Stability and bifurcations of heteroclinic cycles of type Z

Dynamical systems that are invariant under the action of a non-trivial symmetry group can possess structurally stable heteroclinic cycles. In this paper we study stability properties of a class of structurally stable heteroclinic cycles in R^n which we call heteroclinic cycles of type Z. It is well-known that a heteroclinic cycle that is not asymptotically stable can attract nevertheless a positive measure set from its neighbourhood. We say that an invariant set X is fragmentarily asymptotically stable, if for any delta>0 the measure of its local basin of attraction B_delta(X) is positive. A local basin of attraction B_delta(X) is the set of such points that trajectories starting there remain in the delta-neighbourhood of X for all t>0, and are attracted by X as t\to\infty. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearisations near steady states involved in the cycle are negative, then fragmentary asymptotic stability implies asymptotic stability. In the latter case the condition for asymptotic stability is that the transition matrices have an eigenvalue larger than one in absolute value. Finally, we discuss bifurcations occurring when the conditions for asymptotic stability or for fragmentary asymptotic stability are broken.Comment: 38 pp. 26 reference

Simple heteroclinic cycles in R^4

In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of flow-invariant subspaces. For dynamical systems in R^n the minimal dimension for which such robust heteroclinic cycles can exist is n=3. In this case the list of admissible symmetry groups is short and well-known. The situation is different and more interesting when n=4. In this paper we list all finite groups Gamma such that an open set of smooth Gamma-equivariant dynamical systems in R^4 possess a very simple heteroclinic cycle (a structurally stable heteroclinic cycle satisfying certain additional constraints). This work extends the results which were obtained by Sottocornola in the case when all equilibria in the heteroclinic cycle belong to the same Gamma-orbit (in this case one speaks of homoclinic cycles).Comment: 43 pages; submitted to "Nonlinearity

Noise-induced switching near a depth two heteroclinic network and an application to Boussinesq convection

Copyright © 2011 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Chaos Volume 20 (2), article 023133, and may be found at http://dx.doi.org/10.1063/1.3439320We investigate the robust heteroclinic dynamics arising in a system of ordinary differential equations in R4 with symmetry D4⋉(Z2)2. This system arises from the normal form reduction of a 1:2√ mode interaction for Boussinesq convection. We investigate the structure of a particular robust heteroclinic attractor with “depth two connections” from equilibria to subcycles as well as connections between equilibria. The “subcycle” is not asymptotically stable, due to nearby trajectories undertaking an “excursion,” but it is a Milnor attractor, meaning that a positive measure set of nearby initial conditions converges to the subcycle. We investigate the dynamics in the presence of noise and find a number of interesting properties. We confirm that typical trajectories wind around the subcycle with very occasional excursions near a depth two connection. The frequency of excursions depends on noise intensity in a subtle manner; in particular, for anisotropic noise, the depth two connection may be visited much more often than for isotropic noise, and more generally the long term statistics of the system depends not only on the noise strength but also on the anisotropy of the noise. Similar properties are confirmed in simulations of Boussinesq convection for parameters giving an attractor with depth two connections.Royal SocietyAgence Nationale de la Recherche, FranceRussian Foundation for Basic Researc

On local attraction properties and a stability index for heteroclinic connections

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