A formalapproximation of the splitting of separatrices in theclassical Arnold's example of diffusion with two equalparameters

Abstract

We consider the classical Arnold example of diffusionwith two equal parameters. Such a system hastwo-dimensional partially hyperbolic invariant tori. Wemainly focus on the tori whose ratio of frequencies isthe golden mean. We present formal approximations of thethree-dimensional invariant manifolds associated withthis torus and numerical globalization of thesemanifolds. This allows one to obtain the splitting (ofseparatrices) vector and to compute its Fouriercomponents. It is apparent that the Melnikov vectorprovides the dominant order of the splitting provided thecontribution of each harmonic is computed after asuitable number of averaging steps, depending on theharmonic. We carry out the first-order analysis of thesplitting based on that approach, mainly looking forbifurcations of the zero-level curves of the componentsof the splitting vector and of the homoclinic points.