On asymptotic description of passage through a resonance in quasi-linear Hamiltonian systems

TL;DR

This paper derives precise asymptotic formulas for the behavior of quasi-linear Hamiltonian systems as they pass through a resonance, improving previous results with higher accuracy.

Contribution

It provides new asymptotic formulas with $O( ext{}\varepsilon^{3/2})$ accuracy for passage through a resonance in quasi-linear Hamiltonian systems, surpassing earlier work.

Findings

Asymptotic formulas with $O( ext{}\varepsilon^{3/2})$ accuracy for resonance passage

Model describes passage through isolated resonance in multi-frequency systems

Improves upon classical results by Chirikov, Kevorkian, and Bosley

Abstract

We consider a quasi-linear Hamiltonian system with one and a half degrees of freedom. The Hamiltonian of this system differs by a small, $\sim\varepsilon$, perturbing term from the Hamiltonian of a linear oscillatory system. We consider passage through a resonance: the frequency of the latter system slowly changes with time and passes through 0. The speed of this passage is of order of $\varepsilon$. We provide asymptotic formulas that describe effects of passage through a resonance with an accuracy $O(\varepsilon^{\frac32})$. This is an improvement of known results by Chirikov (1959), Kevorkian (1971, 1974) and Bosley (1996). The problem under consideration is a model problem that describes passage through an isolated resonance in multi-frequency quasi-linear Hamiltonian systems.