Integrable Approximation of Regular Islands: The Iterative Canonical Transformation Method

TL;DR

This paper introduces an iterative canonical transformation method to construct integrable approximations of regular regions in mixed Hamiltonian systems, effectively extending regular dynamics into chaotic regions for complex systems.

Contribution

The paper presents a novel iterative method for creating integrable approximations that accurately mimic regular dynamics and extend into chaotic regions in mixed Hamiltonian systems.

Findings

Effective for strongly perturbed systems

Works with arbitrary degrees of freedom

Successfully applied to standard map and cosine billiard

Abstract

Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. We present an iterative method to construct an integrable approximation, which resembles the regular dynamics of a given mixed system and extends it into the chaotic region. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. This method works for strongly perturbed systems and arbitrary degrees of freedom. We apply it to the standard map and the cosine billiard.