Few Islands Approximation of Hamiltonian System with divided Phase Space

TL;DR

This paper introduces a novel approximation method for analyzing Hamiltonian systems with divided phase space by representing them with finite KAM islands, enabling better numerical and analytical understanding.

Contribution

The paper proposes a new approach to approximate Hamiltonian systems with divided phase space using finite KAM islands, facilitating analysis of their complex dynamics.

Findings

The approximation method effectively models the divided phase space.

Statistical characteristics of approximations converge to the original system.

Application to billiards demonstrates the method's practical utility.

Abstract

It is well known that typical Hamiltonian systems have divided phase space consisting of regions with regular dynamics on KAM tori and region(s) with chaotic dynamics called chaotic sea(s). This complex structure makes rigorous analysis of such systems virtually impossible and significantly complicates numerical exploration of their dynamical properties. In this paper we outline a new approach for the analysis of Hamiltonian systems with divided phase space. These systems are approximated by a sequence of Hamiltonian systems having an increasing (but finite) number of KAM islands. The islands in the approximating systems are sub-islands of the islands in the initial system with an infinite number of KAM-islands. We apply this approach to two-dimensional billiards and demonstrate that it works. In particular the statistical characteristics of the approximating systems tend to the ones for the whole system when the number of islands in the approximating systems grows. Therefore our approach opens up a new way for numerical and analytical studies of the dynamics of Hamiltonian systems with divided phase space.