Attractors for Robust Heteroclinic Cycles with Continua of Connections

Abstract

For systems with symmetry (or more generally, systems with invariant subspaces) it is possible to find robust heteroclinic cycles with multi-dimensional connecting manifolds. Motivated by a problem of rotating convection with low Prandtl number, Swift and Barany [23] considered generic Hopf bifurcation with tetrahedral symmetry. In this situation it is possible to get bifurcation from a steady state directly to a homoclinic cycle with a two-dimensional set of connections. We numerically investigate the dynamics near such cycles. We conjecture that if a heteroclinic cycle is asymptotically stable then all connections corresponding to the most positive expanding eigenvalues of the linearisation at the fixed points will generically form part of an attractor. This attractor may fail to be asymptotically stable and is, to our knowledge, the first example of this for a homoclinic (as opposed to a heteroclinic) cycle. We prove this conjecture for homoclinic cycles with distinct real expanding and contracting eigenvalues, and present evidence to support it for other cases. An example due to Kirk and Silber [15] (two competing cycles in R 4 with (Z 2 ) 4 symmetry) is discussed and continua of connections are found in this example.