Stable periodic solutions for delay equations with positive feedback

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October 2006, 14(4): 721-736. doi: 10.3934/dcds.2006.14.721

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

A. Aschwanden ^1, , A. Schulze-Halberg ^1, and D. Stoffer ^1,

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

Received October 2004 Revised July 2005 Published January 2006

Abstract
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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.

Keywords: Stable periodic solution, computer-assisted proof., interval arithmetic.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721

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