-phasespaceon rather long time scales.The time punovexponentsdecayslowerthanexpectedbecausescaleson which an equilibrium density may be of clingingof trajectoriesto stabletori. But thetimeachieved,arethusmuchiargerthan i0~.On theother scaleson which thiseffecthasanimportanteffectonhanddifferencesbetweenthedistributionsoccuronly global averages(i~i1000)are muchsmallerthan theat low valuesof A. This suggeststhat theseregions life timesof peaksat A = 0 discussedabove,andthewhich are nearlydisconnectedfrom that generated numberof iterationsweusedto calculateLyapunovby the hyperbolicfixed point (it, 0)Nare “lagunas” exponents(~80 mio). Sothe resultsofthe first partwith very low maximal Lyapunov exponent. of this paperare not affectedby this.

In an attempt to understandin more detail the Finallywemeasuredthedistributionof the length

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- ..._ •. -—.—.—.- 2~ which are importantat this parametervalue have

55. 5,, ‘ .. S well-definedtime scales[8]. In the caseof coupledxl- x

2 ~- -.F ~. L maps,however,phasespaceis muchlarger Together. .~ . withmuch moretori andcantoritherearemanytime

- 2it scalesinvolved This shouldsmoothendistributions

2it like fig. 7 andcangeneratea powerlaw behaviour.X2-X3 ~ .~. ~ .. .~ ‘,.~,. Sincethe curvesin fit 7 showthe probabilitythata

trajectorystaysa time t closeto sometorus,theyare

-2Tt connectedto the quantityA(t) from eq. (4) [8].Indeedboth resultsare in mutualagreement.

Contraryto otherauthors[6,10], in no casewefound partsof trajectorieswith someeffectiveLya-punov exponentszero and the rest strictly larger,

I I I Jj I which would imply theexistenceof manifoldsin theI di Id~Iii~iIi~ih phasespacewith dimensiondifferentfrom Nand2N.UL This confirmsthe old conjectureof Froeschlé[11],

I that thereareonlyNorzeroconstantsofmotion,butthismaybe specialto our typeof map.It alsomight‘4 !I p1 I~~jI~1:’! meanthat there are such manifolds but that the

0 chanceto stayon themis zerodueto round-offerrorsS _____ T (wemight mentionthatcomputationspresentedhere

weredonewith 64-bitarithmetic),buttheseshouldhaveaffectedthe resultsof refs. [6,10] in the same

~ 2Tt way

4 Conclusions_____ ________ i ~ We presentedthe existenceof an asymptoticdis

tribution oftheNLyapunovexponentstogetherwiththe appropriatescalingbehaviourTheapproachtothis distributionfor finite N is surprisingly fast The

—~ J.~-5-~~4~ - —-—i- function describingits shapeis different from thestraightline founefor randommatrices,anddepends

Fig. 6. Regular part of a chaotic trajectory (N3, k 1.48, on the parametersof the map./3= 0.25). Fluctuationsof theeffectiveLyapunovexponents

are found to be due to the clinging of chaotic tra-jectories to stable tori, but important for global

of partsof atrajectorywith A,,16lessthansomefixed quantitiesonly on finite time scales.Whenlookingbounds (fig. 7 fors= 0.09, k= 1.48, startednearthe at special featureswe do find memory effects ofhyperbolic fixed point) for various N. The ampli- extremelylongtimescales,butin thissystemwecan-tudeof thesenormalizedcurvesis anestimateofthe not find any hint that Arnold diffusion shouldbemeasureof stabletori in phasespace.Fig. 7 shows suppressedat arbitrarily long time scales.that thereis a qualitativedifferencein the decreaseof thesecurvesbetweenthe coupledanduncoupledcase.This might be explainedasfollows: in the lat-ter, there is only a smaller part of phasespaceexploredby any single trajectory.The few cantori

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Volume123, number9 PHYSICSLETTERSA 7 September1987

“.5-- 5Z 10 o N=1, K=1.48, /3=.25

°N=2

~ N=34 ° N=4

N=5• N=3, /3=0.

io2

10

1 I I I I1 2 3 4 5 6 7

Ogit)

Fig. 7. Distributionofthenumberof partsofonetrajectorywith A1, 16lessthan0.09 duringtiterations(k=l.48,fl=0.25, startingpoint

near(x, 0)N, 30mio iterationseach).

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[6] M. PettiniandA.Vulpiani, Phys.Lett. A 106 (1984)204.

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chasticmotion (Springer,Berlin, 1983). [9] G.PaladinandA. Vulpiani, J.Phys.A 19 (1986) 1881.[3]B.V. Chirikov, Phys.Rep. 52 (1979) 1183; Intrinsic ~t~- [l0]A. Malagoli,G. PaladinandA. Vulpiani, Phys.Rev. A 34

chasticity, Novosibirskpreprint (1984); (1986) 1550.B.V. Chirikov andD.L. Shepelyanski,PhysicaD 13 (1984) [11] C.Froeschlé,Astron.Astrophys.16 (1972) 172.395. [12] K. Kaneko, Physica D23 (1986) 436.

[4] P. Mannevllle, in: Macroscopicmodelling of turbulent [13] R. Livi, A. Potiii andS. Ruffo, J. Phys. A 19 (1986) 2033.flows..., ed. 0. Pironneau, Lecture Notes in Physics, [14] C.M. Newman,Commun.Math.Phys.103 (1986) 121.(Springer,Berlin, 1985). [15] R.Livi, A. Politi, S.Roffo andA. Vulpiani, J. Stat. Phys.46

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