Computation of attractor dimension and maximal sums of Lyapunov exponents using polynomial optimization

Abstract

Two approaches are presented for computing upper bounds on Lyapunov exponents and their sums, and on the Lyapunov dimension, among all trajectories of a dynamical system governed by ordinary differential equations. The first approach expresses a sum of Lyapunov exponents as a time average in an augmented dynamical system and then applies methods for bounding time averages. This generalizes the method of Oeri \& Goluskin (Nonlinearity 36:5378--5400, 2023) for bounding the single largest Lyapunov exponent. The second approach considers a different augmented dynamical system, where bounds on sums of Lyapunov exponents are implied by stability of certain sets, and such stability is verified using Lyapunov function methods. Both of our approaches also can be adapted to directly compute bounds on Lyapunov dimension, which in turn imply bounds on the fractal dimension of a global attractor. For systems of ordinary differential equations with polynomial right-hand sides, all of our bounding formulations lead to polynomial optimization problems with sum-of-squares constraints. These sum-of-squares problems can be solved computationally for a chosen system to yield numerical bounds, provided the number of variables and degree of polynomials are not prohibitive. Most of our bounding formulations are proved to be sharp under mild assumptions. In the case of the polynomial optimization problems, sharpness means that upper bounds converge to the quantities being bounded as polynomial degrees are raised. Computational examples demonstrate upper bounds that are sharp to several digits, including for a six-dimensional dynamical system where sums of Lyapunov exponents are maximized on periodic orbits.