Time averages for unpredictable dynamics - Universität Wien

Annals of Operations Research 37(1992)217 -228

217

TIME AVERAGES FOR UNPREDICTABLE ORBITS OF DETERMINISTIC SYSTEMS Karl SIGMUND 4, A-1090Wien,Ausftia Wien,Strudlhofgasse der Universität Institut für Mathematik

Abstract In many cases,the orbits of deterministic systemsdisplaying highly inegular oscillations yield smoothly converging time averages.It may happen,however, that thesetime averages do not converge and themselvesdisplay wild oscillations. This is analyzed for heteroclinic attractors and hyperbolic strange attractors.

1.

Introduction

prices(e.g.I1, 2l), populationdensities[3]' Whetheroneplotsstockexchange variables[6] or medicalindicators[4], chemicalconcentrations [5], atmospheric what not, one regularlyencountersirregularfluctuations.Mankind has long been oscillatingtime series,and recentlymodellerstoo are accustomed to unpredictable gettingusedto them(seee.g.t7-91). In fact,thesearenowadaysenjoyedby most But whenit comesto tasksof evaluationanddecisionmaking,the systemanalysts. to die down and,if they showno sign recipeis usuallyto wait for the fluctuations themout. In mostcases,this is indeed of doingthis of their own accord,to average a sensiblething to do, validatedby somelaw of largenumbersor by an ergodic in theorem(seee.g. t10-121).To quoteRuellet10l: ... thereare manysituations which a geometricdescription[of the attractor] is no longerfeasible,due to the extremecomplicationsof the dynamics... Thesecasesare thosein which statistical analysisbecomesreally releyant... If we analysea sufficientlylong record of a chaotic signal generatedby a deterministictime evolution,we mayfind that the signalamplitudesare within a definiterangefor a well definedfraction of the time ... More precisely,this meansthat the expectedtime averagedoesexistfor suitable initial points,providing a well definedinvariantprobabilitymeasure(in the sense that the time averageof the observableis equal to its spaceavarage). Nevertheless, therearesituationswherethe time averagerefusesto converge, no matterhow long one waits.The purposeof this article is to highlight such dynamics.What we havein situationsby somesimpleexamplesof deterministic J.C. Baltzer AG, Scientific Publishing Company

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K. Sigmund, Time averagesfor unpredictable orbits

mind are modelsdescribingthe repeatedinteractions of playersin somesocial, economicor biologicalgame,or somehypothetical dynamics underlyingthefluctuations of pricesor populationnumbers.In the first part,we dealwith heteroclinicattractors: suchcyclesexemplifyunpredictable behaviourwherealmostall orbitsfail to converge. In the secondpart, we tum to strangeattractorsof hyperbolictype. There is an interesting contrast,here,betweenthe probabilistic pointof view (wherealmostall initial data lead to convergingtime averages) and the topologicalpoint of view (generically, the time averages do not converge:in fact, they exhibit an extreme form of misbehaviour). If the initial conditionsareonly knownstatistically, i.e. as probabilitydistributions, the formerpoint of view becomesmeaningless and the latter one gainsinterest. Thus the two typesof deterministic but unpredictable behaviourwhich we considerhereleadto very differentasymptotic behaviourof the time averages. For heteroclinicattractors, Lebesgue-almost all initial conditionsleadto time averages which diverge,but they all divergein preciselythe sameway. For hyperbolic strangeattractors, Lebesgue-almost all initial conditionslead to convergingtime averages, but thereexistsnevertheless a hugevarietyof differentdivergentbehaviours, and for eachpossibletype of divergence thereis a denseset of initial conditions leadingto it. 2.

Heteroclinicattractors

Let us considera dynamicalsystemon a subsetX of lRn.If the stateat time 0 is given by r e X, then it will be given by r(r) e x at time r. we shall consider bothcontinuous time (r e lR) anddiscretetime (r e Z) dynamical systems. A fixed point r satisfiesr(t) = r for all r. A heterocliniccycle consistsof severalfixed p o i n t lst Q = 7 , . . . , k : k > 2 ) a n do r b i t st + z r ( t ) c o n n e c t i n g t h ei n mt,h es e n s teh a t zr(t)-+J/{or l?-a n dz i ( t ) - y j * I f o r r - + + @ . ( W e i e t y r t , = y l s o t h a tz t connects J* to Jt and therebyclosästhe cycle)The fixed pointsare saddlepoints (sincesomeorbits approachthem for positive,somefor negativetime) and the connectingorbits are saddleconnections. We note in passingthat dynamicalsystemswith heterocliniccyclesare not structurallystable.The saddleconnectionscan be brokenup by arbitrarily small perturbations (where"small"canbe defined,here,in a varietyof ways).This seems to be the reasonwhy heteroclinic cycleshavebeenneglected by themainlinetheory of dynamicalsystems (see,however,Guckenheimer andHolmes[13]).Nevertheless, they occur robustlyin a varietyof models,in the sensethat if the perturbation respects someessential featureor symmetry, it alsopreserves thesaddleconnections and hencethe heterocliniccycles. As an instanceof a systemfeaturingheterocliniccycleswe mentionthe hypercyclewhich modelsthe evolutionof concentrations of severaltypesof selfreplicatingmoleculesin a flow reactor,eachtype acting catalyticallyupon the

K. Sigmund, Time averagesfor unpredictable orbits

219

growth of the next type, therebyforming a closedloop of catalytic actions[14]. from chemicalkineticshasalsobeenusedin economicor social This mechanism of cooperatingunits. Other examplesof contextsto describethe self-organization wherethevariablesdenotethefrequencies game in dynamics, heteroclinic cyclesoccur spread(seee.g.[15]). Whenever strategies andthe successful of certainstrategies, game),heteroclinic (like for thestone-scissors-paper thereexistsa cyclicstructure Prisoner'sDilemma[16]; in the repeated cyclesoccur.This happensfor strategies each,if no pure players strategies with two are two it alsooccurswheneverthere pair of strategies is a Nashequilibrium[17,18].Furtherexamplesof heteroclinic cyclescan be found in populationecology,as when (for instance)two competing species7 and2 are besetby two strainsa andb of predators(or parasites):if the present) are fourequilibria(l,c), (l,b), (2,b)and(2,a)(eachwith onlytwo species of regime a on 4 cyclically connected,in the sensethat predatorb outcompetes prey I if only predatorb is around,and so on ..., this prey 1, that prey 2 replaces cycles: yieldsa heterocliniccycle [19]. For discretedynamicsthereare analogous in a sense,they occurevenmore readily[7]. Now let us assumethat the heterocliniccycle f is a limit in the sensethat pointsof thereexistsan orbit x(l) suchthat the pointsof f are the accumulation of the The continuity .r. f is olimitof i.e. the with that x(ro), t1,i*a, sequences If it is starts. in fits and proceeds dynamicalsystemimpliesthat the motionof r(r) it follows nearone of the fixed pointsof f, it remainstherefor a long time; then the outgoingsaddleconnectionto the next fixed point, whereit hoversfor a much to thefollowingfixed alongthenextsaddleconnection longertime;thenit proceeds point, remainstherefor someStill longerperiodof time, etc. Sincer(f) converges remainsthere to f, it comescloserandcloserto the fixed pointsandconsequently for longer and longertimes.If the fixed pointsare hypcrbolic(the genericcase, increase which we shall assumefrom now on), the periodsof near-stationarity for switching intervals On the otherhand,the lengthsof the time exponentially. from a neighbourhoodof one saddleto a neighbourhoodof the next one do not they are given by the time neededby the saddle changemuch: asymptotically, to theother.Comparedwith the amount connectionto crossfrom oneneighbourhood the durationof the joumeysin between of time spentwithin the neighbourhoods, can be neglected. close-byorbits may leave the The dynamicsis highly unpredictable: neighbourhoods of the fixed pointsat very differenttimes,andevolvequiteout of phase.It is difficult to tell in advance,for somegiven largeT, nearwhich fixed point r(7) will be lingering. thetimeaverages Averagingwill nothelpwith suchhighlyerraticoscillations: ,\

il' IJox t r ) d r

r &l

or *Irftl "t*o

(depending or discrete)will fail to converge. on whetherthedynamicsis continuous

220

K. Sigmund, Time averagesfor unpredictable orbits

As an example,let us considera gamewith threestrategiessuchthat strategy I beats2,2 beats3 and3 beats1. Thepayoffa;1fora playerusingstrategyi against a j-player is given,withoutrestrictinggenerality,by

(o

43

0

l-t\ \ . a r -4

-br)

O,L ;)

with a;,b;> 0. (For the rock-scissors-paper game,we canuS€4; - bi= l.) Let us considera populationconsistingof threetypes,eachplayingone of the strategies. Let rr(t) be the frequencyof the i-playersat time l. The stateof the populationat any time is given by a vectorx = (x1,x2,\) on the simplex53 (sincex; > 0 and Lx;= 1). If we assumerandomencounters, the expectedpayoff for strategyi is (Ar); = 2a;1x1and the averagepayoff for the populationis x 'Ax = Ixt(Ar)i. In gamedynamics,one assumes that (dueto someeffectof inheritance or leaming) t h e r a t e o f i n c r e a*s;el x ; o f s t r a t e g yiis g i v e n b y ( A x ) ; - x . A x , i . e . b y t h e d i f f e r e n c e betweenits payoff and the averagepayoff. Hence,the stateevolvesaccordingto *i = xi [(Ar)i - x'Ax].

(1)

It is easy to see that the simplex 53 and the boundaryfaces are invariant.The comerse; aresaddlepoints,the edgesaresaddleconnections (seefig. l). Thusthe

K. Sigmund, Timc averagesfor unpredictable orbits

221

boundaryis a heterocliniccycle. In [15] it is shown that if bft2fu) ay42a3'. then all orbits r t+ r(r) in the interior of 53 (with the exceptionof the centre m = i0,1,1))convergeto the boundary:it is a heteroclinicattractor. time averages z(I), with In [20] Gaunersdorfer showsthatthe corresponding coordinates 1r

z' ; ( T ) = * l x ( t ) d t lr

by fail to converge. Instead,they approach the boundaryof the trianglespanned + +übzk (b' zh - ,a 1 b 2 ,a1a3) , atctz* aftz

Ar ' = -1

pointsA2andÄ3 definedsimilarly.We note thatA3,Ä1 and andthe corresponding €1arecolinear.Asymptotically, duringthe periodwherer(r) is boggeddown in a which wascloseto Ä3, movesstraighttowardse1. vicinity of e1,the time average, When r(r) jumps over to 02, the motion of time averagechangesits direction towardseztthis happensat the comerA1 of the triangle.(Thisbehaviourseemsto havebeenfirst noticedby E.C. Zeeman;seealso [15].) type of argument,applied One way of proving this usesa Poincard-section to the heterocliniccycleinsteadof a periodicorbit.The saddlee;hasa; aspositive and -b; as negativeeigenvalue.An orbit comingin at a distancex from the orbit convergingto the saddleleavesat a distanceproportionalto *""' from the orbit issuingfrom the saddle.For the time average,one can neglectthe time spenton switchingfrom onesaddleto thenext.Thetime spentby r(t) neare; is approximately the b;-tla;th of the time spentnear the previouscomer and hencethe time spent by r(0 on the nth roundgrowslikep', with p =ll(b,lai). This allowsto compute the asymptoticbehaviourof the time average. The samemethodcanbe usedquite generallyfor heteroclinicattractors,with -b; and 4; &Scorrespondingnegativeand positive eigenvaluesof the ith saddle point.For example,we may considerthe Lotka-Volterracompetitionequation *r = xt(l- xr - Mz - Fxt), iz = x2( - Fq - x2 - ar3),

Q)

*3 = xz1- wt- F*z- xt), wherethex; arepopulationdensities This equationwas of threecompetingspecies. introducedby May and Leonardt2ll and is at the origin of muchrecentwork on competition(seee.g. [22,23D.We assume0 < b< 1 < a and a + b > 2. Thereis a heterocliniccycle on the boundaryof the statespaceIR3.:it consistsof the saddle points e; and connectingorbits. With the exceptionof the orbits on the diagonal xt= x2= rr, äll orbits x(r) in the interior of lR3-convergeto the heterocliniccycle. Again, the time averagespiralscloserand closerto a triangleA1A2A3,slowing down but neverconverging.one has

222

K. Sigmund, Time averagesfor unpredictable orbits

Ä', =

(l - F)'+(l-0)@-1)+(

a- t)-

( a - 1 ) 2, ( t - p ) 2, @ - 1 ) ( 1 -p ) )

and similar expressions for A2 andÄ3. It followsthatbothfor thegamedynamical eq.(1) andthecompetition eq.(2), almostall orbitshave time averages which do not converge.This seemsat first glancesomewhatat oddswith the ergodictheorem,accordingto which almostall the time averages converge.But of coursealmostc// is usedwith two different meanings.In the formercase,it is with respectto Lebesgue measure(on s3 or on IR3*), and in the lattercase,with respectto any invariantmeasure. It is easyto see that an invariantmeasurefor the dynamicalsystem(l) hasits supportcontainedin the setof fixedpoints{yryz,yt,m}:for suchmeasures thestatement of theergodic theoremis obviouslytrivial. Similar behaviouris found for a greatvariety of dynamicalsystems.For example,thefixedpointsof theheteroclinic cyclecanbe replaced by periodicorbits (see[24] for a seven-dimensional example)or by a strangeattractoron somelowerdimensionalsubmanifold.The orbit approaches the attractorfor sometime, then movesoff to visit anotherattractor, but retumsaftera while for a muchlongertime etc. The oscillatoryregimeof such an orbit seemsto switch, at exponentially increasing times,into differentmodes(similarto distinctclimatesin meteorology): the time averages, again,do not converge.Analogousresultsalso hold for the corresponding discretesystems. 3.

Strangeattractors

There is no commonlyagreeddefinition for a strangeattractoryet, but one certainlyexpectsoscillationsthereto exhibita livelier behaviourthanfor heteroclinic attractors: insteadof boggingdownfor longperiodsof near-immobility, theyshould be restlesslywiggling around.It is not paradoxicalto expectthat thesewilder oscillationsin tum shouldmakefor tamertime averages. The simplestexampleis the discretedynamicalsystemon the unit intervai [0,1] givenby the map x v-+4x(1- x).

(3)

This mapplaysa starrole in theoretical population ecologyt3l; if x denotes the size o f t h e p o p u l a t i o n ( r e s c a l e d s o t h a[ 0t x, 1 €])andifthepopulationgrowsforsmal. x, but decreases for largex, thenthebehaviourcanbe modelledby the logisticmap -,r). x t-+ax(l For smallvaluesof c, thereis a uniqueequilibrium(a - l)/a attracting all orbitsin (0,1),but if oneincreases thevalueof a, a sequence of perioddoubling occurs,and ultimatelychaoticmotion. Erratic bchaviourcausedby some very simpleregulatorymechanism is foundnotonly in population ecology,but in economics and other fields as well. It alsounderliesmany continuousdynamicalsystems.

K. Sigmund, Time averagesfor unpredictsble orbits

223

Other examplesof chaoticmotion occur for hyperbolicattractors,whose They arethe prototypesof strangeattractors ergodicbehaviouris well understood. (in spiteof the fact that no differentialequationsarisingin applicationshavebeen shownto belongto this class).Hyperbolicitymeansthat transversalsectionsto the into exponentially contractingandexpandingdirectionsin a smooth orbitsdecompose The Bowen(for precise way more a definitionwe referto [10,11,261). uniform Ruelle-Sinaitheorem[26] implies,as we shallsee,that almosteveryorbit in the In spiteof this, time average. neighbourhood of suchan attractorhasa converging thereremainplentyof orbitswithoutconvergingtime averages. dynamical Let us first fill in somebackground, startingwith a continuously after systemon a compactmetricsetx (seee.g. t27)).A subsetA is transformed, p is saidto be a time t, into a subsetA(t)= {.r(r):r € A}. A probabilitymeasure A cX andall r e IR.The ergodic invariantif 1t(A(t))= tL(A)holdsfor all measurable -+ IR is integrablewith respectto p, there theoremof Birkhoff impliesthat if/:X = exists a set Qf c X with It(Q) 1 suchthat the time average I

1 r ^.

T J0 f f r ? > l a ,

(4)

convergesfor T -+ +€ wheneverr € O1.It doesnot hold, in general,that Qf = X The functionsconverge. iflis continuous, i.e. thatall time averagesof continuous (as previous continuous in can be example the sectionshowshow wrong this function/, we canusein this casethe i th coordinatefunctionx H .r;). On the other hand,sincethe set of continuousfunctionsis separable, thereexistsa set QcX suchthat (4) converges (a) for all x e Q andall continuous/thetime average for 7 -+ +*; (b) tr(Q) = 1 for all invariantprobabilitymeasures /r. The pointsof Q arecalledquasiregular [28] or statisticallyregular[10].The results we mentionedso far showthat pointsare almostsurelystatisticallyregular,both with respectto every invariantprobabilitymeasureand (in the neighbourhoodof Therecanbe manypoints, a hyperbolicattractor)with respectto Lebesguemeasure. nevertheless, which are not statisticallyregular. Let us denoteby M(X) thespaceof probabilitymeasures onX (we alwaysconsider Borelmeasures, i.e.measures definedon the smallesto-algebracontainingall open sets).If p e M(X), the integral f r+ ttI fdtt is a linear functional on the spaceof continuousfunctions on X which maps the positive function l; to 1. In fact the probability measurescorrespondprecisely to these functionals.

224

K. Sigmund, Time averagesfor unpredictable orbits

The spaceM(X) is obviouslyconvex,andX can be imbeddedinto it: to each x e X we associatethepoint measure6, (whereö,(Ä) is I if r e A, and 0 if not). The point measures are, incidentally,just the extremalpointsof M(X). Now the topologyof X extendsin a naturalway to the so-calledweaktopologyin M(X). We p,,converges saythat a sequence of probabilitymeasures to the probabilitymeasure uif 'tt

J fdtr^

-+ fdtt J

holdsfor all continuousfunctions/: X -+ IR. The imbeddingx H ör of X into M(X) is then a continuous mappingonto the closedsetof all point measures. SinceX is compact,M(X) is also compact.The measures with finite support,i.e. the finite convexcombinations of point measures, are densein M(X). Just as the linear functional f t-+f (x) on the spaceof continuousfunctionscorresponds to the point measureör, so the linear functional -f

'tT r - +* l r ' f ( x ( t ) ) d t I

0

also yields a probability measure: it corresponds to the time average of the value

of / alongthe portion of the orbit of x from time 0 to time f > 0. We denotethis averageby 6,(D and define 6,10;= 5,. If r is statisticallyregularthen the limit

/ - ''lT-|1"""0' is also a well-definedlinear functionalon the spaceof continuousfunctionsand thus a probability measure.It is easy to see that it has the property of being invariant,which translatesinto the propertythat the continuousfunctionsx r+ f(x) andx Hflr(t)) havethe sameintegralfor all r. Sincetherealwaysexiststatistically regularpoints,therealwaysexist invariantmeasures. Hence,the spaceM1(X)of invariantprobabilitymeasures is nonempty.It is alsocompactand convex. There are dynamicalsystemssuchthat every x is statisticallyregular:in particular,thereexistsystemsadmittinga uniqueinvariantmeasure pr.In this case all averages ä,(7) converge to 1t, for ?"-+ +-. But in general,all onecansayis that the set (s) pQ) = {tt e M(X): l?"1+ +* suchthat 6,(T) -+ 1tl is a nonempty subsetof the setMr(X) of invariant measureswhich is compact and connected[27]. This latter property is a simple consequenceof the fact that f p ö,(f) is continuous.

K. Sigmund, Time averagesfor unpredictqble orbits

225

In the gamedynamicsexample(1), if the orbit of r hasthe boundaryof 53 as limit set,thenp(x) consistsof threemeasures I ar' -- -----:i--, r^t^ (bzk6", + arbr6r,+ ap36q) atazI apz + bZ4 joining themin defineda2anda3as well asthe line segments andthe analogously (but not convex!). the convexsetM()a).This set is connected If A is a hyperbolicattractor,then the theoremof Bowen-Ruelle-Sinai implies that there existsan invariantprobabilitymeasure,u on A such that for of A, the set p(x) reducesto {p}. all x in a neighbourhood Lebesgue-almost In fact if R is However,thereexistpointsr € ^ withoutconvergingtime average. qny nonemptycompactconnected subsetof the set M/(^) of invariantmeasures, thereexist pointsx suchthat p(x) = R, and the set of thesex is actuallydensein Ä, see [27]. There existsa staggeringvariety of compactconnectedsubsetsof by three convexsubsetspanned M/1t): evenif we look only at thetwo-dimensional This on A, we find a hugediversityof suchsubsets. distinctinvariantmeasures showsthat thereis an enormousrepertoireof possibleasymptoticbehavioursfor the time averages. In particular,if we take R = M//y), we find that the set L of all r e A with = U of A is the union of stable p(r) M/ly) is densein A. Sincea neighbourhood we canfind arbitrarilycloseto any manifoldsof pointsofÄ (seee.g.1L2,p.2621), y e U a point z e U suchthat d(2,,x,)-+0 for somer € L, whered is the distance and r + +*. Hence,the set {x eU : PG)= MrQt)} displaymaximaloscillation)is densein U. (the set of pointswhosetime-averages of opensets:this follows Moreover,it is a G5-Set,i.e. the countableintersection exactlyas in [27,p.207]. Thus,it is not only densebut large,andevenprevalent, is a meagreset(of first category). pointof view:its complement fromthetopological then,thetypicalinitialconditionr leadsto maximaloscillations Topologicallyspeaking, everyinvariantmeasureoccursas a limit point of 6lfo) for of the time averages: -) thatthesetM{lv) is very large.In particular, +€. It should be emphasized some11 propertythatits extremalpointsaredense(see[27]) - something it hastheinteresting that evidentlycan only happenif the convexset is infinite-dimensional. Hencethe questionaboutthe typical averagingbehaviourreceivesdifferent or topologically. on whether"typical"is meantmeasure-theoretically depending answers, The option initial conditions.) (In no caseis therean efficientclassification for the for one or the other point of view is a matterof taste(seeOxtoby t29) for an Probablymost(mayb we shouldsayalmostall) mathematicians discussion). interesting it shouldbe mentioned answer.Nevertheless, measure-thoretic would favour the thattheinitialconditionis perfectly thatit is in someway limitedby the assumption the initial determined-i.e.givenby a pointr. In manycasesof practicalrelevance

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K. Sigmund, Timc averagesfor unpredictqble orbits

conditionis not perfectlyknown, and corresponds to a probabilitydistribution which doesnot reduceto a point measure. This corresponds to modelswherethe dynamicsis perfectlydeterministic but the statestochastic. Thereseemsto be no naturalmeasureon the spaceM(X) of probabilitymeasures-but, as we haveseen, a very sensibletopology,namelythe weakone.Justas we can considerorbitsof points,we canalsoconsiderorbitsof statisticalstates,i.e. of probabilitymeasures. The dynamicalsystemsendsa probabilitymeasurelt e M(X) into the probability measurellt at time r, wherep, is given by Ir,(A)= tt,(A,) for measureable A c X. Thus the dynamicalsystemon X inducesa dynamical systemon the extension M(x). The time average of the orbit r -+ 1t,inM(X)is given bYr 1;

It(T):= * | Pdt. lJ 0

whichcorresponds to thelinearfunctional f ä;)

'tT ,T, f d t t ,= 7 J J r < r r t ) d 1 t d , t . 0

0x

Again, one defines pQ) = {tt e M (X) :3To -->+-

such that v(T) -+ tt\

for v e M(x), and obtains a nonempty compact connected subset of Ml(X). If the dynamicsconvergesto somehyperbolicattractor^, then for everynonemptyR c.M^ ) which is compact and connected,rhe ser {y: p(.v) =R} is densein M(^) t301. Again, the set {v: p(v)=M{L)} is a countableintersectionof open densesets. Hence the typical probability distributionwill have a time averagewhich oscillates maximally (where typical, here, is understoodfrom the topological point of view, while the measure-theoretic point of view does not enter into consideration). In order to verify the extreme misbehaviourof the time averages,one uses propertiesof the dynamics on the basic set Ä (see e.g. {121 or [27D: (a) topological mixing: for every two nonempty open subsets{J, v cÄ, there is a time I such that for all r> I, one finds an x e U with x(r) e V. Thus we can move from every neighbourhoodto every other one, provided we allow enough time for it; (b) a very weak form of the shadowing property: for every €, there is a p with the property that whenever the points y and z in Ä satisfy d(y,z)

0. The orbit of r shadowsthat of y for negative time and that of z for positive time.

K. Sigmund, Time averagesfor unpredictable orbits

227

formof this shadowing arguedthata stronger In [33],it hasbeenconvincingly propertyis fundamental for whatis meantby chaoticdynamics.All chaoticattractors andevengenerically do not converge, of this typehaveorbitswhosetime averages divergein somemaximalsense.Thereare other dynamicalsystemswith only a propertyfor whichthis oscillatorybehaviourof the weakerform of the shadowing (see holds still e.g [3a]).In particularthis is the casefor the discrete time averages p corresponding to theBowen-Ruellesystemx r+ 4x(l .r)on [0,1].Themeasure Sinai theoremhas the density f (x\ =

xlx(l- x)

but again, Lebesgue-almost all pointsin [0,1] havethis measureas time-average: displays whilethesetof pointswhosetime average theyform a setof firstcategory, as limit-points)is generic. maximaloscillation(i.e. has all invariantmeasures do To summarize, thereareimportantclassesof orbitswhosetime averages to havea methodwhich,by evaluatingtime not converge.It shouldbe interesting thetypeof underlyingattractor. in concretesituations, allowsto characterize averages of the asymptotic This does not seemto exist yet. There can be combinations heterocycles visiting strangeattractors; regimesdescribed in sections2 and3, with there could be altogetherdifferent types of behaviour.Furthermore,the transient This motion can subsistfor a very long while, thus affectingthe time averages. occurs,e.g. in non-linearnetworks,wherethc attractorsare fixed pointsand the trajectoriesare due to the complicatcd irregularoscillationsof the approaching geometryof the basinsof attraction. Acknowledgements Part of this work was supportedby the AustrianForschungsförderungsfonds M. SievekingandE.C. P8043.I wouldlike to thankJ. Hofbauer,A. Gaunersdorfer, Zeemanfor helpful discussions. References tll

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t8l t91 [10] [11] [12] [13] [14] [15] [16] [17] [18] t19l t20] [21] l22l I23l l24l l25l t26l

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