October  2005, 12(5): 827-852. doi: 10.3934/dcds.2005.12.827

Topological horseshoes and delay differential equations

1. 

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

Received  January 2004 Revised  October 2004 Published  February 2005

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 $x'(t) \in f(x(t)) + \bar B(0,\delta)$.
Citation: Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827
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