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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Bifurcation from codimension one relative homoclinic cycles


Authors: Ale Jan Homburg, Alice C. Jukes, Jürgen Knobloch and Jeroen S.W. Lamb
Journal: Trans. Amer. Math. Soc. 363 (2011), 5663-5701
MSC (2010): Primary 34C37, 37G40, 34C23
DOI: https://doi.org/10.1090/S0002-9947-2011-05193-7
Published electronically: June 20, 2011
MathSciNet review: 2817404
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Abstract: We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in the unfolding. Descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided.


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Additional Information

Ale Jan Homburg
Affiliation: KdV Institute for Mathematics, University of Amsterdam, Science Park 904,1098 XH Amsterdam, The Netherlands – and – Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
Email: a.j.homburg@uva.nl

Alice C. Jukes
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
Email: alice.jukes@gmail.com

Jürgen Knobloch
Affiliation: Department of Mathematics, TU Ilmenau, Postfach 100565, 98684 Ilmenau, Germany
Email: juergen.knobloch@tu-ilmenau.de

Jeroen S.W. Lamb
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
Email: jeroen.lamb@imperial.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2011-05193-7
Received by editor(s): September 14, 2008
Received by editor(s) in revised form: July 22, 2009
Published electronically: June 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.