Lyapunov exponents in constrained and unconstrained ordinary differential equations

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.