A Stable Measure of Chaos in Dynamical Systems using Persistent Homology

TL;DR

This paper introduces a new, stable measure of chaos in dynamical systems called the 0-persistence exponent, based on persistent homology, which is less sensitive to noise than traditional Lyapunov exponents.

Contribution

The authors propose and theoretically validate a novel chaos measure using persistent homology, demonstrating its stability and correlation with Lyapunov exponents on various systems.

Findings

The 0-persistence exponent is non-negative for chaotic systems.

It exhibits greater stability than Lyapunov exponents on noisy data.

Experimental results show strong correlation with Lyapunov exponents on multiple systems.

Abstract

Many real-world dynamics exhibit chaos, a phenomenon in which neighboring trajectories in the state space of a dynamical system diverge exponentially over time. A common measure used for quantifying the degree of this divergence is the maximal Lyapunov exponent, which relies on pairwise Euclidean distances between the trajectories at each time. The main limitation of the maximal Lyapunov exponent in practice is its sensitivity to small perturbations in system trajectories. Persistent homology, the study of holes that appear in different dimensions as the points of a data set are thickened over time, has guaranteed theoretical stability under such added noise. As such, we propose a novel, 0-dimensional persistent homology based measure of chaos termed the 0-persistence exponent and prove its theoretical stability. We show that if a system is chaotic, then the 0-persistence exponent is non-negative by proving that positive Lyapunov measures imply non-negative 0-persistence measures, and further discuss when strict positivity of 0-persistence measures occur. Additionally, we prove the existence of an upper bound on our measure, and show its greater experimental stability on the Lorenz and Rossler systems describing fluid convection and taffy pulling. We present an algorithm for computing the 0-persistence exponent given a single univariate time series with N points from a dynamical system that runs in O(N^2 log N) time. We finally show the greater experimental stability of the 0-persistence exponent on time series data depicting a Belousov-Zhabotinsky chemical reaction, which transitions from periodicity to chaos and back as the system evolves in time. We present experimental results which verify that positive Lyapunov exponents imply positive 0-persistence exponents under sufficient conditions through high correlation between both measures on the Lorenz and Rossler systems.