Santiago Ibáñez

Cocoon bifurcation in three-dimensional reversible vector fields

The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed by Lau (1992 Int. J. Bifurc. Chaos 2 543–58) in the Michelson system, a three-dimensional ODE system describing travelling waves of the Kuramoto–Sivashinsky equation. In this paper, we present an organizing centre of the principal part of the cocoon bifurcation in more general terms in the setting of reversible vector fields on . We prove that in a generic unfolding of an organizing centre called the cusp-transverse heteroclinic chain, there is a cascade of heteroclinic bifurcations with an increasing length close to the organizing centre, which resembles the principal part of the cocoon bifurcation.We also study a heteroclinic cycle called the reversible Bykov cycle. Such a cycle is believed to occur in the Michelson system, as well as in a model equation of a Josephson Junction (van den Berg et al 2003 Nonlinearity 16 707–17). We conjecture that a reversible Bykov cycle is, in its unfolding, an accumulation point of a sequence of cusp-transverse heteroclinic chains. As a first result in this direction, we show that a reversible Bykov cycle is an accumulation point of reversible generic saddle-node bifurcations of periodic orbits, the main ingredient of the cusp-transverse heteroclinic chain.

On the Dynamics Near a Homoclinic Network to a Bifocus: Switching and Horseshoes

We study a homoclinic network associated to a nonresonant hyperbolic bifocus. It is proved that on combining rotation with a nondegeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, switching behavior is created: close to the network, there are trajectories that visit the neighborhood of the bifocus following connections in any prescribed order. We discuss the existence of suspended horseshoes which accumulate on the network and the relation between these horseshoes and the switching behavior.

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in

Complexity and Dynamical Uncertainty

{\mathbb{R}^4}

Shil'nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on R3☆

unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in

Nilpotent Singularities in Generic 4-Parameter Families of 3-Dimensional Vector Fields

{\mathbb{R}^3}