论文标题
带有应用的哈密顿系统的当地功能
The local period function for Hamiltonian systems with applications
论文作者
论文摘要
在本文的第一部分中，我们制定了一种建设性的程序，以在任何平面分析性汉密尔顿系统的非分离中心的时期函数方面获得泰勒扩展。我们将其应用于几个示例，包括旋转摆和立方哈密顿系统。对该系统的周期功能扩展的了解是研究Abelian积分的零数的关键点之一，该积分控制着从平面汉密尔顿系统的周期性轨道分叉的极限周期数的数量，该系统受到毛细管性启发的启发。其他几种经典工具，例如使用Chebyshev系统来研究这一数量的零。引入的方法也可以在其他情况下应用。
In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations.
