Dynamics of conservative Bykov cycles: Tangencies, generalized Cocoon bifurcations and elliptic solutions

•Heteroclinic tangencies densely occur near a conservative Bykov cycle.•Elliptic solutions and invariant tori arise near heteroclinic tangencies.•Hyperbolic and non-hyperbolic sets coexist in a degenerate class of vector fields.•Chirality is an ingredient for the non-hyperbolic dynamics near the cycle.•Generalization of the Cocoon Bifurcations is achieved.

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 non-uniformly hyperbolic suspended horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.