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

In recent years much progress has been made in pling time & and construct vectors of the form understanding and characterizing chaotic dynamical X~=(x1, X•+L, ..., X~(rn—I )L), with m the embedding systems. Much ofthe progresshas been brought about dimension and L delay time. Hence a dynamics F: by the important discovery that an appropriate state X1—*X~+1is defined, which is assumed to be represpace can be reconstructed from a scalar time series sentative of the original system. The distance be[1]. Calculation of the correlation dimensionand of tween X~and X~,denoted by dis(X~,X), is mapped the K2 entropy by the Grassberger—Procaccia algoto dis(X,+k, ~÷k) after k iterations of F. The local rithm [2], and estimation of the Lyapunov expoexponential divergence plot is defined by plotting nents [3—6] have become standard procedures for ln[dis(X,+k, A+k)/dis(X, X~)]versus ln[dis(X1, analyzing chaotic signals. However, one may not gain Xi)] when dis(X,, X~)is smaller than a prescribed much understanding by routine calculations under small distance r*. If we assume that most of these certain circumstances, since there are difficulties in sufficiently small distances dis(X~,X~)can be reinterpreting correlation integral results [71,which 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 toview the dynamics, especiallythe local Figure 1 shows divergence plots with different m exponential divergencedominated 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