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

Abstract

Moser derived a normal form for the family of four-dimensional, 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 codimensionthree 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 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 Hénon maps near their saddle-center bifurcations. The quadfurcation also occurs in the creation of accelerator modes in a 4d standard map.

Moser’s Quadratic, Symplectic Map

“…There are at most four fixed points except when ε 2 = 0, which has a line of fixed points when a = b = c = 0. For simplicity we will assume in this paper that ε 2 = 0 (the case ε 2 = 0 is discussed in [3]).…”

“…The different stability combinations that arise from quadfurcations along lines in (a, b, c) space, like that in (14), are summarized in Tab. I, see [3] for details.…”

“…In contrast, the dynamics of the Moser map for n ≥ 2 have not yet been explored. In [3] we started the investigation of the n = 2 case, i.e., the four-dimensional quadratic symplectic map. Perhaps one of the most intriguing of its features is the possibility of a quadfurcation, in which four new solutions are created out-of-nowhere under parameter variation.…”

“…By applying a similar coordinate change, Moser showed in [1] that the quadratic symplectic map in R 4 can generically be written in an analogous normal form. Transforming this slightly by shifting the coordinates and parameters [3], this map can be written as (ξ , η ) = M (ξ, η) = (ξ + C −T (−η + Cξ + ∇U (ξ)), Cξ).…”

“…As shown in [3] the choice of sign for β and γ is unimportant. Note that (8) has real solutions only when µ 2 ≥ 4(αδ − ε 1 ), and that C is symmetric, β = γ = µ/2, only at the lower bound of this inequality.…”