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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multiple homoclinic orbits in conservative and reversible systems


Authors: Ale Jan Homburg and Jürgen Knobloch
Journal: Trans. Amer. Math. Soc. 358 (2006), 1715-1740
MSC (2000): Primary 34Cxx, 37Cxx, 37Gxx
Published electronically: November 1, 2005
MathSciNet review: 2186994
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Abstract: We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times.

In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist.

A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.


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Additional Information

Ale Jan Homburg
Affiliation: KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Email: alejan@science.uva.nl

Jürgen Knobloch
Affiliation: Institute for Mathematics, Technical University Ilmenau, Postbox 10 05 65, D-98684 Ilmenau, Germany
Email: knobi@mathematik.tu-ilmenau.de

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03793-1
PII: S 0002-9947(05)03793-1
Received by editor(s): September 19, 2002
Received by editor(s) in revised form: July 1, 2004
Published electronically: November 1, 2005
Article copyright: © Copyright 2005 American Mathematical Society