Complicated histories close to a homoclinic loop generated by variable delay

Abstract

There exist a delay functional $d$ on an open set $U \subset C([-2,0],$ $\mathbb R)$ and a parameter $\alpha\gt 0$ so that the equation $$ x'(t)=-\alpha\,x(t-d(x_t)) $$ has a solution which is homoclinic to zero, the zero equilibrium is hyperbolic with 2-dimensional unstable manifold, and the stable manifold and unstable manifold intersect transversely along the homoclinic flowline. We prove that close to the homoclinic loop there exists a kind of complicated motion, which involves partially unstable behavior. This does not require any relation between the growth rate in the unstable space and the decay rate in the leading stable plane of the linearized semiflow.