Stability and bifurcations of heteroclinic cycles of type Z

Abstract

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 which we call heteroclinic cycles of type Z. It is well known that a heteroclinic cycle that is not asymptotically stable can nevertheless attract a positive measure set from its neighbourhood. We say that an invariant set X is fragmentarily asymptotically stable, if for any δ > 0 the measure of its local basin of attraction is positive. A local basin of attraction is the set of such points that trajectories starting there remain in the δ-neighbourhood of X for all t > 0, and are attracted by X as t → ∞. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearizations 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.