Global bifurcations close to symmetry

TL;DR

This paper investigates the complex dynamics near heteroclinic cycles in symmetric differential equations on the 3-sphere, revealing how symmetry breaking leads to intricate bifurcations, tangencies, and the emergence of attracting periodic solutions.

Contribution

It provides a detailed analysis of heteroclinic bifurcations near symmetry, including the coexistence of tangencies, hyperbolic sets, and the destruction of horseshoes as symmetry is broken.

Findings

Existence of n-pulse heteroclinic tangencies.

Coexistence of tangencies with hyperbolic dynamics.

Destruction of horseshoes leading to attracting periodic solutions.

Abstract

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies - trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions.