On asymptotic description of passage through a resonance in quasi-linear Hamiltonian systems
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,
∼ε
, 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
ε
. We provide asymptotic formulas that describe effects of passage through a resonance with an accuracy
O(
ε
3
2
)
. 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.