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Homoclinic and heteroclinic orbits of reversible vectorfields under perturbation

Published online by Cambridge University Press:  14 November 2011

Richard C. Churchill  and
David L. Rod
Richard C. Churchill
Affiliation:
Department of Mathematics, Hunter College (CUNY), New York, New York 10021, U.S.A.
David L. Rod
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N1N4, Canada
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Synopsis

Averaging techniques on Hamiltonian dynamical systems can often be used to establish the existence of hyperbolic periodic orbits. In equilibrium situations, it is then often difficult to show that there are homoclinic/heteroclinic connections between these hyperbolic orbits in the original unaveraged system. This existence problem is solved in this paper for a class of Hamiltonian systems admitting a sufficient number of symmetries (including reversing symmetries). Under isoenergetic reduction, the problem is reduced to one involving reversible vector fields under time-dependent perturbations admitting the same reversing symmetries. Applications are made to the one-parameter Hénon-Heiles family. The paper concludes with remarks on the problem of showing transversality of these homoclinic/heteroclinic orbits.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 345 - 363
Copyright
Copyright © Royal Society of Edinburgh 1986

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