Bifurcations of families of 1D-tori in 4D symplectic maps

Onken, Franziska; Lange, Steffen; Ketzmerick, Roland; Bäcker, Arnd

doi:10.1063/1.4954024

Cited by 20 publications

(19 citation statements)

References 60 publications

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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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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%

“…Lower inset: trapped orbit (blue dots) and regular phase space structures (gray) in a 3d phase-space slice representation. zation in terms of families of elliptic 1d tori [56].…”

Section: Introductionmentioning

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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“…The families of the elliptic and hyperbolic 1d-tori build up the skeleton of the resonance channel [72]. The regular structures form a tilted tube in phase space as shown in Fig.…”

Section: A Single Coupled Resonancementioning

confidence: 99%

Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians

et al. 2019

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Nonlinear resonances in the classical phase space lead to a significant enhancement of tunneling. We demonstrate that the double resonance gives rise to a complicated tunneling peak structure. Such double resonances occur in Hamiltonian systems with an at least four-dimensional phase space. To explain the tunneling peak structure, we use the universal description of single and double resonances by 4d normal-form Hamiltonians. By applying perturbative methods, we reveal the underlying mechanism of enhancement and suppression of tunneling and obtain excellent quantitative agreement. Using a minimal matrix, we obtain model an intuitive understanding.

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Bifurcations of families of 1D-tori in 4D symplectic maps

Cited by 20 publications

References 60 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

Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians

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