Nonmonotonic twist maps

Howard, James E.; Humpherys, Jeffrey

doi:10.1016/0167-2789(94)00180-x

Cited by 86 publications

(73 citation statements)

References 15 publications

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“…When the map is perturbed by a small periodic perturbation interesting dynamics is expected when the fold is near a resonant frequency such that the resonance is in the image of the frequency map. This situation is well studied for the case n = 1 where the map is area-preserving and leads to a simple one-degreeof freedom Hamiltonian model [HH95,Sim98]. The canonical example is called the "standard nontwist map," for reviews see [AWM03,WAM04,WAFM05].…”

Section: Introductionmentioning

confidence: 99%

Normal forms for 4D symplectic maps with twist singularities

Dullin

Ivanov

Meiss

2006

Physica D: Nonlinear Phenomena

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We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time-T mapping of a two-degree-of freedom Hamiltonian flow. Consequently there is an energy-like invariant. The fold Hamiltonian is similar to the well-studied, one-degree-of freedom case but is essentially nonintegrable when the direction of the singular curve in action does not coincide with curves of the resonance module. We show that many familiar features, such as multiple island chains and reconnecting invariant manifolds, are retained even in this case. The cusp Hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module. Using averaging, we approximately reduced this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.

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Section: Introductionmentioning

confidence: 99%

Normal forms for 4D symplectic maps with twist singularities

Dullin

Ivanov

Meiss

2006

Physica D: Nonlinear Phenomena

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“…also studies of related maps [7,8]) i.e. systems in which the frequency of eigenoscillation possesses an extremum (typically, a local maximum or minimum) as a function of its energy.…”

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confidence: 99%

“…In such systems, there are typically two resonances of one and the same order [9], and their overlap in energy does not result in the onset of global chaos [5][6][7][8]. Even their overlap in phase space results typically, instead of the onset of global chaos, only in reconnection of the thin chaotic layers associated with the resonances while, as the amplitude of the time-periodic perturbation grows further, the layers separate again (with a different topology [5][6][7][8]) despite the growth of the width of the overall relevant range of energy.The major idea of the present work is to limit the energy range relevant to the overlap of resonances of the same order by chaotic layers relating either to resonances of a different order [10] or to separatrices of the unperturbed system (these scenarios immediately suggest the corresponding substitute for the Chirikov criterion in a ZD system in the range of extremum). The onset of global chaos occurs respectively in a broader energy range or at unusually small magnitudes of the perturbation.…”

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confidence: 99%

“…systems in which the frequency of eigenoscillation possesses an extremum (typically, a local maximum or minimum) as a function of its energy. In such systems, there are typically two resonances of one and the same order [9], and their overlap in energy does not result in the onset of global chaos [5][6][7][8]. Even their overlap in phase space results typically, instead of the onset of global chaos, only in reconnection of the thin chaotic layers associated with the resonances while, as the amplitude of the time-periodic perturbation grows further, the layers separate again (with a different topology [5][6][7][8]) despite the growth of the width of the overall relevant range of energy.…”

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Drastic Facilitation of the Onset of Global Chaos

2003

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The Chirikov resonance-overlap criterion predicts the onset of global chaos if nonlinear resonances overlap in energy, which is conventionally assumed to require a non-small magnitude of perturbation. We show that, for a time-periodic perturbation, the onset of global chaos may occur at unusually small magnitudes of perturbation if the unperturbed system possesses more than one separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with separatrices of the unperturbed system. One of the most important manifestations of this effect is a drastic increase of the energy range involved into the unbounded chaotic transport in spatially periodic system driven by a rather weak time-periodic force, provided the driving frequency approaches the extremal eigenfrequency or its harmonics. We develop the asymptotic theory and verify it in simulations. 05.45. Ac, 05.45.Pq, 75.70.cn A weak perturbation of a Hamiltonian system causes the onset of chaotic layers in the phase space around separatrices of the unperturbed system and/or separatrices surrounding nonlinear resonances generated by the perturbation itself [1][2][3]: the system may be transported along the layer random-like. Chaotic transport plays an important role in many physical phenomena [3]. If the perturbation is weak enough, then the layers are thin and such chaos is called local [1][2][3]. As the magnitude of the perturbation increases, the widths of the layers grow and the layers corresponding to adjacent separatrices (either to those of the unperturbed system, or to separatrices surrounding different nonlinear resonances) reconnect at some, typically non-small, critical value of the magnitude, which conventionally marks the onset of global chaos [1][2][3] i.e. chaos in a large range of the phase space which increases with a further increase of the magnitude of perturbation. Such scenario often correlates with the overlap in energy between neighbouring resonances calculated independently in the resonant approximations of the corresponding orders: this constitutes the celebrated empirical Chirikov resonance-overlap criterion [1][2][3].But the Chirikov criterion may fail in time-periodically perturbed zero-dispersion (ZD) systems [4-6] (cf. also studies of related maps [7,8]) i.e. systems in which the frequency of eigenoscillation possesses an extremum (typically, a local maximum or minimum) as a function of its energy. In such systems, there are typically two resonances of one and the same order [9], and their overlap in energy does not result in the onset of global chaos [5][6][7][8]. Even their overlap in phase space results typically, instead of the onset of global chaos, only in reconnection of the thin chaotic layers associated with the resonances while, as the amplitude of the time-periodic perturbation grows further, the layers separate again (with a different topology [5][6][7][8]) despite the growth of the width of the overa...

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“…Howard and Humpherys extended the study to cubic and quartic nontwist maps. [27] These reconnection scenarios had been conjectured by Stix [8] in the context of the evolution of magnetic surfaces in the nonlinear double-tearing instability, and were seen by Gerasimov et al [28] in a two-dimensional model of the beam-beam interaction in a storage ring.…”

Section: Introductionmentioning

confidence: 99%

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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Nonmonotonic twist maps

Cited by 86 publications

References 15 publications

Normal forms for 4D symplectic maps with twist singularities

Normal forms for 4D symplectic maps with twist singularities

Drastic Facilitation of the Onset of Global Chaos

Meanders and reconnection–collision sequences in the standard nontwist map

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