A new test for chaos
in deterministic systems
By Georg A.Gottwald†and Ian Melbourne‡
We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic.In contrast to the usual method of computing the maximal Lyapunov exponent,our method is applied directly to the time series data and does not require phase space reconstruction.Moreover,the dimension of the dynamical system and the form of the underlying equations is irrelevant.The input is the time series data and the output is0or1depending on whether the dynamics is non-chaotic or chaotic.The test is universally applicable to any deterministic dynamical system,in particular to ordinary and partial differential equations,and to maps.
Our diagnostic is the real valued function p(t)= t0φ(x(s))cos(θ(s))d s whereφis an observable on the underlying dynamics x(t)andθ(t)=ct+ t0φ(x(s))d s.The constant c>0isfixed arbitrarily.We define the mean-square-displacement M(t) for p(t)and set K=lim t→∞log M(t)/log t.Using recent developments in ergodic theory,we argue that typically K=0signifying nonchaotic dynamics or K=1 signifying chaotic dynamics.
Keywords:Chaos,deterministic dynamical systems,Lyapunov exponents,
mean square displacement,Euclidean extension
1.Introduction
The usual test of whether a deterministic dynamical system is chaotic or nonchaotic is the calculation of the largest Lyapunov exponentλ.A positive largest Lyapunov exponent indicates chaos:ifλ>0,then nearby trajectories separate exponentially and ifλ<0,then nearby trajectories stay close to each other.This approach has been widely used for dynamical systems whose equations are known(Abarbanel et al.1993;Eckmann et al.1986;Parker&Chua1989).If the equations are not known or one wishes to examine experimental data,this approach is not directly applicable. However Lyapunov exponents may be estimated(Wolf et al.1985;Sana&Sawada 1985;Eckmann et al.1986;Abarbanel et al.1993)by using the embedding theory of Takens(1981)or by approximating the linearisation of the evolution operator. Nevertheless,the computation of Lyapunov exponents is greatly facilitated if the underlying equations are known and are low-dimensional.
In this article,we propose a new0–1test for chaos which does not rely on knowing the underlying equations,and for which the dimension of the equations is irrelevant.The input is the time series data and the output is either a0or a1 depending on whether the dynamics is nonchaotic or chaotic.Since our method is †School of Mathematics and Statistics,University of Sydney,NSW2006,Australia
‡Department of Mathematics and Statistics,University of Surrey,Guildford,Surrey GU27XH, UK
Article submitted to Royal Society T E X Paper
2G.Gottwald and I.Melbourne
applied directly to the time series data,the only difference in difficulty between analysing a system of partial differential equations or a low-dimensional system of ordinary differential equations is the effort required to generate sufficient data.(As with all approaches,our method is impracticable if there are extremely long tran-sients or once the dimension of the attractor becomes too large.)With experimental data,there is the additional effect of noise to be taken into consideration.We briefly discuss this important issue at at the end of this paper.However,our aim in this paper is to present ourfindings in the situation of noise-free deterministic data.
2.Description of the0–1test for chaos
To describe the new test for chaos,we concentrate on the continuous time case and denote a solution of the underlying system by x(t).The discrete time case is handled analogously with the obvious modifications.Consider an observableφ(x) of the underlying dynamics.The method is essentially indep
endent of the actual form ofφ—almost any choice ofφwill suffice.For example if x=(x1,x2,...,x n) thenφ(x)=x1is a possible and simple choice.Choose c>0arbitrarily and define
θ(t)=ct+ t0φ(x(s))d s,
p(t)= t0φ(x(s))cos(θ(s))d s.(2.1)
(Throughout the examples in§3and§4wefix c=1.7once and for all.)We claim that
(i)p(t)is bounded if the underlying dynamics is nonchaotic and
(ii)p(t)behaves asymptotically like a Brownian motion if the underlying dynam-ics is chaotic.
The definition of p in(2.1),which involves only the observableφ(x),highlights the universality of the test.The origin and nature of the data which is fed into the system(2.1)is irrelevant for the test,and so is the dimension of the underlying dynamics.
Later on,we briefly explain the justification behind the claims(i)and(ii).For the moment,we suppose that the claims are correct and show how to proceed.
To characterise the growth of the function p(t)defined in(2.1),it is natural to look at the mean square displacement(MSD)of p(t),defined to be
1
M(t)=lim
T→∞
A new test for chaos3 which obviously does not change the slope K.)This allows for a clear distinction of a nonchaotic and a chaotic system as K may only take values K=0or K=1. We have lost though the possibility of quantifying the chaos by the magnitude of the largest Lyapunov exponentλ.
Numerically one has to make sure that initial transients have died out so that the trajectories are on(or close to)the attractor at time zero,and that the integration time T is long enough to ensure T t.
3.An example:the forced van der Pol oscillator
We illustrate the0–1test for chaos with the help of a concrete example,the forced van der Pol system,
˙x1=x2
˙x2=−d(x21−1)x2−x1+a cosωt(3.1) which has been widely studied in nonlinear dynamics(van der Pol1927;Guck-enheimer&Holmes1990).Forfixed a and d,the dynamics may be chaotic or nonchaotic depending on the parameterω.Following Parlitz&Lauterborn(1987), we take a=d=5and letωvary from2.457to2.466in increments of0.00001. Chooseφ(x)=x1+x2and c=1.7.We stress that the results are independent of these choices for all practical purposes.As described below in§5,almost all choices will work.(Deliberately poor choices such as c=0,orφ=7orφ=t obviously fail;sensible choices that fail are virtually impossible tofind.)
Infigure1we show a plot of K versusω.The periodic windows are clearly seen. As a comparison we show infigure2the largest Lyapunov exponentλversusω. Since the onset of chaos does not occur until afterω=2.462we display the results only for the range2.462<ω<2.466infigures1and2.(Both methods easily indicate regular dynamics for2.457<ω<2.462.)
Naturally we do not obtain the values K=0and K=1exactly–the mathe-matical results that underpin our method predict these values in the limit of infi-nite integration time.(The same caveat applies equally to the Lyapunov exponent method.)In producing the data forfigures1and2,we allowed for a transient of 200,000units of time and then integrated up to time T=2,000,000.As can be seen infigure3,for most of the400data points in the range ofω,we obtain K>0.8or K<0.01.
Next,we carry out the0–1test for the forced van der Pol system in the situation of a more limited quantity of data.The results are shown infigure4for2.463<ω<2.465.We again allow for a transient time200,000but then integrate only for T=50,000.The transitions between chaotic dynamics and periodic windows are almost as clear with T=50,000as they are with T=2,000,000even though the convergence of K to0or1is better with T=2,000,000.
Article submitted to Royal Society
4G.Gottwald and I.Melbourne 0
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Figure 1.Asymptotic growth rate K of the mean square displacement (2.2)versus ωfor the van der Pol system (3.1)determined by a numerical simulation of the skew product system (3.1)and (2.1)with a =d =5,c =1.7,φ(x )=x 1+x 2and ωvarying from 2.462to 2.466.The integration interval is T =2,000,000after an initial transient of 200,000units of time.-0.1-0.08
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Figure 2.Largest Lyapunov exponent λversus ωfor the van der Pol system (3.1)with a =d =5and ωvarying from 2.462to 2.466(cf Parlitz &Lauterborn 1987).The integration interval is T =2,000,000after an initial transient of 200,000units of time.Article submitted to Royal Society
A new test for chaos 50
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Figure 3.Asymptotic growth rate K versus ωfor the van der Pol system (3.1)as in figure 1with T =2,000,000.The horizontal lines represent K =0.01and K =0.8.
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2.463  2.464  2.46500.20.40.60.811.22.463  2.464  2.465Figure 4.Asymptotic growth rate K versus ωvarying from 2.463to 2.465for the van der Pol system as in figures 1and 3but with integration interval T =50,000after an initial transient of 200,000units of time.The horizontal lines represent K =0.01and K =0.8.Article submitted to Royal Society

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