Computation of Lyapunov functions for systems with multiple local attractors

Abstract

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graph-theoretic approach.Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicialcomplex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov functiongives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors.We develop the theory in detail and present numerical examples demonstrating the applicability of our method.