Strange attractors and non wandering domains near a homoclinic cycle to a bifocus

TL;DR

This paper investigates the complex dynamics near a homoclinic cycle to a bifocus in a three-dimensional system, revealing the emergence of strange attractors, non-trivial wandering domains, and bifurcations through heteroclinic tangencies.

Contribution

It extends previous work by analyzing heteroclinic tangencies and bifurcations near a homoclinic cycle to a bifocus, leading to new insights into chaotic dynamics and strange attractors.

Findings

Existence of hyperbolic horseshoes near the cycle.

Approximation of the first return map by maps with heteroclinic tangencies.

Generation of strange attractors and wandering domains through bifurcations.

Abstract

In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits. When the cycle is broken, under an additional hypothesis about the coexistence of two heteroclinically related periodic points (one without dominated splitting into one-dimensional sub-bundles), the heteroclinic tangencies can be slightly modified in order to satisfy Tatjer's conditions for a generalized tangency of codimension two. This configuration may be seen as the organizing center, by which one can obtain Bogdanov-Takens bifurcations and therefore, strange attractors, infinitely many sinks and non-trivial contracting wandering domains.