Global structure of regular tori in a generic 4D symplectic map

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.

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“…As grows these loops thicken into annuli in the slice. For further examples and detailed discussion see [41][42][43][44][45][46].…”

Section: Elliptic Bubblesmentioning

confidence: 99%

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

Bäcker

Meiss²

2020

SIAM J. Appl. Dyn. Syst.

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“…For further illustrations and discussions of the 3d phase-space slice representation see Refs. [53,55,56,114]. Fig.…”

Section: B 3d Phase Space Slicementioning

confidence: 99%

“…The regular 2d tori in the phase space of a 4d map are organized around families of elliptic 1d tori [55,56]. Most prominently one has the so-called Lyapunov families [115][116][117] of elliptic 1d tori which emanate from the central elliptic-elliptic fixed point u fp .…”

Section: B 3d Phase Space Slicementioning

confidence: 99%

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

et al. 2018

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The dynamics in three-dimensional (3D) billiards leads, using a Poincaré section, to a four-dimensional map, which is challenging to visualize. By means of the recently introduced 3D phase-space slices, an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincaré recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories, which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps, we find that (i) trapping takes place close to regular structures outside the Arnold web, (ii) trapping is not due to a generalized island-around-island hierarchy, and (iii) the dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.

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“…Whenever the points of an orbit, given by a sequence of points (ξ 1 , ξ 2 , χ 1 , χ 2 ), lie in this slice, the three nontrivial coordinates (ξ 1 , ξ 2 , χ 1 ) are displayed in a 3d plot. For further examples and detailed discussion see [33][34][35][36]. Figure 4 shows a 3d phase space slice plot combined with an escape time plot for a grid of initial conditions in the 2d-plane χ = 0, for the quadfurcation of Fig.…”

Section: Geometry -Elliptic Bubblesmentioning

confidence: 99%

Moser’s Quadratic, Symplectic Map

Bäcker

Meiss

2018

Regul. Chaot. Dyn.

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In 1994, Jürgen Moser generalized Hénon'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 Hénon 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.MSC2010 number: 37J40, 70H08, 34C28, 37C05

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Global structure of regular tori in a generic 4D symplectic map

Cited by 34 publications

References 65 publications

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

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

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

Moser’s Quadratic, Symplectic Map

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