A unified theory of chaos linking nonlinear dynamics and statistical physics

TL;DR

This paper challenges traditional criteria for identifying chaos in real-world data, proposing a new approach that accounts for noise and external influences to better distinguish chaotic from stochastic signals.

Contribution

It introduces a noise titration assay to assess nonautonomous chaos, expanding the tools for analyzing complex systems beyond Lyapunov exponents.

Findings

Positive Lyapunov exponent is not definitive proof of chaos.

Multiple forms of nonautonomous chaos can be identified.

New methodology improves analysis of complex physical, biological, and socioeconomic data.

Abstract

A fundamental issue in nonlinear dynamics and statistical physics is how to distinguish chaotic from stochastic fluctuations in short experimental recordings. This dilemma underlies many complex systems models from stochastic gene expression or stock exchange to quantum chaos. Traditionally, deterministic chaos is characterized by "sensitive dependence on initial conditions" as indicated by a positive Lyapunov exponent. However, ambiguity arises when applying this criterion to real-world data that are corrupted by measurement noise or perturbed nonautonomously by exogenous deterministic or stochastic inputs. Here, we show that a positive Lyapunov exponent is surprisingly neither necessary nor sufficient proof of deterministic chaos, and that a nonlinear dynamical system under deterministic or stochastic forcing may exhibit multiple forms of nonautonomous chaos assessable by a noise titration assay. These findings lay the foundation for reliable analysis of low-dimensional chaos for complex systems modeling and prediction of a wide variety of physical, biological, and socioeconomic data.