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

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


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