Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof

A. Aschwanden; A. Schulze-Halberg; D. Stoffer

doi:10.3934/dcds.2006.14.721

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2006, Volume 14, Issue 4: 721-736. Doi: 10.3934/dcds.2006.14.721

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Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof

1.
Department of Mathematics, ETH Zürich, CH-8092 Zürich

Received: September 30, 2004

Revised: June 30, 2005

Published: September 2006

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Abstract

We study the delay equation $\dot{x}(t)=-\mu x(t)+f(x(t-1))$ with $\mu>0$ and a nonmonotone $C^1$-function $f$ obeying $x f(x)>0$ (positive feedback) outside a small neighbourhood of zero. By means of a computer-assisted method we prove the existence of asymptotically orbitally stable periodic solutions. The main idea behind our proof is the reduction of the infinite-dimensional dynamics to a finite-dimensional map. In particular, for two classes of nonlinearities $f$ we construct two types of solutions, the dynamics of which is reduced to a one- and a two-dimensional map, respectively.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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