A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

TL;DR

This paper introduces a novel Lyapunov function approach using Sum-of-Squares programming to find the minimal attractor set of nonlinear systems, providing tight outer bounds and improved approximations.

Contribution

It proposes a new Lyapunov characterization for attractor sets and a determinant maximization SOS programming method for minimal attractor set approximation.

Findings

The method accurately approximates the Lorenz attractor.

It provides minimal volume outer bounds for attractor sets.

Numerical examples demonstrate effectiveness on classical nonlinear systems.

Abstract

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose 1-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.