abstr18

The first part of this paper is a general approach towards chaotic dynamics for a continuous
map $f:X \supset M\to X$ which employs the fixed point index and continuation. The second part deals with the differential equation
$$ x'(t)=-\alpha\,x(t-d_{\Delta}(x_t)). $$
with state-dependent delay. For a suitable parameter $\alpha$ close to $5\pi/2$ we construct a delay functional $d_{\Delta}$,
constant near the origin, so that the previous equation has a homoclinic solution, $h(t)\to0$ as $t\to\pm\infty$,
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.

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