  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.
