Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

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Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

Show simple item record Ashwin, Peter en_GB Burylko, Oleksandr en_GB Maistrenko, Yuri en_GB
dc.contributor.department University of Exeter; National Academy of Sciences of Ukraine; Research Centre Jülich, Germany en_GB 2008-03-19T12:04:17Z en_GB 2011-01-25T10:33:29Z en_US 2013-03-20T12:30:55Z 2008 en_GB
dc.description.abstract We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking). For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S3-transcritical homoclinic bifurcation, (b) a saddle–node/heteroclinic bifurcation and (c) a Z3-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry. For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions. en_GB
dc.identifier.citation 237 (4), pp. 454-466 en_GB
dc.identifier.doi 10.1016/j.physd.2007.09.015 en_GB
dc.identifier.uri en_GB
dc.language.iso en en_GB
dc.publisher Elsevier en_GB
dc.relation.url en_GB
dc.subject coupled phase oscillator en_GB
dc.subject bifurcation en_GB
dc.subject heteroclinic cycle en_GB
dc.title Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators en_GB
dc.type Article en_GB 2008-03-19T12:04:17Z en_GB 2011-01-25T10:33:29Z en_US 2013-03-20T12:30:55Z
dc.identifier.issn 0167-2789 en_GB
dc.description Copyright © 2008 Elsevier. NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D, Vol 237, Issue 4, 2008, DOI: 10.1016/j.physd.2007.09.015 en_GB
dc.identifier.journal Physica D en_GB

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