Computation and verification of contraction metrics for periodic orbits

Abstract

Exponentially stable periodic orbits of ordinary differential equations and their basins' of attraction are characterized by contraction metrics. The advantages of a contraction metric over a Lyapunov function include its insensitivity to small perturbations of the dynamics and the exact location of the periodic orbit. We present a novel algorithm to rigorously compute contraction metrics, that combines the numerical solving of a first order partial differential equation with rigorous verification of the conditions for a contraction metric. Further, we prove that our algorithm is able to compute a contraction metric for any ordinary differential equation possessing an exponentially stable periodic orbit. We demonstrate the applicability of our approach by computing contraction metrics for three systems from the literature.