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.
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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.