A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

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