一般四维辛映射中正则环面的整体结构
Global structure of regular tori in a generic 4D symplectic map
摘要：
对于混合相空间的一般四维辛映射的情况下，我们研究了正则环面的全局组织。为此，我们计算椭圆1-环面的两个耦合的标准地图，并显示它们在一个3D相空间切片。这可视化了在4D相空间中，所有规则的2-环面是如何围绕椭圆1-环面的骨架组织的。1-环面出现在两种类型的单参数族中：（a）从椭圆-椭圆周期轨道产生的李雅普诺夫族，观察到甚至存在于远离它们和超越主要共振间隙的地方，以及（b）从秩-1共振产生的族。在两种类型的族的共振间隙处，（i）存在周期轨道，类似于2D映射的庞加莱-伯克霍夫定理，或者（ii）族可以形成大弯曲。结合这些结果允许描述的4D相空间中的定期环面的层次结构类似于在2D地图中的岛屿周围的岛屿层次。
英文摘要：
For the case of generic 4D symplectic maps with a mixed phase space we investigate the global organization of regular tori. For this we compute elliptic 1-tori of two coupled standard maps and display them in a 3D phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4D phase space. The 1-tori occur in two types of one-parameter families: (a) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (b) families originating from rank-1 resonances. At resonance gaps of both types of families either (i) periodic orbits exist, similar to the Poincare-Birkhoff theorem for 2D maps, or (ii) the family may form large bends. In combination these results allow for describing the hierarchical structure of regular tori in the 4D phase space analogously to the islands-around-islands hierarchy in 2D maps.
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