Abstract

The paper studies the problem of approximate construction of eigenvalues and multipliers of linear (autonomous and periodic) Hamiltonian systems depending on a small parameter in the main critical cases. New formulas for the asymptotic (in powers of a small parameter) representation of the eigenvalues and multipliers of the task are proposed. The obtained formulas allow us to effectively study the problems of stability and hyperbolicity of linear systems, equilibrium points and periodic solutions of nonlinear Hamiltonian systems, the problem of constructing the boundaries of the regions of stability and hyperbolicity, problems of local bifurcations of nonlinear dynamical systems, etc.

中文翻译：

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临界情况下周期性和自治哈密顿系统扰动问题的近似研究

摘要

本文研究了在主要危急情况下依靠小参数的线性（自治和周期性）哈密顿系统特征值和乘数的近似构造问题。提出了特征值和任务乘数的渐近（小参数幂）表示的新公式。所获得的公式使我们能够有效地研究线性系统的稳定性和双曲性问题，非线性哈密顿系统的平衡点和周期解，构造稳定性和双曲率区域的边界问题，非线性动力系统的局部分叉问题等