"Large" strange attractors in the unfolding of a heteroclinic attractor

TL;DR

This paper investigates how strange attractors emerge in a 3D dynamical system near a heteroclinic network, showing that small parameter changes can lead to complex chaotic behavior with SRB measures.

Contribution

It introduces a mechanism for the formation of strange attractors in a 3D sphere system near a Bykov heteroclinic network, including the transition from stable sinks to chaos.

Findings

Existence of strange attractors with SRB measures for positive measure parameter sets

Identification of parameter values with superstable sinks

Characterization of the transition from heteroclinic network to chaos

Abstract

In this paper we present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations acting on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between hyperbolic saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around an annulus in the phase space, supporting Sinai-Ruelle-Bowen (SRB) measures. We prove the existence of a sequence of parameter values for which the family exhibits a superstable sink. We also characterise the transition from a Bykov network to a strange attractor.