Abstract

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.

PDF file of article.


For (p)reprint, e-mail: mikefield@gmail.com or Michael.J.Field@rice.edu

Professor Mike Field
Department of Mathematics
Rice University
6100 S Main St
Houston
TX 77005-1892