On the Basin of Attraction of Limit Cycles in Periodic Differential Equations

We consider a general system of ordinary dierential equations ˙ x = f(t,x), where x 2 R n , and f(t + T,x) = f(t,x) for all (t,x) 2 R ◊ R n is a periodic function. We give a sucient and necessary condition for the existence and uniqueness of an exponentially asymptotically stable periodic orbit. Moreover, this condition is su- cient and necessary to prove that a subset belongs to the basin of attraction of the periodic orbit. The condition uses a Riemannian metric, and we present methods to construct such a metric explicitly.