On Asymptotic Description of Passage through a Resonance in Quasilinear Hamiltonian Systems

Abstract

We consider a quasilinear 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 of $O(\varepsilon^{\frac32})$. This is an improvement of known results by Chirikov [Soviet Physics Dokl., 4 (1959), pp. 390--394], Kevorkian [SIAM J. Appl. Math., 20 (1971), pp. 364--373; 26 (1974), p. 686], Bosley and Kevorkian [SIAM J. Appl. Math., 52 (1992), pp. 494--527], and Bosley [SIAM J. Appl. Math., 56 (1996), pp. 420--445]. The problem under consideration is a model problem that describes passage through an isolated resonance in multifrequency quasilinear Hamiltonian systems.