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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 23, Issue 3, 2004, pp. 547–576**

**DOI: 10.4171/ZAA/1210**

Published online: 2004-09-30

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

Peter Giesl^{[1]}(1) TU München, Germany

We consider a general system of ordinary differential equations \begin{eqnarray*}\dot{x}=f(t,x)\mbox{,}\end{eqnarray*} 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.

*Keywords: *Dynamical system, periodic differential equation, periodic orbit, asymptotic stability, basin of attraction

Giesl Peter: On the Basin of Attraction of Limit Cycles in Periodic Differential Equations. *Z. Anal. Anwend.* 23 (2004), 547-576. doi: 10.4171/ZAA/1210