Dynamical Systems: An International Journal 25 (2010) 323-349. doi:10.1080/14689361003779134

A mechanism for switching near a heteroclinic network

V. Kirk(1), Emily Lane, Claire M. Postlethwaite(1), A.M. Rucklidge(2) and Mary Silber(3)

(1) Department of Mathematics,
University of Auckland, Private Bag 92019,
Auckland, New Zealand
(2) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK
(3) Department of Engineering Sciences and Applied Mathematics and Northwestern Institute on Complex Systems,
Northwestern University, Evanston, IL 60208, USA

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.

gzipped PostScript version of this paper (0.3MB)