Fast drift and diffusion in an example of isochronous system through Windows Method

TL;DR

This paper investigates Arnold's diffusion in an isochronous system using the Windows Method, overcoming specific challenges to demonstrate the existence of fast drifting orbits with times comparable to variational approaches.

Contribution

It introduces adaptations of the Windows Method for isochronous systems, including non-uniform transition chains and variable windows, to establish diffusion results.

Findings

Successfully applied the Windows Method to an isochronous system.

Obtained fast drifting orbits with times comparable to variational methods.

Addressed challenges due to lack of anisochrony in the system.

Abstract

We study the problem of Arnold's diffusion in an example of isochronous system by using a geometrical method known as Windows Method. Despite the simple features of this example, we show that the absence of an anisochrony term leads to several substantial difficulties in the application of the method, requiring some additional devices as non-equally spaced transition chains and variable windows. In this way we are able to obtain a set of fast orbits whose drifting time matches, up to a constant, the time obtained via variational methods.