Lyapunov exponents in constrained and unconstrained ordinary differential equations

TL;DR

This paper reviews numerical methods for calculating Lyapunov exponents in both constrained and unconstrained ODE systems, emphasizing techniques to handle constraints and evaluating their accuracy and reliability.

Contribution

It introduces and compares deviation vector and Jacobian methods specifically adapted for constrained systems, addressing unique challenges and potential errors.

Findings

Deviation vector and Jacobian methods are effective for Lyapunov exponent calculation.

Constraints introduce specific complications that require tailored numerical techniques.

Assessment of method accuracy highlights best practices for reliable chaos quantification.

Abstract

We discuss several numerical methods for calculating Lyapunov exponents (a quantitative measure of chaos) in systems of ordinary differential equations. We pay particular attention to constrained systems, and we introduce a variety of techniques to address the complications introduced by constraints. For all cases considered, we develop both deviation vector methods, which follow the time-evolution of the difference between two nearby trajectories, and Jacobian methods, which use the Jacobian matrix to determine the true local behavior of the system. We also assess the merits of the various methods, and discuss assorted subtleties and potential sources of error.