Information Flow in Chaotic Symbolic Dynamics for Finite and Infinitesimal Resolution

Abstract

We introduce an information-theory-based concept for the characterization of the information flow in chaotic systems in the framework of symbolic dynamics for finite and infinitesimal measurement resolutions. The information flow characterizes the loss of information about the initial conditions, i.e. the decay of statistical correlations (i.e. nonlinear and non-Gaussian) between the entire past and a point p steps into the future as a function of p. In the case where the partition generating the symbolic dynamics is finite, the information loss is measured by the mutual information that measures the statistical correlations between the entire past and a point p steps into the future. When the partition used is a generator and only one step ahead is observed (p = 1), our definition includes the Kolmogorov–Sinai entropy concept. The profiles in p of the mutual information describe the short- and long-range forecasting possibilities for the given partition resolution. For chaos it is more relevant to study the information loss for the case of infinitesimal partitions which characterizes the intrinsic behavior of the dynamics on an extremely fine scale. Due to the divergence of the mutual information for infinitesimal partitions, the "intrinsic" information flow is characterized by the conditional entropy which generalizes the Kolmogorov–Sinai entropy for the case of observing the uncertainty more than one step into the future. The intrinsic information flow offers an instrument for characterizing deterministic chaos by the transmission of information from the past to the future.