Parameter-dependent behaviour of periodic channels in a locus of boundary crisis
- 133 Downloads
A boundary crisis occurs when a chaotic attractor outgrows its basin of attraction and suddenly disappears. As previously reported, the locus of a boundary crisis is organised by homo- or heteroclinic tangencies between the stable and unstable manifolds of saddle periodic orbits. In two parameters, such tangencies lead to curves, but the locus of boundary crisis along those curves exhibits gaps or channels, in which other non-chaotic attractors persist. These attractors are stable periodic orbits which themselves can undergo a cascade of period-doubling bifurcations culminating in multi-component chaotic attractors. The canonical diffeomorphic two-dimensional Hénon map exhibits such periodic channels, which are structured in a particular ordered way: each channel is bounded on one side by a saddle-node bifurcation and on the other by a period-doubling cascade to chaos; furthermore, all channels seem to have the same orientation, with the saddle-node bifurcation always on the same side. We investigate the locus of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We find that the Ikeda map features periodic channels with a richer and more general organisation than for the Hénon map. Using numerical continuation, we investigate how the periodic channels depend on a third parameter and characterise how they split into multiple channels with different properties.
- 3.R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, Inc., 1987)Google Scholar
- 4.A. Dhooge, W. Govaerts, Y.A. Kuznetsov, W. Mestrom, A.M. Riet, Cl_matcont: a Continuation Toolbox in Matlab, in Proceedings of the ACM Symposium on Applied Computing, 2003 (ACM New York, 2003), pp. 161–166Google Scholar
- 19.H.M. Osinga, J. Rankin, Two-parameter locus of boundary crisis: mind the gaps, in Proceedings of The 8th AIMS International Conference, 2010, edited by W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, Stefan Siegmund, J. Voigt (American Institue of Mathematical Sciences, 2011), pp. 1148–1157Google Scholar
- 20.J. Palis, F. Takens, Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge University Press, 1993)Google Scholar
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.