Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems

Peter Giesl

doi:10.3934/dcds.2007.18.355

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2007, Volume 18, Issue 2&3: 355-373. Doi: 10.3934/dcds.2007.18.355

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Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems

Peter Giesl^1,

1.
Zentrum Mathematik, TU München, Boltzmannstr. 3, D-85747 Garching bei München

Received: February 28, 2006

Revised: June 30, 2006

Published: May 2007

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Abstract

We study a time-periodic non-smooth differential equation $\dot{x}=f(t,x)$, $x\in \mathbb R$. In [4] we have presented a sufficient condition for existence, uniqueness, stability and the basin of attraction of a periodic orbit in such a system, which is a generalized Borg's condition. In this paper we prove that this condition is necessary. The proof involves a generalization of Floquet exponents for periodic orbits of non-smooth differential equations.

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Mathematics Subject Classification: 34C25, 34A36.

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Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems

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