Fast Drift and Diffusion in a Class of Isochronous Systems with the Windows Method

The aim of the paper is to deal with some peculiar difficulties arising from the use of the geometrical tool known as windows method in the context of the well known problem of Arnold's diffusion for isochronous nearly-integrable Hamiltonian systems. Despite the simple features of the class of systems at hand, it is possible to show how the absence of an anisochrony term leads to several substantial differences in the application of the method, requiring some additional devices, such as non-equally spaced transition chains and variable windows. As a consequence, we show the existence of a set of unstable orbits, whose drifting time matches, up to a constant, the one obtained via variational methods.