A competition between heteroclinic cycles

Abstract

Competition between co-existing heteroclinic cycles that have a common heteroclinic connection is considered. A simple model problem, consisting of a system of ordinary differential equations in R4 with Z24 symmetry, is analysed. The differential equations possess four hyperbolic fixed-points xi 1, xi 2, xi 3, and xi 4, with heteroclinic connections joining pairs of fixed points to form a 'heteroclinic network'. The network contains two heteroclinic cycles xi 1 to xi 2 to xi 3 to xi 1 and xi 1 to xi 2 to xi 4 to xi 1, each of which is structurally stable with respect to perturbations that preserve the Z24 symmetry of the problem. Local analysis, valid in the vicinity of the heteroclinic cycles, shows that while neither cycle can be asymptotically stable, there are conditions under which both cycles have strong attractivity properties simultaneously. For example, it is possible for both cycles to have the property that trajectories that pass through an open neighbourhood of one or more (but not all) of the heteroclinic connections in the given cycle are asymptotic to that cycle. The stability results depend on the strengths of the contracting and expanding eigenvalues of the flow linearized about each of the fixed points and on the validity of certain nondegeneracy conditions. The possible stability properties of the network and the cycles within it are determined.