Stability and bifurcations of heteroclinic cycles of type Z

Details Stability and bifurcations of heteroclinic cycles of type Z Stability and bifurcations of heteroclinic cycles of type Z

2011-08-21

arxiv.org/abs/1108.4204v2

Nonlinear Sciences

Chaotic Dynamics

35 pages, submitted to Nonlinearity

Scientific paper

Dynamical systems, which are equivariant 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, which 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 {\cal B}_{\delta}(X) is positive. A local basin of attraction {\cal B}_{\delta}(X) is the set of points, such 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 the 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.

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