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

We consider the classical Arnold example of diffusion with two equal parame ters. 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 manif olds associated with this torus and numerical globalization of these manifo lds. This allows one to obtain the splitting (of separatrices) vector and t o compute its Fourier components. It is apparent that the Melnikov vector p rovides the dominant order of the splitting provided the contribution of ea ch harmonic is computed after a suitable number of averaging steps, dependi ng 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 c urves of the components of the splitting vector and of the homoclinic point s.