Topological horseshoes and delay differential equations

Klaudiusz Wójcik; Piotr Zgliczyński

doi:10.3934/dcds.2005.12.827

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2005, Volume 12, Issue 5: 827-852. Doi: 10.3934/dcds.2005.12.827

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Topological horseshoes and delay differential equations

Klaudiusz Wójcik^1, and
Piotr Zgliczyński^1,

1.
Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow

Received: January 2004

Revised: October 2004

Published: October 2005

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Abstract

We show that if an ordinary differential equation $x'=f(x)$, where $x\in \mathbb R^n$ and $f \in \mathcal C^1$, has a topological horseshoe, then the corresponding delay equation $x'(t)=f(x(t-h))$ for small $h >0$ also has a topological horseshoe, i.e. symbolic dynamics and an infinite number of periodic orbits. A method of computation of $h$ is given in terms of topological properties of solutions of differential inclusion $x'(t) \in f(x(t)) + \bar B(0,\delta)$.

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Mathematics Subject Classification: 34K23, 34K13.

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