DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES
Physica 16D (1985)285-317North-Holland, AmsterdamDETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIESAlan WOLF -, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANODepartment of Physics, University of Texas, Austin, Texas 78712, USAReceived 18 October 1984We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental timeseries. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related tothe exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunovexponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only beapplied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor.The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskiireaction and Couette-Taylor flow.Contents1.2.3.4.5.6.7.8.9.IntroductionThe Lyapunov spectrum definedCalculation of Lyapunov spectra from differential equationsAn approach to spectral estimation for experimental dataSpectral algorithm implementation*Implementation details*Data requirements and noise*ResultsConclusionsAppendices*A. Lyapunov spectrum program for systems of differentialequationsB. Fixed evolution time program for '11. IntroductionConvincing evidence for deterministic chaos hascome from a variety of recent experiments [1-6]on dissipative nonlinear systems; therefore, thequestion of detecting and quantifying chaos hasbecome an important one. Here we consider thespectrum of Lyapunov exponents [7-10], whichhas proven to be the most useful dynamical diagnostic for chaotic systems. Lyapunov exponentsare the average exponential rates of divergence ortPresent address: The Cooper Union, School of Engineering,N.Y., NY 10003, USA.*The reader may wish to skip the starred sections at a firstreading.0167-2789/85/ 03.30 Elsevier Science Publishers(North-H01!and Physics Publishing Division)convergence of nearby orbits in phase space. Sincenearby orbits correspond to nearly identical states,exponential orbital divergence means that systemswhose initial differences we may not be able toresolve will soon behave quite differently-predictive ability is rapidly lost. Any system containingat least one positive Lyapunov exponent is definedto be chaotic, with the magnitude of the exponentreflecting the time scale on which system dynamicsbecome unpredictable [10].For systems whose equations of motion are explicitly known there is a straightforward technique[8, 9] for computing a complete Lyapunov spectrum. This method cannot be applied directly toexperimental data for reasons that will be discussed later. We will describe a technique whichfor the first time yields estimates of the non-negative Lyapunov exponents from finite amounts ofexperimental data.A less general procedure [6, 11-14] for estimating only the dominant Lyapunov exponent in experimental systems has been used for some time.This technique is limited to systems where a welldefined one-dimensional (l-D) map can be recovered. The technique is numerically unstableand the literature contains several examples of itsimproper application to experimental data. A discussion of the 1-D map calculation may be found
286A. Wolf et al./ Determining Lyapunov exponentsfrom a time seriesin ref. 13. In ref. 2 we presented an unusuallyrobust 1-D map exponent calculation for experimental data obtained from a chemical reaction.Experimental data inevitably contain externalnoise due to environmental fluctuations and limitedexperimental resolution. In the limit of an infiniteamount of noise-free data our approach wouldyield Lyapunov exponents by definition. Our ability to obtain good spectral estimates from experimental data depends on the quantity and qualityof the data as well as on the complexity of thedynamical system. We have tested our method onmodel dynamical systems with known spectra andapplied it to experimental data for chemical [2, 13]and hydrodynamic [3] strange attractors.Although the work of characterizing chaotic datais still in its infancy, there have been many approaches to quantifying chaos, e.g., fractal powerspectra [15], entropy [16-18, 3], and fractal dimension [proposed in ref. 19, used in ref. 3-5, 20, 21].We have tested many of these algorithms on bothmodel and experimental data, and despite theclaims of their proponents we have found thatthese approaches often fail to characterize chaoticdata. In particular, parameter independence, theamount of data required, and the stability of resuits with respect to external noise have rarelybeen examined thoroughly.The spectrum of Lyapunov exponents will bedefined and discussed in section 2. This sectionincludes table I which summarizes the model systems that are used in this paper. Section 3 is areview of the calculation of the complete spectrumof exponents for systems in which the definingdifferential equations are known. Appendix A contains Fortran code for this calculation, which toour knowledge has not been published elsewhere.In section 4, an outline of our approach to estimating the non-negative portion of the Lyapunovexponent spectrum is presented. In section 5 wedescribe the algorithms for estimating the twolargest exponents. A Fortran program for determining the largest exponent is contained inappendix B. Our algorithm requires input parameters whose selection is discussed in section 6. Sec-tion 7 concerns sources of error in the calculationsand the quality and quantity of data required foraccurate exponent estimation. Our method is applied to model systems and experimental data insection 8, and the conclusions are given insection 9.2. The Lyapunov spectrum definedWe now define [8, 9] the spectrum of Lyapunovexponents in the manner most relevant to spectralcalculations. Given a continuous dynamical system in an n-dimensional phase space, we monitorthe long-term evolution of an infinitesimal n-sphereof initial conditions; the sphere will become ann-ellipsoid due to the locally deforming nature ofthe flow. The ith one-dimensional Lyapunov exponent is then defined in terms of the length of theellipsoidal principal axis p i ( t ) :h lim 1 log 2 p c ( t )t--,oo tpc(O)'(1)where the )h are ordered from largest to smallestt.Thus the Lyapunov exponents are related to theexpanding or contracting nature of different directions in phase space. Since the orientation of theellipsoid changes continuously as it evolves, thedirections associated with a given exponent vary ina complicated way through the attractor. One cannot, therefore, speak of a well-defined directionassociated with a given exponent.Notice that the linear extent of the ellipsoidgrows as 2 htt, the area defined by the first twoprincipal axes grows as 2 (x *x2)t, the volume defined by the first three principal axes grows as2 (x' x2 x )t, and so on. This property yieldsanother definition of the spectrum of exponents:tWhile the existenceof this limit has been questioned [8, 9,22], the fact is that the orbital divergenceof any data set maybe quantified.Even if the limit does not exist for the underlyingsystem, or cannot be approacheddue to having finite amountsof noisy data, Lyapunovexponent estimates could still providea useful characterizationof a givendata set. (See section 7.1.)
A. Wolf et aL / Determining Lyapunov exponents from a time seriesthe sum of the first j exponents is defined by thelong term exponential growth rate of a j-volumeelement. This alternate definition will provide thebasis of our spectral technique for experimentaldata.Any continuous time-dependent dynamical system without a fixed point will have at least onezero exponent [22], corresponding to the slowlychanging magnitude of a principal axis tangent tothe flow. Axes that are on the average expanding(contracting) correspond to positive (negative) exponents. The sum of the Lyapunov exponents isthe time-averaged divergence of the phase spacevelocity; hence any dissipative dynamical systemwill have at least one negative exponent, the sumof all of the exponents is negative, and the posttransient motion of trajectories will occur on azero volume limit set, an attractor.The exponential expansion indicated by a positive Lyapunov exponent is incompatible with motion on a bounded attractor unless some sort offolding process merges widely separated trajectories. Each positive exponent reflects a "direction"in which the system experiences the repeatedstretching and folding that decorrelates nearbystates on the attractor. Therefore, the long-termbehavior of an initial condition that is specifiedwith any uncertainty cannot be predicted; this ischaos. An attractor for a dissipatiVe system withone or more positive Lyapunov exponents is saidto be "strange" or "chaotic".The signs of the Lyapunov exponents provide aqualitative picture of a system's dynamics. Onedimensional maps are characterized by a singleLyapunov exponent which is positive for chaos,zero for a marginally stable orbit, and negative fora periodic orbit. In a three-dimensional continuousdissipative dynamical system the only possiblespectra, and the attractors they describe, are asfollows: ( , 0 , - ) , a strange attractor; ( 0 , 0 , - ) , atwo-toms; (0, - , - ) , a limit cycle; and ( - , - , - ) ,a fixed point. Fig. 1 illustrates the expanding,"slower than exponential," and contracting character of the flow for a three,dimensional system,the Lorenz model [23]. (All of the model systems287that we will discuss are defined in table I.) SinceLyapunov exponents involve long-time averagedbehavior, the short segments of the trajectoriesshown in the figure cannot be expected to accurately characterize the positive, zero, and negativeexponents; nevertheless, the three distinct types ofbehavior are clear. In a continuous four-dimensional dissipative system there are three possibletypes of strange attractors: their Lyapunov spectraare ( , , 0 , - ) , ( , 0 , 0 , - ) , and ( , 0 , - , - ) .An example of the first type is Rossler's hyperchaos attractor [24] (see table I). For a givensystem a change in parameters will generallychange the Lyapunov spectrum and may alsochange both the type of spectrum and type ofattractor.The magnitudes of the Lyapunov exponentsquantify a n attractor's dynamics in informationtheoretic terms. The exponents measure the rate atwhich system processes create or destroy information [10]; thus the exponents are expressed in bitsof information/s or bits/orbit for a continuoussystem and bits/iteration for a discrete system.For example, in the Lorenz attractor the positiveexponent has a magnitude of 2.16 bits/s (for theparameter values shown in table I). Hence if aninitial point were specified with an accuracy of onepart per million (20 bits), the future behaviorcould not be predicted after about 9 s [20 bits/(2.16bits/s)], corresponding to about 20 orbits. Afterthis time the small initial uncertainty will essentially cover the entire attractor, reflecting 20 bits ofnew information that can be gained from an ad:ditional measurement of the system. This newinformation arises from scales smaller than ourinitial uncertainty and results in an inability tospecify the state of the system except to say that itis somewhere on the attractor. This process issometimes called an information gain- reflectingnew information from the heat bath, and sometimes is called an information loss-bits shiftedout of a phase space variable "register" when bitsfrom the heat bath are shifted in.The average rate at which information contained in transients is lost can be determined from
A. Wolf et al. / Determining Lyapunov exponents from a time series288 . o . "." ." '. .-. . . ".-'. . .;.;.,.".:,v':.: .:"""'. '." .' "." "". .time- . ". . . . . . . . . . . " .". "-." x . -' - i l -. .I.'. - .: " . . .,.,.-.: .:-.:." . . . . : : . : . . " .'."'".". .,. .' :'-:, :. "r. :-::::,.!.":"""-.--,.-,. . . . , : , , . : :o. ,-:.IIIIIl[l l -.i:'.".- """:'"'"'"" " , ' . : ' ". ., .-. . , .:' .?'-,',; ' xxx ". " : : : , . f , , , , , . :L . -" .'. ' . ., . . . . ." ." .'. . .',. '." ::.:.':. "." : . . . ., .,.:.startN " "- '.:,.": /" % V 4 " ;.' : ' " : "t, :. .,--.:-:.::-:. '"" """(b).".':':"'( (, '." ,m- . . .-. . . .,:'. .--.: ::.-.:.:'.:.:.-. ." " ":'" : " : : " ' " ;.": " [l,,m,,,,l,,,,,llli,,lllli,,dlltime-- " : start .-.-.:; .- --- .- : . . ,. ;,.':.::. :" -:.v. ;;. . .::. s,,,,. time:'" .,, ",',.: .'.-2 'W '". -.".".'."'::.':"'.:" ,' . - - " ". "- : "" Fig. 1. The short term evolution of the separation vector between three carefully chosen pairs of nearby points is shown for theLorenz attractor, a) An expanding direction ( 1 0); b) a "slower than exponential" direction ( '2 0); C) a contracting direction(X3 0).the negative exponents The asymptotic decay of aperturbation to the attractor is governed by theleast negative exponent, which should therefore bethe easiest of the negative exponents to estimatet.For the Lorenz attractor the negative exponent isso large that a perturbed orbit typically becomesindistinguishable from the attractor, by "eye", inless than one mean orbital period (see fig. 1).t W e have been quite successful with an algorithm for determiuing the dominant (smallest magnitude) negative exponent from pseudo-experimental data (a single time series extracted from the solution of a model system and treated as anexperimental observable) for systems that are nearly integerdimensional. Unfortunately, our approach, which involves measuring the mean decay rate of many induced perturbations ofthe dynamical system, is unlikely to work on many experimental systems. There are several fundamental problems with thecalculation of negative exponents from experimental data, butof greatest importance is that post-transient data may notcontain resolvable negative exponent information and perturbed data must refl t properties of the unperturbed system,that is, perturbations must only change the state of the system(current values of the dynamical variables). The response of aphysical system to a non-delta function perturbation is difficultto interpret, as an orbit separating from the attractor mayreflect either a locally repelling region of the attractor (apositive contribution to the negative exponent) or the finite risetime of the perturbation.
A. Wolf et al. / Determining Lyapunov exponents from a time series289Table IThe model systems considered in this paper and their Lyapunov spectra and dimensions as computed from the equations of )tLyapunovdimension*{ b 1.4 1 0.603h 2 - 2.34H non: [25]X. 1 1 - aX;. YnY. 1 bX.1.26(bits/iter.) 0.3Rossler-chaos: [26]0.13)( - (Y Z)[ a 0.15)k 1 )' X aY b Z(X-I b 0.20 2 0.00h 3 - 14.1c)c 10.02.01Lorenz: [23])( o(Y-X) ' X ( R - Z ) X Y - bZY[ o 16.0h 1 2.16I R 45.92b 4.0X2 0.00;k3 - 32.4( a 0.25[ b 3.0 c 0.05k d 0.5At 0.16X2 0.03h 3 0.00h4 - 39.0( a 0.2h t 6.30E-3/ b 0.1) c 10.0s 31.8) 2 2.62E-3IX31 8.0E-6)'4 - 1.39E-22.07Rossler-hyperchaos: [24]Jr' - ( Y Z ))' X aY W b XZif" c W - d Z3.005Mackey-Glass: [27]j( aX(t s )- bX(t)1 [ X(t s)] c3.64t A mean orbital period is well defined for Rossler chaos (6.07 seconds) and for hyperchaos (5.16 seconds) for the parameter valuesused here. For the Lorenz attractor a characteristic time (see footnote- section 3) is about 0.5 seconds. Spectra were computed foreach system with the code in appendix A. As defined in eq. (2).The Lyapunov spectrum is closely related to thefractional dimension of the associated strange attractor. There are a number [19] of different fractional-dimension-like quantities, including thefractal dimension, information dimension, and thecorrelation exponent; the difference between themis often small. It has been conjectured by Kaplanand Yorke [28, 29] that the information dimensiond r is related to the Lyapunov spectrum by theequationE i - - 1 idf J I? j il '(2)where j is defined by the condition thatjE) i 0i--1j landEX, O.(3)i--1The conjectured relation between d r (a staticproperty of an attracting set) and the Lyapunov
290,4. 14/olfet aL / Determining Lyapunov exponents from a time seriesexponents appears to be satisfied for some modelsystems [30]. The calculation of dimension fromthis equation requires knowledge of all but themost negative Lyapunov exponents.3. Calculation of Lyapunov spectra from differentialequationsOur algorithms for computing a non-negativeLyapunov spectrum from experimental data areinspired by the technique developed independently by Bennetin et al. [8] and by Shimada andNagashima [9] for determining a complete spectrum from a set of differential equations. Therefore, we describe their calculation (for brevity, theODE approach) in some detail.We recall that Lyapunov exponents are definedby the long-term evolution of the axes of an infinitesimal sphere of states. This procedure could beimplemented by defining the principal axes withinitial conditions whose separations are as small ascomputer limitations allow and evolving these withthe nonlinear equations of motion. One problemwith this approach is that in a chaotic system wecannot guarantee the condition of small separations for times on the order of hundreds oforbital periodst, needed for convergence of thespectrum.This problem may be avoided with the use of aphase space plus tangent space approach. A "fiducial" trajectory (the center of the sphere) is definedby the action of the nonlinear equations of motionon some initial condition. Trajectories.of points onthe surface of the sphere are defined by the actionof the linearized equations of motion on pointsinfinitesimally separated from the fiducial trajectory. In particular, the principal axes are definedby the evolution via the linearized equations of aninitially orthonormal vector frame anchored to thet S h o u l d the mean orbital period not be well-defined, acharacteristic time can be either the mean time between intersections of a Poincar6 section or the time corresponding to ad o m i n a n t power spectral feature.fiducial trajectory. By definition, principal axesdefined by the linear system are always infinitesimalrelative to the attractor. Even in the linear system,principal axis vectors diverge in magnitude, butthis is a problem only because computers have alimited dynamic range for storing numbers. Thisdivergence is easily circumvented. What has beenavoided is the serious problem of principal axesfinding the global "fold" when we really only wantthem to probe the local "stretch."To implement this procedure the fiducial trajectory is created by integrating the nonlinear equations of motion for some post-transient initialcondition. Simultaneously, the linearized equations of motion are integrated for n different initial conditions defining an arbitrarily orientedframe of n orthonormal vectors. We have alreadypointed out that each vector will diverge in magnitude, but there is an additional singularity-in achaotic system, each vector tends to fall along thelocal direction of most rapid growth. Due to thefinite precision of computer calculations, the collapse toward a common direction causes the tangent space orientation of all axis vectors to becomeindistinguishable. These two problems can beovercome by the repeated use of the GramSchmidt reorthonormalization (GSR) procedure onthe vector frame:Let the linearized equations of motion act onthe initial frame of orthonormal vectors to give aset of v e c t o r s { v 1. . . . . Vn). (The desire of eac
286 A. Wolf et al. / Determining Lyapunov exponents from a time series in ref. 13. In ref. 2 we presented an unusually robust 1-D map exponent calculation for experi- mental data obtained from a chemical reaction.
The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifur-cation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer .
largest nonzero Lyapunov exponent λm among the n Lyapunov exponents of the n-dimensional dynamical system. A.2.1 Computation of Lyapunov Exponents To compute the n-Lyapunov exponents of the n-dimensional dynamical system (A.1), a reference trajectory is created by integrating the nonlinear equations of motion (A.1).
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The Lyapunov theory of dynamical systems is the most useful general theory for studying the stability of nonlinear systems. It includes two methods, Lyapunov’s indirect method and Lyapunov’s direct method. Lyapunov’s indirect method states that the dynamical system x f(x), (1)
Lyapunov exponents may provide a more useful characterization of chaotic systems. For time series produced by dynamical systems, the presence of a positive characteristic exponent indicates chaos. Furthermore, in many applications it is sufficient to calculate only the largest Lyapunov exponent (λ1).
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