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Physica 16D (1985)285-317

North-Holland, Amsterdam

DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES

Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO

Department of Physics, University of Texas, Austin, Texas 78712, USA

Received 18 October 1984

We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time

series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to

the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov

exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be

applied 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-Zhabotinskii

reaction and Couette-Taylor flow.

Contents

1. Introduction

2. The Lyapunov spectrum defined

3. Calculation of Lyapunov spectra from differential equations

4. An approach to spectral estimation for experimental data

5. Spectral algorithm implementation*

6. Implementation details*

7. Data requirements and noise*

8. Results

9. Conclusions

Appendices*

A. Lyapunov spectrum program for systems of differential

equations

B. Fixed evolution time program for ~'1

1. Introduction

Convincing evidence for deterministic chaos has

come from a variety of recent experiments [1-6]

on dissipative nonlinear systems; therefore, the

question of detecting and quantifying chaos has

become an important one. Here we consider the

spectrum of Lyapunov exponents [7-10], which

has proven to be the most useful dynamical di-

agnostic for chaotic systems. Lyapunov exponents

are the average exponential rates of divergence or

tPresent address: The Cooper Union, School of Engineering,

N.Y., NY 10003, USA.

*The reader may wish to skip the starred sections at a first

reading.

convergence of nearby orbits in phase space. Since

nearby orbits correspond to nearly identical states,

exponential orbital divergence means that systems

whose initial differences we may not be able to

resolve will soon behave quite differently-predic-

tive ability is rapidly lost. Any system containing

at least one positive Lyapunov exponent is defined

to be chaotic, with the magnitude of the exponent

reflecting the time scale on which system dynamics

become unpredictable [10].

For systems whose equations of motion are ex-

plicitly known there is a straightforward technique

[8, 9] for computing a complete Lyapunov spec-

trum. This method cannot be applied directly to

experimental data for reasons that will be dis-

cussed later. We will describe a technique which

for the first time yields estimates of the non-nega-

tive Lyapunov exponents from finite amounts of

experimental data.

A less general procedure [6, 11-14] for estimat-

ing only the dominant Lyapunov exponent in ex-

perimental systems has been used for some time.

This technique is limited to systems where a well-

defined one-dimensional (l-D) map can be re-

covered. The technique is numerically unstable

and the literature contains several examples of its

improper application to experimental data. A dis-

cussion of the 1-D map calculation may be found

0167-2789/85/$03.30 © Elsevier Science Publishers

(North-H01!and Physics Publishing Division)

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.

Experimental data inevitably contain external

noise due to environmental fluctuations and limited

experimental resolution. In the limit of an infinite

amount of noise-free data our approach would

yield Lyapunov exponents by definition. Our abil-

ity to obtain good spectral estimates from experi-

mental data depends on the quantity and quality

of the data as well as on the complexity of the

dynamical system. We have tested our method on

model dynamical systems with known spectra and

applied it to experimental data for chemical [2, 13]

and hydrodynamic [3] strange attractors.

Although the work of characterizing chaotic data

is still in its infancy, there have been many ap-

proaches to quantifying chaos, e.g., fractal power

spectra [15], entropy [16-18, 3], and fractal dimen-

sion [proposed in ref. 19, used in ref. 3-5, 20, 21].

We have tested many of these algorithms on both

model and experimental data, and despite the

claims of their proponents we have found that

these approaches often fail to characterize chaotic

data. In particular, parameter independence, the

amount of data required, and the stability of re-

suits with respect to external noise have rarely

been examined thoroughly.

The spectrum of Lyapunov exponents will be

defined and discussed in section 2. This section

includes table I which summarizes the model sys-

tems that are used in this paper. Section 3 is a

review of the calculation of the complete spectrum

of exponents for systems in which the defining

differential equations are known. Appendix A con-

tains Fortran code for this calculation, which to

our knowledge has not been published elsewhere.

In section 4, an outline of our approach to estimat-

ing the non-negative portion of the Lyapunov

exponent spectrum is presented. In section 5 we

describe the algorithms for estimating the two

largest exponents. A Fortran program for de-

termining the largest exponent is contained in

appendix B. Our algorithm requires input parame-

ters whose selection is discussed in section 6. Sec-

tion 7 concerns sources of error in the calculations

and the quality and quantity of data required for

accurate exponent estimation. Our method is ap-

plied to model systems and experimental data in

section 8, and the conclusions are given in

section 9.

2. The Lyapunov spectrum defined

We now define [8, 9] the spectrum of Lyapunov

exponents in the manner most relevant to spectral

calculations. Given a continuous dynamical sys-

tem in an n-dimensional phase space, we monitor

the long-term evolution of an

infinitesimal

n-sphere

of initial conditions; the sphere will become an

n-ellipsoid due to the locally deforming nature of

the flow. The ith one-dimensional Lyapunov expo-

nent is then defined in terms of the length of the

ellipsoidal principal axis

pi(t):

h~ = lim 1 log 2

pc(t)

t--,oo t pc(O)'

(1)

where the )h are ordered from largest to smallestt.

Thus the Lyapunov exponents are related to the

expanding or contracting nature of different direc-

tions in phase space. Since the orientation of the

ellipsoid changes continuously as it evolves, the

directions associated with a given exponent vary in

a complicated way through the attractor. One can-

not, therefore, speak of a well-defined direction

associated with a given exponent.

Notice that the linear extent of the ellipsoid

grows as 2 htt, the area defined by the first two

principal axes grows as 2 (x~*x2)t, the volume de-

fined by the first three principal axes grows as

2 (x'+x2+x~)t, and so on. This property yields

another definition of the spectrum of exponents:

tWhile the existence of this limit has been questioned [8, 9,

22], the fact is that the orbital divergence of

any

data set may

be quantified. Even if the limit does not exist for the underlying

system, or cannot be approached due to having finite amounts

of noisy data, Lyapunov exponent estimates could still provide

a useful characterization of a given data set. (See section 7.1.)

A. Wolf et aL / Determining Lyapunov exponents from a time series

287

the sum of the first j exponents is defined by the

long term exponential growth rate of a j-volume

element. This alternate definition will provide the

basis of our spectral technique for experimental

data.

Any continuous time-dependent dynamical sys-

tem without a fixed point will have at least one

zero exponent [22], corresponding to the slowly

changing magnitude of a principal axis tangent to

the flow. Axes that are on the average expanding

(contracting) correspond to positive (negative) ex-

ponents. The sum of the Lyapunov exponents is

the time-averaged divergence of the phase space

velocity; hence any dissipative dynamical system

will have at least one negative exponent, the sum

of all of the exponents is negative, and the post-

transient motion of trajectories will occur on a

zero volume limit set, an attractor.

The exponential expansion indicated by a posi-

tive Lyapunov exponent is incompatible with mo-

tion on a bounded attractor unless some sort of

folding

process merges widely separated trajecto-

ries. Each positive exponent reflects a "direction"

in which the system experiences the repeated

stretching and folding that decorrelates nearby

states on the attractor. Therefore, the long-term

behavior of an initial condition that is specified

with

any

uncertainty cannot be predicted; this is

chaos. An attractor for a dissipatiVe system with

one or more positive Lyapunov exponents is said

to be "strange" or "chaotic".

The signs of the Lyapunov exponents provide a

qualitative picture of a system's dynamics. One-

dimensional maps are characterized by a single

Lyapunov exponent which is positive for chaos,

zero for a marginally stable orbit, and negative for

a periodic orbit. In a three-dimensional continuous

dissipative dynamical system the only possible

spectra, and the attractors they describe, are as

follows: (+,0,-), a strange attractor; (0,0,-), a

two-toms; (0, -, -), a limit cycle; and (-, -, -),

a fixed point. Fig. 1 illustrates the expanding,

"slower than exponential," and contracting char-

acter of the flow for a three,dimensional system,

the Lorenz model [23]. (All of the model systems

that we will discuss are defined in table I.) Since

Lyapunov exponents involve long-time averaged

behavior, the short segments of the trajectories

shown in the figure cannot be expected to accu-

rately characterize the positive, zero, and negative

exponents; nevertheless, the three distinct types of

behavior are clear. In a continuous four-dimen-

sional dissipative system there are three possible

types of strange attractors: their Lyapunov spectra

are (+, +,0,-), (+,0,0,-), and (+,0,-,-).

An example of the first type is Rossler's hyper-

chaos attractor [24] (see table I). For a given

system a change in parameters will generally

change the Lyapunov spectrum and may also

change both the type of spectrum and type of

attractor.

The magnitudes of the Lyapunov exponents

quantify

an attractor's dynamics in information

theoretic terms. The exponents measure the rate at

which system processes create or destroy informa-

tion [10]; thus the exponents are expressed in bits

of information/s or bits/orbit for a continuous

system and bits/iteration for a discrete system.

For example, in the Lorenz attractor the positive

exponent has a magnitude of 2.16 bits/s (for the

parameter values shown in table I). Hence if an

initial point were specified with an accuracy of one

part per million (20 bits), the future behavior

could not be predicted after about 9 s [20 bits/(2.16

bits/s)], corresponding to about 20 orbits. After

this time the small initial uncertainty will essen-

tially cover the entire attractor, reflecting 20 bits of

new information that can be gained from an ad:

ditional measurement of the system. This new

information arises from scales smaller than our

initial uncertainty and results in an inability to

specify the state of the system except to say that it

is somewhere on the attractor. This process is

sometimes called an information gain- reflecting

new information from the heat bath, and some-

times is called an information loss-bits shifted

out of a phase space variable

"register"

when bits

from the heat bath are shifted in.

The average rate at which information con-

tained in transients is lost can be determined from

288

A. Wolf et al. / Determining Lyapunov exponents from a time series

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Fig. 1. The short term evolution of the separation vector between three carefully chosen pairs of nearby points is shown for the

Lorenz 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 a

perturbation to the attractor is governed by the

least negative exponent, which should therefore be

the easiest of the negative exponents to estimatet.

tWe have been quite successful with an algorithm for de-

termiuing the dominant (smallest magnitude) negative expo-

nent from pseudo-experimental data (a single time series ex-

tracted from the solution of a model system and treated as an

experimental observable) for systems that are nearly integer-

dimensional. Unfortunately, our approach, which involves mea-

suring the mean decay rate of many induced perturbations of

the dynamical system, is unlikely to work on many experimen-

tal systems. There are several fundamental problems with the

calculation of negative exponents from experimental data, but

For the Lorenz attractor the negative exponent is

so large that a perturbed orbit typically becomes

indistinguishable from the attractor, by "eye", in

less than one mean orbital period (see fig. 1).

of greatest importance is that

post-transient

data may not

contain resolvable negative exponent information and

per-

turbed

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 a

physical system to a non-delta function perturbation is difficult

to interpret, as an orbit separating from the attractor may

reflect either a locally repelling region of the attractor (a

positive contribution to the negative exponent) or the finite rise

time of the perturbation.

A. Wolf et al. / Determining Lyapunov exponents from a time series

289

Table I

The model systems considered in this paper and their Lyapunov spectra and dimensions as computed from the equations of motion

Lyapunov Lyapunov

System Parameter spectrum dimension*

values (bits/s)t

H~non:

[25]

~1 = 0.603

X. +1 = 1 -

aX;. + Yn

{ b = 1.4 h 2 = - 2.34

Y. + 1

bX.

= 0.3

(bits/iter.)

Rossler-chaos:

[26]

)( = - (Y + Z) [ a = 0.15

)k 1 =

0.13

)'= X+ aY I

b = 0.20 ~2 =0.00

= b + Z(X-

c) c = 10.0 h 3 = - 14.1

Lorenz:

[23]

)(= o(Y- X) [ o = 16.0 h 1 = 2.16

~'= X( R- Z)- Y I

R=45.92 X 2 =0.00

= XY - bZ b =

4.0 ;k 3 = - 32.4

Rossler-hyperchaos:

[24]

Jr'= - (Y+ Z) ( a = 0.25 A t = 0.16

)'= X+ aY+

W [ b= 3.0 X 2 =0.03

= b + XZ | c =

0.05 h 3 = 0.00

if" = cW - dZ

k d = 0.5 h4 = - 39.0

Mackey-Glass:

[27]

( a = 0.2 h t = 6.30E-3

j( = aX(t + s ) - bX(t)

/ b = 0.1 )~2 = 2.62E-3

1 + [ X(t +

s)] c ) c = 10.0

IX31 <

8.0E-6

s = 31.8 )'4 = - 1.39E-2

1.26

2.01

2.07

3.005

3.64

tA mean orbital period is well defined for Rossler chaos (6.07 seconds) and for hyperchaos (5.16 seconds) for the parameter values

used here. For the Lorenz attractor a characteristic time (see footnote- section 3) is about 0.5 seconds. Spectra were computed for

each system with the code in appendix A.

~As defined in eq. (2).

The Lyapunov spectrum is closely related to the

fractional dimension of the associated strange at-

tractor. There are a number [19] of different frac-

tional-dimension-like quantities, including the

fractal dimension, information dimension, and the

correlation exponent; the difference between them

is often small. It has been conjectured by Kaplan

and Yorke [28, 29] that the information dimension

d r is related to the Lyapunov spectrum by the

equation

Ei-- 1~i

df=J+

I?~j+il ' (2)

where j is defined by the condition that

j j+l

E)~i> 0 and

EX,<O.

(3)

i--1 i--1

The conjectured relation between d r (a

static

property of an attracting set) and the Lyapunov

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