Global bifurcation of homoclinic solutions of Hamiltonian systems

S. Secchi; C. A. Stuart

doi:10.3934/dcds.2003.9.1493

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2003, Volume 9, Issue 6: 1493-1518. Doi: 10.3934/dcds.2003.9.1493

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Global bifurcation of homoclinic solutions of Hamiltonian systems

S. Secchi^1, and
C. A. Stuart^2,

1.
Scuola Internazionale Superiore de Studi Avanzati, via Beirut 2-4, 34014 Trieste
2.
Institut de Analyse et de Calcul Scientifique, Section de mathmatiques, Ecole Polytechnique Fédérale Lausanne, CH - 1015 Lausanne

Received: June 30, 2002

Revised: March 31, 2003

Published: October 2003

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Abstract

The main results give hypotheses ensuring that a non-autonomous first order Hamiltonian system has a global branch of homoclinic solutions bifurcating from an eigenvalue of odd multiplicity of the linearization. The system is required to be asymptotically periodic (as time goes to plus and minus infinity) and these limit problems should have no homoclinic solutions. Furthermore, the asymptotic limits of the linearization should have no characteristic multipliers on the unit circle. The proof uses the topological degree for proper Fredholm maps of index zero.

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Mathematics Subject Classification: 37J45, 43C37, 34C23.

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