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Abstract
We describe an example of a robust heteroclinic network for which nearby orbits exhibit irregular but sustained switching between the various sub-cycles in the network. The mechanism for switching is the presence of spiralling due to complex eigenvalues in the flow linearized about one of the equilibria common to all cycles in the network. We construct and use return maps to investigate the asymptotic stability of the network, and show that switching is ubiquitous near the network. Some of the unstable manifolds involved in the network are two-dimensional; we develop a technique to account for all trajectories on those manifolds. A simple numerical example illustrates the rich dynamics that can result from the interplay between the various cycles in the network.
Acknowledgements
This research has been supported by the University of Auckland Research Committee, the Engineering and Physical Sciences Research Council (EP/G052603/1) and the National Science Foundation (DMS-0709232). We are grateful for the hospitality of the Department of Mathematics at the University of Auckland, the Department of Engineering Sciences and Applied Mathematics at Northwestern University, and the School of Mathematics at the University of Leeds.