On the dynamics of certain homoclinic tangles

TL;DR

This paper analyzes the complex dynamics of homoclinic tangles in perturbed dissipative systems, demonstrating the existence of horseshoe maps, attracting periodic solutions, and strange attractors with SRB measures across parameter ranges.

Contribution

It establishes the presence of infinitely many horseshoe maps, attracting periodic solutions, and strange attractors in the dynamics of homoclinic tangles under perturbation.

Findings

Existence of infinitely many disjoint parameter intervals with horseshoe dynamics.

Presence of parameters with attracting periodic solutions.

Parameters where homoclinic tangles admit non-degenerate transversal tangencies.

Abstract

In this paper we study homoclinic tangles formed by transversal intersections of the stable and the unstable manifold of a {\it non-resonant, dissipative} homoclinic saddle point in periodically perturbed second order equations. We prove that the dynamics of these homoclinic tangles are that of {\it infinitely wrapped horseshoe maps}. Using $\mu$ as a parameter representing the magnitude of the perturbations, we prove that (a) there exist infinitely many disjoint open intervals of $\mu$, accumulating at $\mu = 0$, such that the entire homoclinic tangle of the perturbed equation consists of one single horseshoe of infinitely many symbols, (b) there are parameters in between each of these parameter intervals, such that the homoclinic tangle contains attracting periodic solutions, and (c) there are also parameters in between where the homoclinic tangles admit non-degenerate transversal homoclinic tangency of certain dissipative hyperbolic periodic solutions. In particular, (c) implies the existence of strange attractors with SRB measures for a positive measure set of parameters.