Converse theorem on a contraction metric for a periodic orbit

TL;DR

This paper establishes that for any exponentially stable periodic orbit, a contraction metric exists within its basin of attraction, enabling the quantification of exponential convergence rates.

Contribution

It proves the converse theorem, showing the existence of a contraction metric for stable periodic orbits and recovering the exponential attraction rate.

Findings

Existence of contraction metric for stable periodic orbits

Bound on exponential attraction rate obtained

Contraction analysis extended to converse scenario

Abstract

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. A contraction metric is 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 and we obtain a bound on the rate of exponential attraction. In this paper we study the converse question and show that, given an exponentially stable periodic orbit, a contraction metric exists on its basin of attraction and we can recover the bound on the rate of exponential attraction.