Asymptotic behavior of the time-dependent divergence exponent.

Abstract

The divergence rate method, which is used to determine the maximum Lyapunov exponent out of time series, is based on the evaluation of the time-dependent divergence exponent. For chaotic systems and in the small time regime, this exponent grows linearly in time. The asymptotic regime is instead characterized by a time-independent behavior due to the system eventually losing its memory of the starting conditions. The amplitude of this "plateau"-like divergence exponent depends both on the choice of the embedding dimension and lag and on the maximum distance of nearby starting trajectories in a way that is characteristic of the underlying dynamical system. In this paper, upon introducing the basic mathematical tools, we address the plateau evaluation for two classes of time series, those generated by a white noise source and those generated by a finite-dimensional chaotic system. The different behavior provides a novel tool to distinguish purely stochastic sources from deterministic ones, as well as to provide a precise estimate of the correlation dimension in the latter case. The method is also sensitive to correlated noise sources.