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Many dynamical systems described by nonlinear ODEs are unstable. Theirassociated solutions do not converge towards an equilibrium point, but ratherconverge towards some invariant subset of the state space called an attractorset. For a given ODE, in general, the existence, shape and structure of theattractor sets of the ODE are unknown. Fortunately, the sublevel sets ofLyapunov functions can provide bounds on the attractor sets of ODEs. In thispaper we propose a new Lyapunov characterization of attractor sets that is wellsuited to the problem of finding the minimal attractor set. We show ourLyapunov characterization is non-conservative even when restricted toSum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOSprogramming problem based on determinant maximization that yields an SOSLyapunov function whose 1-sublevel set has minimal volume, is an attractor setitself, and provides an optimal outer approximation of the minimal attractorset of the ODE. Several numerical examples are presented including the Lorenzattractor and Van-der-Pol oscillator.

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