JournalszaaVol. 23 , No. 3DOI 10.4171/zaa/1210

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

  • Peter Giesl

    TU München, Germany
On the Basin of Attraction of Limit Cycles in Periodic Differential Equations cover

Abstract

We consider a general system of ordinary differential equations \begin{eqnarray*}\dot{x}=f(t,x)\mbox{,}\end{eqnarray*} where xRnx\in\mathbb R^n, and f(t+T,x)=f(t,x)f(t+T,x)=f(t,x) for all (t,x)R×Rn(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.