A competition between heteroclinic cycles

Abstract

Competition between co-existing heteroclinic cycles that have a c o m o n hetemclinic connection is considered. A simple model problem, consisting of a system of ordinary 'difimtial equations in R4 with 2; symmetry, is analysed. The differential equations possess four hyperbolic fixedpoints el. &. tj, and 54, with heteroclinic mnnections joining pairs of fixed points to form a 'hetemclinic network'. The network contains two heteroclinic cycles tt + 52 + b + PI and 61 + & -+ 64 + el, each of which is strucnually stable with respect to perturbations that p m m e the 2 ; 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 ihrough an open neighbourhwd of one or more (but not all) of the heteroclinic comedons 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 lin& about each of the fixed points and on the validity of certain nondegeneracy conditions. The possible stability propelries of the network and the cycles within it are determined.

Oscillatory long-wave Marangoni convection in a layer of a binary liquid: Hexagonal patterns

“…Of course, each AR pattern, either AR-A, or AR-B, or AR-C, forms its own heteroclinic cycle, but following Ref. [19] we treat the group orbit of these heteroclinic connections as a single heteroclinic cycle.…”

Construction of heteroclinic networks in ${{\mathbb{R}}^{4}}$

“…Then, the study of the asymptotic stability of cycles led to results in Krupa and Melbourne [19,20]. When addressing the stability of cycles in networks, it is clear that no individual cycle can be asymptotically stable and intermediate notions of stability appeared in Melbourne [22], Brannath [4], Kirk and Silber [18], Driesse and Homburg [9] and Podvigina and Ashwin [25]. Of these essential asymptotic stability (e.a.s.)…”

Bifurcation from codimension one relative homoclinic cycles

“…Such connections are clearly robust. There has been much interest in the existence, asymptotic stability and also bifurcations of robust heteroclinic cycles or networks; see [10,11,26,5,29,36,1,22,2] and the references therein.…”