A connection between Lyapunov exponents and sensitive dependence on parameters of chaotic systems

TL;DR

This paper establishes a direct link between Lyapunov exponents and the sensitive dependence on parameters in chaotic systems, showing that variations in exponents lead to exponential divergence in orbits.

Contribution

It demonstrates that changes in Lyapunov exponents precisely characterize the sensitive dependence on parameters in chaotic dynamical systems.

Findings

Lyapunov exponents vary with system parameters

Exponential divergence correlates with Lyapunov exponent changes

Characterization of sensitivity depends on Lyapunov exponent computation

Abstract

The sensitive dependence of chaos on parameters is a topic of great interest in the study of integrability and stability of dynamical systems. Previous work has proposed ways to identify the sensitive dependence on parameters by topological criteria or large numerical simulations. In this paper, we show that when the Lyapunov exponents of the system vary with a change in the parameters, the system diverges exponentially in the orbits associated with the considered parameters. We use this result to explore the sensitive dependence on parameters in an uncertainty interval and conclude that the characterization of this phenomenon is directly related to our ability to determine the Lyapunov exponents of the system for different parameters.