SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 442-479, January 2020.
Moser derived a normal form for the family of four-dimensional (4D), quadratic, symplectic maps in 1994. This six-parameter family generalizes Hénon's ubiquitous 2D map and provides a local approximation for the dynamics of more general 4D maps. We show that the bounded dynamics of Moser's family is organized by a codimension-three bifurcation that creates four fixed points---a bifurcation analogous to a doubled, saddle-center---which we call a quadfurcation. In some sectors of parameter space a quadfurcation creates four fixed points from none, and in others it is the collision of a pair of fixed points that re-emerge as two or possibly four. In the simplest case the dynamics is similar to the cross product of a pair of Hénon maps, but more typically the stability of the created fixed points does not have this simple form. Up to two of the fixed points can be doubly elliptic and surrounded by bubbles of invariant two-tori; these dominate the set of bounded orbits. The quadfurcation can also create one or two complex-unstable (Krein) fixed points. Special cases of the quadfurcation correspond to a pair of weakly coupled Hénon maps near their saddle-center bifurcations.

中文翻译：

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摩泽尔（Moser）的4D二次图中的椭圆形气泡：四叉形

SIAM应用动力系统杂志，第19卷，第1期，第442-479页，2020年1月。
Moser于1994年推导了四维（4D）二次辛映射图族的标准形式。该六参数族概括了Hénon普遍存在的2D图，并为更通用的4D图的动力学提供了局部近似。我们证明了Moser家族的有界动力学是由一个三维的分叉组织的，它产生了四个固定点-一个类似于加倍的鞍形中心的分叉-我们称之为四分叉。在参数空间的某些扇区中，四叉形从一个都不产生四个固定点，而在另一些参数空间中，则是一对固定点的碰撞重新出现为两个或四个。在最简单的情况下，动力学类似于一对Hénon映射的叉积，但是更典型地，创建的固定点的稳定性没有这种简单形式。最多两个固定点可以是双椭圆形，并由不变的两个托里的气泡围绕。这些支配着有界轨道的集合。四叉也可以创建一个或两个复数不稳定（Krein）不动点。四分叉的特殊情况对应于鞍中心分叉附近的一对弱耦合的Hénon映射。