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

摘要

The first part of this paper is a general approach towards chaotic dynamics for a continuous map which employs the fixed point index and continuation. The second part deals with the differential equation >with state-dependent delay. For a suitable parameter close to we construct a delay functional , constant near the origin, so that the previous equation has a homoclinic solution, as , with certain regularity properties of the linearization of the semiflow along the flowline . 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.