Moser's quadratic, symplectic map

TL;DR

This paper studies Moser's four-dimensional quadratic symplectic map, revealing a complex bifurcation structure called quadfurcation that creates multiple fixed points and analyzing the associated invariant tori and dynamics.

Contribution

It identifies and characterizes a new codimension-three bifurcation, quadfurcation, that organizes the bounded dynamics of Moser's map, including the creation of fixed points.

Findings

Bounded dynamics are organized by a quadfurcation bifurcation.

Moser's map can have two doubly elliptic fixed points, unlike uncoupled Hénon maps.

Invariant tori form Cantor families around fixed points.

Abstract

In 1994, J\"urgen Moser generalized H\'enon's area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser's six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none. The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser's map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled H\'enon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2D planes through the phase space and by 3D slices through the tori.