On the basin of attraction of limit cycles in periodic differential equations

We consider a general system of ordinary differential equations (x) over dot = f (t, x), where x is an element of R-n, and f (t + T, x) = f (t, x) for all (t, x) is an element of R x R-n is a periodic function. We give a sufficient and necessary condition for the existence and uniqueness of an exponentially asymptotically stable periodic orbit. Moreover, this condition is sufficient 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.