A Shilnikov Phenomenon Due to State-Dependent Delay, by Means of the Fixed Point Index

Abstract

The first part of this paper is a general approach towards chaotic dynamics for a continuous map $f:X\supset M\rightarrow X$ which employs the fixed point index and continuation. The second part deals with the differential equation

$$\begin{aligned} x'(t)=-\alpha \,x(t-d_{{\varDelta }}(x_t)). \end{aligned}$$

with state-dependent delay. For a suitable parameter $\alpha$ close to $5\pi /2$ we construct a delay functional $d_{{\varDelta }}$, constant near the origin, so that the previous equation has a homoclinic solution, $h(t)\rightarrow 0$ as $t\rightarrow \pm \infty$, with certain regularity properties of the linearization of the semiflow along the flowline $t\mapsto h_t$. The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion.