Stability in Chaos

TL;DR

This paper reveals that chaotic systems can exhibit unexpectedly long-term stability in trajectories, analyzed through a novel large-deviation approach linking Lyapunov exponents to Schrödinger equations.

Contribution

It introduces a quantitative method to describe long-term stability in chaotic systems using large-deviation theory and semi-classical analysis of Lyapunov exponents.

Findings

Chaotic trajectories can be stable over long timescales.

The distribution of finite-time Lyapunov exponents is characterized by an entropy function.

A Schrödinger equation approach is used to analyze stability properties.

Abstract

Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed quantitative description of this effect for a one-dimensional model of inertial particles in a turbulent flow using large-deviation theory. Specifically, the determination of the entropy function for the distribution of finite-time Lyapunov exponents reduces to the analysis of a Schr\"odinger equation, which is tackled by semi-classical methods.