Convex computation of maximal Lyapunov exponents

TL;DR

This paper introduces a polynomial optimization method to compute tight upper bounds on the maximal Lyapunov exponent of polynomial ODE systems, demonstrated on chaotic Lorenz and Hénon-Heiles systems.

Contribution

It formulates a convex optimization approach using sum-of-squares relaxations to accurately bound Lyapunov exponents for polynomial dynamical systems.

Findings

Upper bounds converge with increasing polynomial degree.

Bounds are sharp to at least five digits.

Method successfully applied to chaotic Lorenz and Hénon-Heiles systems.

Abstract

We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent, provided that trajectories of interest remain in a compact set. The minimization is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case -- i.e., when the ODE's right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials -- the minimization can be relaxed into a computationally tractable polynomial optimization problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimization problems of increasing computational cost whose infima converge from above to the maximal Lyapunov exponent, at least when the set of interest is compact. For illustration, we carry out such polynomial optimization computations for two chaotic examples: the Lorenz system and the H\'enon-Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.