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September  2010, 14(2): 367-387. doi: 10.3934/dcdsb.2010.14.367

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  June 2010

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.
Keywords: Melnikov function, Singular perturbation, homoclinic bifurcation, invariant manifolds, Sil'nikov orbits..
Mathematics Subject Classification: Primary: 34E15, 34C23; Secondary: 37G2.
Citation: Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete and Continuous Dynamical Systems - Series B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367
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