Computation of a contraction metric for a periodic orbit using meshfree collocation

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. We consider a contraction metric, i.e. a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable. In this paper we propose a construction method using meshfree collocation to approximately solve a matrix-valued PDE problem. We derive error estimates and show that the approximation is itself a matrix-valued PDE problem. We derive error estimates and show that the approximation is itself a contraction metric if the collocation points are sufficiently dense. We apply the method to several examples.