Abstract

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Gröbner bases and for the work with polynomial ideals in the computer algebra system Maple.

中文翻译：

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具有离散对称群的哈密顿系统的周期解的分歧

摘要

研究了具有两个自由度的自治哈密顿系统，该系统在系统扩展相空间的Klein四组线性规范自同构下是不变的。构造了该系统对称周期解的单峰矩阵的辛变换序列。使用这些变换，研究了该溶液附近相流的结构和分叉。结果表明，对于具有双对称性的解和具有单个对称性的解，对应于多个周期增加的分叉是不同的。讨论了平面圆形希尔问题的双对称轨道族的临界周期解的一个例子。枫木。