Local exponential divergence plot and optimal embedding of a chaotic time series

Abstract

We propose here a local exponential divergence plot which is capable ofproviding a new means ofcharacterizing chaotic time series. The suggested plot defines a time dependent exponent A and a "plus" exponent A÷ which servesas a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time and the largest Lyapunov exponent, garded as distances between orbits, then if the moare intimately related to the problem of how to distion is truly chaotic, points with dis(Xa+k, XJ+k)> tinguish chaos from stochastic processes. Therefore, dis(X1, X~)will dominate and lie above the zero level it would be very helpful if a geometric method could line in the plot. be devised to view the dynamics, especially the local Figure 1 shows divergence plots with different m exponential divergence dominated behaviorof a time and L for the Rössler attractor (a = 0.15, b = 0.20, series, so that a glance at this divergence plot would c= 10.0, &t= 7t/25, the dynamics is reconstructed provide some insight into the dynamic system.from the x component of the flow). We will show We report here a kind of local exponential diverbelow that the difference between these plots gives gence plot which enables one to view the dynamics a hint to optimal embedding, and m =3, L =8 coron a chaotic attractor. The simple plot provides a respond to optimal parameter values. The zero level criterion for the selection of the minimal acceptable line is added to fig. lb for a clear view of the diverembedding dimension and an optimal delay time.gence dominated behavior. When the unstable motion on the chaotic attractor A problem of significant practical importance is to only is extracted, a proper estimation of the largest determine the minimum acceptable embedding dipositive Lyapunov exponent can also be obtained. mension m~. A basic idea is that in the passage from Assume we have a time series x1, x2, ..., with samdimension m to m + 1 one can differentiate between 0375-9601/93/S 06.00

Estimating Lyapunov Exponents from Time Series

“…Different implementations of the direct approach have been suggested in the past 25 years [18,25,30,38,41] that are based on the following considerations. Let x.m.n// be a neighbour of the reference state x.n/ (with respect to the Euclidean norm or any other norm) and let both states be temporally separated jm.n/ nj > w (1.26) where w is a characteristic time scale (e.g., a mean period) of the time series.…”

2000 Physical Acoustics Summer School (PASS 00). Volume III: Background Materials

“…Most of the methods for determining d are based on continuity tests for the induced flow in the reconstruction space [Cenys & Pyragas, 1988;Buzug et al, 1990;Alecsic, 1991;Buzug & Pfister, 1992;Gao & Zheng, 1993Huerta et al, 1995] or for the embedding itself [Liebert et al, 1991;Kennel et al, 1992]. The main idea is to check whether closely neighboring points are mapped to neighboring points.…”

“…The most popular approach consists in a minimization of the redundancy of the coordinates of the reconstructed states using (linear) autocorrelation functions or information theoretic concepts like mutual information [Frazer & Swinney, 1986;Frazer, 1989aFrazer, , 1989bLiebert & Schuster, 1989;Martinerie et al, 1992]. Since bad values for ij may destroy the embedding, the continuity tests for estimating d may also be used to determine fy [Buzug et al, 1990;Buzug & Pfister, 1992;Gao & Zheng, 1993Liebert et al, 1991].…”

Direct Dynamical Test for Deterministic Chaos

“…Kaplan and Glass[9] have used a coarse-grained directional vector, and Wayland et al have employed ((phase space continuity,, [lo], which is a variant of the Kaplan-Glass method, while Sugihara and May [ll] and Kennel and Isebelle [12] have explored prediction to detect determination in a time series. In this paper we shall extend the concept of the time-dependent exponent [13] and develop a direct and dynamical test for distinguishing chaos from noise. The angle brackets denote ensemble average of all possible pairs of (Xi, X j ) , Y* is a prescribed sufficiently small distance, and k8t is the evolution time.…”