Transversal periodic-to-periodic homoclinic orbits in singularlyperturbed systems

Flaviano Battelli; Ken Palmer

doi:10.3934/dcdsb.2010.14.367

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2010, Volume 14, Issue 2: 367-387. Doi: 10.3934/dcdsb.2010.14.367

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Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems

Flaviano Battelli^1, and
Ken Palmer^2,

1.
Dipartimento di Scienze Matematiche, Facoltà di Ingegneria, Università, Via Brecce Bianche, 1, 60100 Ancona
2.
Department of Mathematics, National Taiwan University, Taipei 106

Received: July 2009

Revised: December 2009

Published: September 2010

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Abstract

We consider a singularly perturbed system with a normally hyperbolic centre manifold. Assuming the existence of a fast homoclinic orbit to a point of the centre manifold belonging to a hyperbolic periodic solution for the slow system, we prove an old and a new result concerning the existence of solutions of the singularly perturbed system that are homoclinic to a periodic solution of the system on the centre manifold. We also give examples in dimensions greater than three of Sil'nikov [16] periodic-to-periodic homoclinic orbits.

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Mathematics Subject Classification: Primary: 34E15, 34C23; Secondary: 37G20.

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