Entropies in case of continuous time

TL;DR

This paper extends information theory to continuous-time systems, enabling the distinction of chaos and noise through new entropy measures derived from dynamical models like Lorenz and Ornstein-Uhlenbeck.

Contribution

It introduces a novel framework for continuous-time entropies, connecting entropy, chaos, and noise analysis in a unified manner using a three-dimensional representation.

Findings

Continuous-time entropies can distinguish chaos from noise.

A threshold for joint uncertainty in deterministic systems is identified.

Entropy rate is obtained as a derivative of entropy in the limit of high sampling.

Abstract

Information theory on a time-discrete setting in the framework of time series analysis is generalized to the time-continuous case. Considerations of the Roessler and Lorenz dynamics as well as the Ornstein-Uhlenbeck process yield for time-continuous entropies a new possibility for the distinction of chaos and noise. In the deterministic case an upper threshold of the joint uncertainty in the limit of infinitely high sampling rate can be found and the entropy rate can be calculated as a usual time derivative of the entropy. In a three-dimensional representation the dependence of the joint entropy on space resolution, discretization time step length and uncertainty-assessed time is shown in a unified manner. Hence the dimension and the Kolmogorov-Sinai entropy rate of any dynamics can be read out as limit cases from one single graph.