双频系统在分界面附近的共振平均和通过
Averaging and passage through resonances in two-frequency systems near separatrices
摘要：
平均方法是动力系统扰动理论中一个经典的有力工具。应用平均方法有两个主要障碍，共振和分离线。在本文中，我们得到了现实的渐近估计，证明使用平均方法在一般情况下，这两个障碍是目前在同一时间，通过一个分界线的时间周期扰动的单频哈密顿系统。作为一种普遍现象，共振在分离面处积累。哈密顿量取决于一个参数，该参数对于受扰系统缓慢变化（因此具有2.5个自由度的慢-快哈密顿系统包括在我们的类中）。我们的结果也可以应用到一般的两个频率的可积系统附近的分离线的扰动，因为它们可以减少到一个频率系统的周期扰动。
英文摘要：
The averaging method is a classical powerful tool in perturbation theory of dynamical systems. There are two major obstacles to applying the averaging method, resonances and separatrices. In this paper we obtain realistic asymptotic estimates that justify the use of averaging method in a generic situation where both these obstacles are present at the same time, passage through a separatrix for time-periodic perturbations of one-frequency Hamiltonian systems. As a general phenomenon, resonances accumulate at separatrices. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so slow-fast Hamiltonian systems with two and a half degrees of freedom are included in our class). Our results can also be applied to perturbations of generic two-frequency integrable systems near separatrices, as they can be reduced to periodic perturbations of one-frequency systems.
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