标题： 对称1:1哈密顿共振中的分支序列 显示英文标题

标题： Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance

摘要： 我们综述了一族具有近似相等未扰频率且在$Z_2 \times Z_2$对称性下不变的共振哈密顿标准形式中，一般位置周期轨道的分岔序列。 这些经典系统的丰富结构通过几何方法被研究，并且与奇点理论方法的关系也被突出显示。 几何方法是获得该族系统相空间动力学全貌的最直接方式，这由控制参数空间中的一个完整子集定义，且满足对称性约束。 展示了如何找到描述该族每个成员相空间结构的能量-动量映射、捕捉其全局特性的灾变映射以及作用-角度变量的形式表达式。 多个例子，主要来自天体力学，被用作应用实例。 显示英文摘要

摘要： We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.