多参数哈密顿系统周期解歧点集的结构
On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems
摘要：
本文研究了含多参数的哈密顿方程的周期解。证明了周期为$2π/j,$ $j\in\mathbb{N},$的解从一个驻点全局分支的定理。哈密顿量在驻点处的黑塞矩阵可以是奇异的。然而，它是假设的局部拓扑度的梯度的哈密顿在驻点是非零的。证明了具有给定周期解的（整体）歧点可以通过参数空间上适当的连续函数的零点来识别。当驻点处哈密顿量的黑塞矩阵为块对角矩阵时，给出了这类函数的显式表达式。得到了不同最小周期解分支的对称破缺结果。给出了歧点集的几何描述。给出了将所证明的定理应用于解析和数值研究的实例，并给出了给定区域内所有歧点的可视化。本文是在作者论文[W. Radzki，"自主哈密顿系统周期解的分支点"（波兰文），博士论文，哥白尼大学数学和计算机科学系，托鲁托，2005年]。
英文摘要：
This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2π/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toruń, 2005].
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