Reversible relative periodic orbits

AbstractWe study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along group orbits. Our results extend those recently obtained in the equivariant context by B. Sandstede et al. (1999, J. Nonlinear Sci.9, 439–478) and C. Wulff et al. (2001, Ergodic Theory Dynam. Systems21, 605–635)

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

AbstractWe study the dynamics near a symmetric Hopf-zero (also known as saddle-node Hopf or fold-Hopf) bifurcation in a reversible vector field in R3, 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 saddle-foci. 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

On Symmetric Attractors in Reversible Dynamical Systems

Let \Gamma ae O(n) be a finite group acting orthogonally on IR n . We say that \Gamma is a reversing symmetry group of a homeomorphism, diffeomorphism or flow f t : IR n 7! IR n (t 2 Z Z or t 2 IR) if \Gamma has an index two subgroup ~ \Gamma whose elements commute with f t and for all elements ae 2 \Gamma \Gamma ~ \Gamma and all t , f t ffi ae(x) = ae ffi f \Gammat (x). We give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors (Liapunov stable !-limit sets). These conditions arise due to topological obstructions. In dimensions 1 and 2 we present a complete description of possible symmetry groups of asymptotically stable attractors for homeomorphisms and diffeomorphisms (these attractors cannot possess reversing symmetries). We also have a fairly complete description in the context of subgroups which contain reversing symmetries. For all dimensions n we present complete re..

Reversible Relative Periodic Orbits

We study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along group orbits. Our results extend those recently obtained in the equivariant context by Sandstede et al. [24] and Wulff et al. [29]. Contents 1 Introduction 2 2 Setting of the problem 3 3 Reversible relative equilibria 6 4 Reversible relative periodic orbits 9 4.1 Some general properties of reversible RPOs : : : : : : : : : : : : : : : : : : : : : : : 10 4.2 Drift of reversible RPOs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 4.3 Bundle structure near reversible RPOs : : : : : : : : : : : : : : : : : : : : : : : : : : 14 4.4 Differential equations on the bundle : : : ..

Pinning and Locking of Discrete Waves

We discuss the behavior of rotating and traveling waves -- such as traveling fronts and pulses -- in partial differential equations with continuous symmetry groups under perturbations that break the continuous symmetries but preserve some discrete (lattice) symmetries. Our theory provides a rigorous and universal framework for understanding pinning, depinning and locking of discrete rotating and traveling waves