Convex computation of maximal Lyapunov exponents

Abstract

We describe an approach for finding upper bounds on an ODE dynamical system’s maximal Lyapunov exponent (LE) among all trajectories in a specified set. A minimisation problem is formulated whose infimum is equal to the maximal LE, provided that trajectories of interest remain in a compact set. The minimisation 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 minimisation can be relaxed into a computationally tractable polynomial optimisation problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimisation problems of increasing computational cost whose infima converge from above to the maximal LE, at least when the set of interest is compact. For illustration, we carry out such polynomial optimisation computations for two chaotic examples: the Lorenz system and the Hénon–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.