Dynamics of conservative Bykov cycles: tangencies, generalized cocoon bifurcations and elliptic solutions

TL;DR

This paper investigates the complex dynamics near conservative Bykov cycles, revealing dense manifold tangencies, coexistence of elliptic and hyperbolic behaviors, and introducing a generalized Cocoon bifurcation in divergence-free vector fields.

Contribution

It introduces a new mechanism for coexistence of hyperbolic and non-hyperbolic dynamics in conservative systems, including generalized Cocoon bifurcations.

Findings

Dense tangencies of invariant manifolds occur in the class of divergence-free vector fields.

Existence of infinitely many elliptic points alongside hyperbolic horseshoes.

Persistent heteroclinic tangencies dominate the global dynamics.

Abstract

This paper presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov cycle where trajectories turn in opposite directions near the two saddle-foci. We show that {within the class of divergence-free vector fields that preserve the cycle,} tangencies of the invariant manifolds of two hyperbolic saddle-foci densely occur. The global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with suspended hyperbolic horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.