A Stable Measure of Chaos in Dynamical Systems Using Persistent Homology
Bala Krishnamoorthy, Elizabeth Thompson
TL;DR
This work addresses the fragility of the Lyapunov exponent in noisy real-world data by introducing the persistence exponent, a chaos measure grounded in 0-dimensional persistent homology. The method analyzes the union of neighboring trajectories through Vietoris-Rips filtrations, tracks Betti-curve evolution via interleaving distance, and defines $\beta_{\text{exp}}(X) = \lim_{r \to \infty} -\log(\beta_0^{\epsilon_r}(Z)/\beta_0^{\epsilon_1}(Z))$, with stability guaranteed by persistent-homology theory. An $O(N^2 \log N)$ algorithm computes the exponent from a single time series embedding, and the authors establish upper bounds and practical estimates. Empirical results on the Lorenz and Rossler systems, as well as real-time Belousov–Zhabotinsky data, show the persistence exponent outperforms the Lyapunov exponent in robustness to noise and its ability to detect periodicity, highlighting its potential as a reliable chaos measure in noisy dynamical systems. The work also discusses extensions to higher-dimensional and multiparameter persistent homology to capture richer dynamical structures.
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 measuring the degree of this divergence is the Lyapunov exponent, which relies on pairwise Euclidean distances between the trajectories at each time. The main limitation of the Lyapunov exponent is its sensitivity to small perturbations in input data. Since many real-world dynamics are likely to contain some degree of inherent noise, we are motivated to construct a chaos measure that is robust to small trajectory perturbations. Persistent homology has been used for proving theoretical stability guarantees under such added noise in the data. As such, we introduce a novel, persistent homology based measure of chaos termed the persistence exponent and prove its theoretical stability. We also 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 persistence exponent given a single 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 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.