Defining Chaos

TL;DR

This paper introduces a general, computationally feasible definition of chaos using an entropy-like measure called expansion entropy, applicable to various dynamical systems, and compares it to topological entropy.

Contribution

It proposes a new entropy-based chaos definition, relates it to topological entropy, and discusses computational methods and issues in defining chaos.

Findings

Expansion entropy is positive in chaotic systems.

Expansion entropy is equivalent to topological entropy under certain conditions.

The proposed definition is broadly applicable and computationally feasible.

Abstract

In this paper we propose, discuss and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered, including attractors, repellers and non-periodically forced systems. This definition is based on an entropy-like quantity, which we call "expansion entropy", and we define chaos as occurring when this quantity is positive. We relate and compare expansion entropy to the well-known concept of topological entropy, to which it is equivalent under appropriate conditions. We also present example illustrations, discuss computational implementations, and point out issues arising from attempts at giving definitions of chaos that are not entropy-based.