Describing chaotic systems

TL;DR

This paper explores the Lyapunov exponent as a measure of chaos, providing methods for its calculation in one- and multi-dimensional systems, and applies these to analyze the logistic map and Hénon map.

Contribution

It introduces practical techniques for calculating Lyapunov exponents and spectrum in complex systems, extending existing methods to higher dimensions.

Findings

Calculated Lyapunov exponents for logistic map and Hénon map.

Compared Lyapunov exponent variations with bifurcation diagrams.

Outlined numerical methods for Lyapunov spectrum computation.

Abstract

In this paper, we discuss the Lyapunov exponent definition of chaos and how it can be used to quantify the chaotic behavior of a system. We derive a way to practically calculate the Lyapunov exponent of a one-dimensional system and use it to analyze chaotic behavior of the logistic map, comparing the $r$-varying Lyapunov exponent to the map's bifurcation diagram. Then, we generalize the idea of the Lyapunov exponent to an $n$-dimensional system and explore the mathematical background behind the analytic calculation of the Lyapunov spectrum. We also outline a method to numerically calculate the maximal Lyapunov exponent using the periodic renormalization of a perturbation vector and a method to numerically calculate the entire Lyapunov spectrum using QR factorization. Finally, we apply both these methods to calculate the Lyapunov exponents of the H\'enon map, a multi-dimensional chaotic system.