Elliptic Bubbles in Moser's 4D Quadratic Map: the Quadfurcation

TL;DR

This paper analyzes a specific bifurcation in 4D quadratic symplectic maps, called quadfurcation, revealing how it organizes bounded dynamics and creates fixed points with complex stability properties.

Contribution

It introduces the concept of quadfurcation in Moser's 4D quadratic map, detailing its role in fixed point creation and stability, extending understanding of 4D symplectic dynamics.

Findings

Quadfurcation creates four fixed points with diverse stability.

Doubly-elliptic fixed points can be surrounded by invariant tori.

The phenomenon relates to bifurcations in coupled Hénon maps and 4D standard maps.

Abstract

Moser derived a normal form for the family of four-dimensional, quadratic, symplectic maps in 1994. This six-parameter family generalizes H\'enon'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 H\'enon 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 be 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 H\'enon maps near their saddle-center bifurcations. The quadfurcation also occurs in the creation of accelerator modes in a 4D standard map.