Heteroclinic cycles may occur as asymptotically stable attractors in
a structurally stable manner if there are invariant subspaces or
symmetries of a dynamical system. Even for cycles between equilibria,
it may be difficult to obtain results on the generic
behavior of trajectories converging to the cycle. For more
complicated cycles, for example between chaotic sets,
the nontrivial dynamics of the `nodes' can interact
with that of the `connections'. This paper focuses on some of the
simplest problems for such dynamics where there are direct products of
an attracting homoclinic cycle with various types of dynamics. Using a precise
analytic description of a general planar homoclinic attractor, we are able
to obtain a number of results for direct product systems.
We show that for flows that are a product of a homoclinic attractor
and a periodic orbit or a chaotic attractor,
the product of the attractors is an attractor for the product.
In the case of a periodic orbit we show that the product is a minimal attractor.
On the other hand, we present evidence to show that for the product of two homoclinic
attractors, typically only a small subset of the product of the attractors is
an attractor for the product system.
We comment on possible extensions of these results to skew product systems,
as well as relationships with the problem of connection selection for
higher dimensional connections in attracting heteroclinic networks.
e-mail: firstname.lastname@example.org or Michael.J.Field@rice.edu Professor Mike Field
Department of Mathematics
6100 S Main St