Chaotic Measures and Real-World Systems

Abstract

Direct estimation of the largest Lyapunov exponent as a measure of exponential divergence of nearby trajectories is well established in the case of deterministic dynamical systems. Questions are naturally raised about applicability of Lyapunov exponents and other “chaotic measures” when analyzing data from real-world systems, which are either stochastic or affected by numerous external influences, which cannot be described in any other way than a stochastic component in system dynamics. In a series of numerical experiments, Gaussian random deviates were added to a set of chaotic time series with different Lyapunov exponents. It is demonstrated that the estimated Lyapunov exponents fail to distinguish different noisy chaotic time series when relatively small scales are used. The distinction can be reestablished by using larger scales. Using larger scales, however, the estimated Lyapunov exponent is determined by macroscopic statistical properties of the series and provides the same information as the autocorrelation function and/or coarse-grained mutual information.