ATTRACTORS FOR ROBUST HETEROCLINIC CYCLES WITH CONTINUE OF CONNECTIONS

For systems with symmetry (or more generally, systems with invariant s ubspaces) 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 g eneric Hopf bifurcation with tetrahedral symmetry. In this situation i t is possible to get bifurcation from a steady state directly to a hom oclinic cycle with a two-dimensional set of connections. We numericall y investigate the dynamics near such cycles. We conjecture that if a h eteroclinic cycle is asymptotically stable then all connections corres ponding to the most positive expanding eigenvalues of the linearisatio n at the fixed points will generically form part of an attractor. This attractor may fail to be asymptotically stable and is, to our knowled ge, the first example of this for a homoclinic (as opposed to a hetero clinic) cycle. We prove this conjecture for homoclinic cycles with dis tinct 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 an d continua of connections are found in this example.