Deciphering Complexity: Machine Learning Insights into Chaotic Dynamical Systems

TL;DR

This paper presents novel machine learning methods for analyzing chaotic dynamical systems, including a simple Lyapunov exponent calculation, phase transition graph analysis, and identification of integrable regions, demonstrated on two example systems.

Contribution

It introduces new machine learning techniques for analyzing chaos, notably a simple Lyapunov exponent method and phase transition analysis, advancing understanding of complex dynamical behaviors.

Findings

Effective Lyapunov exponent calculation from minimal data

Identification of phase transitions from regular to chaotic dynamics

Detection of integrable regions within chaotic trajectories

Abstract

We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory data points unlike traditional methods that require an averaging procedure, the exploration of phase transition graphs from regular periodic to chaotic dynamics to identify "almost integrable" trajectories where conserved quantities deviate from whole numbers, and the identification of "integrable regions" within chaotic trajectories. These methods are applied and tested on two dynamical systems: "Two objects moving on a rod" and the "Henon-Heiles" systems.