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

Abstract

We consider a general system of ordinary differential equations

where $x\in {\mathbb{R}}^n$, and $f(t+T,x)=f(t,x)$ for all $(t,x)\in \mathbb{R}\times {\mathbb{R}}^n$ 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.