Partial symmetry breaking and heteroclinic tangencies

TL;DR

This paper investigates how breaking symmetry in volume-contracting vector fields on the 3-sphere leads to complex dynamics like Bykov cycles, heteroclinic tangencies, and persistent horseshoes.

Contribution

It reveals the mechanisms behind heteroclinic tangencies and complex dynamics emerging from symmetry breaking in equivariant vector fields.

Findings

Existence of Bykov cycles in symmetry-broken systems

Emergence of heteroclinic tangencies near symmetric configurations

Persistence of horseshoes and attracting periodic orbits

Abstract

We study some global aspects of the bifurcation of an equivariant family of volume-contracting vector fields on the three-dimensional sphere. When part of the symmetry is broken, the vector fields exhibit Bykov cycles. Close to the symmetry, we investigate the mechanism of the emergence of heteroclinic tangencies coexisting with transverse connections. We find persistent suspended horseshoes accompanied by attracting periodic trajectories with long periods.