  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.
