99-437 Carles Sim\'o, Claudia Valls

A formal approximation of the splitting of separatrices in the Classical Arnold's example of diffusion with two equal parameters. (936K, postscript (gzipped + uuencoded)) Nov 17, 99

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Abstract. We consider the classical Arnold's example of diffusion with two equal parameters. Such system has two dimensional normally hyperbolic invariant tori. We focus on the torus whose ratio of frequencies is the golden mean. We present formal approximations of the three dimensional invariant manifolds associated to this torus and numerical globalization of these manifolds. This allows to obtain the splitting (of separatrices) vector and to compute its Fourier components. It is apparent that the Melnikov vector provides the dominant order of the splitting provided it is computed after a suitable number of averaging steps. We carry out the first order analysis of the splitting based on that approach, mainly looking for bifurcations of the zero level curves of the components of the splitting vector and of the homoclinic points.

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