Homoclinic points of 2D and 4D maps via the parametrization method

Anastassiou, Stavros; Bountis, Tassos; Bäcker, Arnd

doi:10.1088/1361-6544/aa7e9b

Cited by 13 publications

(11 citation statements)

References 45 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%

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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“…that satisfies the condition (5) globally, over all lattice sites. For a finite lattice L one needs to specify the boundary conditions (bc's).…”

Section: Deterministic Lattice Field Theorymentioning

confidence: 99%

“…Each lattice state is a distinct deterministic solution Φ c to the discretized Euler-Lagrange equations (5), so its probability is a N L -dimensional Dirac delta function (that's what we mean by the system being deterministic)…”

Section: Deterministic Lattice Field Theorymentioning

confidence: 99%

“…As is case for a WKB approximation [92], the deterministic field theory partition sum has support only on lattice field values that are solutions to the saddle-point condition (5), and the partition function ( 3) is now a sum over configuration state space (2) points, what in theory of dynamical systems is called the 'deterministic trace formula' [49],…”

Section: Deterministic Lattice Field Theorymentioning

confidence: 99%

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A chaotic lattice field theory in one dimension

Liang,

Cvitanović

2022

Preprint

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Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB 'skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no 'Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations.The reformulation aligns 'chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the timeperiodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells.The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, 'time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

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Homoclinic points of 2D and 4D maps via the parametrization method

Cited by 13 publications

References 45 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

A chaotic lattice field theory in one dimension

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