The Breakup Condition of Shearless KAM Curves in the Quadratic Map

Shinohara, Susumu; Aizawa, Yōji

doi:10.1143/ptp.97.379

Cited by 42 publications

(36 citation statements)

References 5 publications

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“…[46] and ours deliver similar results. However, it seems that the computation of the winding number provides better means of monitoring and controlling its accuracy (aside from giving us the winding number of the shearless curve as a useful side-product).…”

Section: E Break-up Diagramsupporting

confidence: 83%

“…[16,17,47] Though the method is very precise, it is not suitable for an exploration of all parameter space. Shinohara and Aizawa [46] obtained a rough estimate for the break-up threshold of many shearless curves by investigating for a range of parameter values whether iterates of one of the indicator points remain bounded.…”

Section: E Break-up Diagrammentioning

confidence: 99%

“…The analysis of Refs. [16,17,46] indicates that the highest peaks correspond to the break-up of shearless invariant tori with noble winding numbers (see Refs. [16,47] for discussion).…”

Section: E Break-up Diagrammentioning

confidence: 99%

See 2 more Smart Citations

Meanders and reconnection–collision sequences in the standard nontwist map

Wurm

Apte

Fuchss

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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New global periodic orbit collision/separatrix reconnection scenarios in the standard nontwist map in different regions of parameter space are described in detail, including exact methods for determining reconnection thresholds that are implemented numerically. The results are compared to a break-up diagram of shearless invariant curves. The existence of meanders (invariant tori that are not graphs) is demonstrated numerically for both odd and even period reconnection for certain regions in parameter space, and some of the implications on transport are discussed.In recent years, area-preserving maps that violate the twist condition locally in phase space have been the object of interest in several studies in physics and mathematics. These nontwist maps show up in a variety of physical models, e.g., in magnetic field line models for reversed magnetic shear tokamaks. An important problem is the determination and understanding of the transition to global chaos (global transport) in these models. Nontwist maps exhibit several different mechanisms: the break-up of invariant tori and separatrix reconnections. The latter may or may not lead to global transport depending on the region of parameter space. In this paper we conduct a detailed study of newly discovered reconnection scenarios in the standard nontwist map, investigating their location in parameter space and their impact on global transport.

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Section: E Break-up Diagramsupporting

confidence: 83%

Section: E Break-up Diagrammentioning

confidence: 99%

See 1 more Smart Citation

Meanders and reconnection–collision sequences in the standard nontwist map

Wurm

Apte

Fuchss

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…13 shows that within (assumed) numerical uncertainty these values are the same as those for 1/γ, as predicted. The a c values used to determine δ 2 were a c [26] and a c [14] , which explains the larger discrepancy. Work is under way to improve this result.…”

Section: Eigenvaluesmentioning

confidence: 99%

Renormalization and destruction of 1/γ2 tori in the standard nontwist map

Apte

Wurm

Morrison

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D 91, 1 (1996) and 100, 311 (1997) on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator.In recent years, area-preserving maps that violate the twist condition locally in phase space have been the object of interest in several studies in physics and mathematics. These nontwist maps show up in a variety of physical models. An important problem from the physics point of view is the understanding of the breakup of invariant tori, which show remarkable resilience in the region where the twist condition is violated, called shearless tori. In terms of the physical system modelled, these tori represent transport barriers, and their breakup corresponds to the transition to global chaos. Mathematically, nontwist maps present a challenge since the standard proofs of celebrated theorems in the theory of area-preserving maps rely heavily on the twist condition. In this paper, we study the breakup of the shearless torus with winding number 1/γ 2 , where γ is the golden mean. This torus serves as a test case for improved techniques we developed. At the point of breakup the shearless torus exhibits universal scaling behavior which leads to a renormalization group interpretation.

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“…However, for another class of dynamical systems described by "non-twist maps", this is not generally the case. A new class of tori, termed "shearless tori" [2] can be present. The classical properties of these shearless tori are attracting considerable attention, due in part to their possible relevance to improved confinement of fusion plasmas in tokamaks [3], but also because the properties of non-twist maps are less well studied.…”

mentioning

confidence: 99%

Quantum transport and spin dynamics on shearless tori

Kudo

Monteiro

2008

Phys. Rev. E

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We investigate quantum dynamics in phase-space regions containing "shearless tori". We show that the properties of these peculiar classical phase-space structures -important to the dynamics of tokamaks -may be exploited for quantum information applications. In particular we show that shearless tori permit the non-dispersive transmission of localized wavepackets. The quantum manybody Hamiltonian of a Heisenberg ferromagnetic spin chain, subjected to an oscillating magnetic field, can be reduced to a classical one-body "image" dynamical system which is the well-studied Harper map. The Harper map belongs to a class of Hamiltonian systems (non-twist maps) which contain shearless tori. We show that a variant with sinusoidal time driving "driven Harper model" produces shearless tori which are especially suitable for quantum state transfer. The behavior of the concurrence is investigated as an example. For a Hamiltonian system, the onset of classical chaotic dynamics is associated with the disappearance of phasespace barriers termed invariant tori: when the last invariant torus disappears, the chaotic diffusive motion is unbounded ("global"). For a wide class of classical dynamical systems, described by so-called "twist-maps" and exemplified by the all-important Standard Map [1], there is a single threshold for this process, i.e., once the last invariant torus breaks, the dynamics is globally diffusive for all parameters above the threshold. However, for another class of dynamical systems described by "non-twist maps", this is not generally the case. A new class of tori, termed "shearless tori" [2] can be present. The classical properties of these shearless tori are attracting considerable attention, due in part to their possible relevance to improved confinement of fusion plasmas in tokamaks [3], but also because the properties of non-twist maps are less well studied. New dynamical phenomena such as separatrix reconnection lead to the intermittent re-appearance of the shearless tori which, when present, separate different regions of the chaotic phase space.In a quite different context, studies of quantum dynamics of spin chains, for example of quantum state transfer and entanglement generation, now play a central role in the burgeoning field of quantum information [4]. There is also growing interest in potential applications of nonlinear dynamics in quantum information. In particular, the realization that a many-body Hamiltonian can be analyzed with the dynamics of "one-body image" quantum and classical Hamiltonians [5,6] has proved very insightful. For example, the Heisenberg ferromagnetic spin chain, in a pulsed parabolic magnetic field, has the * Electronic address: kudo.kazue@ocha.ac.jp; Present address: Ochadai Academic Production, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan Standard Map as its classical image [6], provided position coordinates in the many-body Hamiltonian are mapped onto momenta of the image system and vice-versa. If, instead, a pulsed sinusoidal external magnetic fiel...

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The Breakup Condition of Shearless KAM Curves in the Quadratic Map

Cited by 42 publications

References 5 publications

Meanders and reconnection–collision sequences in the standard nontwist map

Meanders and reconnection–collision sequences in the standard nontwist map

Renormalization and destruction of 1/γ2 tori in the standard nontwist map

Quantum transport and spin dynamics on shearless tori

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