Quantifying Chaos: A tale of two maps

TL;DR

This paper emphasizes the importance of convergence evidence in Lyapunov exponent estimates for chaos detection, illustrating this with two maps and highlighting cautious interpretation of real data.

Contribution

It introduces two maps to demonstrate the necessity of convergence evidence in Lyapunov exponent estimation for chaos detection.

Findings

Convergence evidence is crucial for reliable Lyapunov exponent estimates.

Illustrates the pitfalls of interpreting Lyapunov exponents without convergence.

Provides two maps as case studies for chaos analysis.

Abstract

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This paper presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.