Asymptotic description of nonlinear resonance

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

We consider a simple model of an open partially expanding map. Its trapped set K in phase space is a fractal set. We first show that there is a well defined discrete spectrum of Ruelle resonances which describes the asymptotics of correlation functions for large time and which is parametrized by the Fourier component νon the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call "minimal captivity". This hypothesis is stable under perturbations and means that the dynamics is univalued on a neighborhood of K. Under this hypothesis we show the existenc…

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

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), Kevorki…

含衰减扰动的振荡系统中的非线性共振

Nonlinear resonance in oscillatory systems with decaying perturbations

本文研究平面非等时非线性振荡系统的时间衰减扰动问题，结合平均法与稳定性分析给出锁相状态存在稳定的扰动条件，并应用于受扰Duffing振子。

The capture into parametric autoresonance

In this work we show that the capture into parametric resonance may be explained as the pitchfork bifurcation in the primary parametric resonance equation. We prove that the solution close to the moment of the capture is described by the Painleve-2 equation. We obtain the connection formulas for the asymptotic solution of the primary parametric resonance equation before and after the capture using the matching of the asymptotic expansions.

带衰减啁啾频率激励的渐近自治系统中的共振

Resonances in asymptotically autonomous systems with a decaying chirped-frequency excitation

该研究针对带衰减啁啾频率振荡扰动的平面自治强非线性系统，结合平均法与Lyapunov函数法分析轨迹长期行为，发现锁相、相位漂移两种渐近态，且锁相下存在能量无界增长的共振解并探究其稳定性与渐近特性。

Resonances and O-curves in Hamiltonian systems

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.