A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters

摘要

We consider the classical Arnold example of diffusion with two equal parameters. Such a system has two-dimensional partially hyperbolic invariant tori. We mainly focus on the tori whose ratio of frequencies is the golden mean. We present formal approximations of the three-dimensional invariant manifolds associated with this torus and numerical globalization of these manifolds. This allows one 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 the contribution of each harmonic is computed after a suitable number of averaging steps, depending on the harmonic. 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. [References: 18]