Dense heteroclinic tangencies near a Bykov cycle

This article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes --- we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere

Spiralling dynamics near heteroclinic networks

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a {two parameter family of vector fields} on the three-dimensional sphere \EU^3, whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic trajectories connecting them {transversely} and a non-trivial hyperbolic, invariant and transitive set. The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The vector field is the restriction to \EU^3 of a polynomial vector field in \RR^4. In this article, we also identify global bifurcations that induce chaotic dynamics of different types.Comment: change in one figur

Global bifurcations close to symmetry

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