Integrable approximation of regular regions with a nonlinear resonance chain

TL;DR

This paper introduces a method to construct integrable approximations of regular regions with nonlinear resonance chains in Hamiltonian systems, improving the modeling of mixed phase space dynamics.

Contribution

It generalizes the iterative canonical transformation method to include nonlinear resonance chains in integrable approximations of Hamiltonian systems.

Findings

Successfully applied to the standard map at various parameters.

Accurately reproduces the shape of regular tori.

Enhances understanding of mixed phase space structures.

Abstract

Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.