SIAM J. Applied Dynamical Systems 11 (2012) 1360-1401. doi:10.1137/120864684

Resonance bifurcations of robust heteroclinic networks

V. Kirk(1), Claire M. Postlethwaite(1), and A.M. Rucklidge(2)

(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

Abstract. Robust heteroclinic cycles are known to change stability in resonance bifurcations, which occur when an algebraic condition on the eigenvalues of the system is satisfied and which typically result in the creation or destruction of a long-period periodic orbit. Resonance bifurcations for heteroclinic networks are potentially more complicated because different subcycles in the network can undergo resonance at different parameter values, but have, until now, not been systematically studied. In this article we present the first investigation of resonance bifurcations in heteroclinic networks. Specifically, we study two heteroclinic networks in $\R^4$ and consider the dynamics that occurs as various subcycles in each network change stability. The two cases are distinguished by whether or not one of the equilibria in the network has real or complex contracting eigenvalues. We construct two-dimensional Poincar\'e return maps and use these to investigate the dynamics of trajectories near the network; a complicating feature of the analysis is that at least one equilibrium solution in each network has a two-dimensional unstable manifold. We use the technique developed in [18] to keep track of all trajectories within these two-dimensional unstable manifolds. In the case with real eigenvalues, we show that the asymptotically stable network loses stability first when one of two distinguished cycles in the network goes through resonance and two or six periodic orbits appear. In some circumstances, asymptotically stable periodic orbits can bifurcate from the network even though the subcycle from which they bifurcate is never asymptotically stable. In the complex case, we show that an infinite number of stable and unstable periodic orbits are created at resonance, and these may coexist with a chaotic attractor. In both cases, we show that near to the parameter values where individual cycles go through resonance, the periodic orbits created in the different resonances do not interact, i.e., the periodic orbits created in the resonance of one cycle are not involved in the resonance of the other cycle. However, there is a further resonance, for which the eigenvalue combination is a property of the entire network, after which the periodic orbits which originated from the individual resonances may interact. We illustrate some of our results with a numerical example.



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Movies relevant to this paper: real case

The red solid curves are the small nullclines in region~$1$, the green solid curves are the small nullclines in region~$2$, and the blue solid curves are the $\theta_3$ nullclines. The red and green dashed curves are the approximate small nullclines. Cf figs 5.5-5.7.

Movies relevant to this paper: complex case

The red solid curves are the small nullclines in region~$1$, the green solid curves are the small nullclines in region~$2$, and the blue solid curves are the $\theta_3$ nullclines. Cf figs 5.10-5.13.