Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R3

Abstract

We study the dynamics near a symmetric Hopf-zero (also known as saddle-node Hopf or fold-Hopf) bifurcation in a reversible vector field in R 3 , with involutory an reversing symmetry whose fixed point subspace is one-dimensional. We focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddlefoci. We study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets (horseshoes), independent of the eigenvalues of the saddle-foci, as well as cascades of bifurcations of periodic, heteroclinic and homoclinic orbits.Finally, we discuss the application of our results to the Michelson system, describing stationary states and travelling waves of the Kuramoto-Sivashinsky PDE.

Asymptotic analysis of the Michelson system

“…At c = 0, F + = F − = (0, 0, 0), and the eigenvalues of the linearization of the origin are {0, ±i}. Thus, at c = 0, there is a 'Hopf-zero' bifurcation [1,12]. This is a generic codimension-1 bifurcation in the space of reversible volume-preserving systems; it is the bifurcation that we analyse in this paper.…”

Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits

“…In the literature, homoclinic and heteroclinic orbits in reversible systems have most frequently been studied in the context of ODEs, where the connecting orbits arise between equilibria (see for instance [2,21,23]). Our study relates to the study of dynamics near heteroclinic orbits in reversible systems between a pair of asymmetric periodic solutions of saddle type.…”

“…Our study relates to the study of dynamics near heteroclinic orbits in reversible systems between a pair of asymmetric periodic solutions of saddle type. In fact, an important class of reversible ODEs are those derived from a Z 2 -symmetric PDE [2,21,23], by restriction to travelling and/or standing waves. In this context, our results can be interpreted to shed light on the complicated structure of the solution set of the PDE in the neighbourhood of heteroclinic cycles between periodic waves.…”

“…Two-dimensional reversible diffeomorphisms appear naturally as Poincaré maps for symmetric periodic orbits in three-dimensional reversible vector fields. Such a situation arises, for instance, in the travelling/standing wave reduction of the Kuramoto-Shivashinsky PDE [24,23]. Alternatively, such reversible maps can appear naturally when periodically forced reversible flows are considered, where the forcing has a special form (e.g.…”

Degenerate with respect to parameters fold-Hopf bifurcations

“…An algorithm for computing the coefficients of a center manifold for the fold-Hopf singularity in a class of autonomous differential equations is presented in [15], while in [13] the normal forms theory is applied to analyze the stability conditions of a fold-Hopf bifurcation in an electromechanical model. In [9] a study of the dynamics near a symmetric fold-Hopf bifurcation in a reversible vector field in R 3 is presented. Insights on the case when the normal form of this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle-foci are obtained.…”