Name
Abstract | ||
|---|---|---|
Robust heteroclinic cycles are known to change stability in resonance bifurcations, which occur when a certain ratio of eigenvalues of the system passes through one and which typically result in the creation or destruction of a long-period periodic orbit. Resonance bifurcations for heteroclinic networks are more complicated because different subcycles in the network can undergo resonance at different parameter values. In this article 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 Poincare 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. 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 not 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. 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. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1137/120864684 | SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS |
Keywords | Field | DocType |
heteroclinic cycle,heteroclinic network,resonance,resonance bifurcation | Topology,Heteroclinic network,Poincaré conjecture,Control theory,Heteroclinic cycle,Mathematical analysis,Heteroclinic bifurcation,Resonance,Mathematics,Heteroclinic orbit,Eigenvalues and eigenvectors,Manifold | Journal |
Volume | Issue | ISSN |
11 | 4 | 1536-0040 |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Vivien Kirk | 1 | 22 | 5.61 |
| Claire M. Postlethwaite | 2 | 6 | 3.53 |
| Alastair M. Rucklidge | 3 | 52 | 6.52 |