On reconnection phenomena in the standard nontwist map

Wurm, Alexander; Apte, Amit; Morrison, Philip

doi:10.1590/s0103-97332004000800035

Cited by 25 publications

(21 citation statements)

References 24 publications

(40 reference statements)

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“…The behavior is different according to the period p of the orbit related with the island. 23 If the orbits have even p, then the orbits on each side of the shearless curve have the same stability type, i.e., if one is a center ͑saddle͒ the other is a also a center ͑saddle͒. However, if p is odd, then the stability types are different on both sides of the shearless curve, as exemplified by Fig.…”

Section: Reconnection and Barrier Destructionmentioning

confidence: 98%

“…22 Since the SNM describes a nonintegrable system, chaotic orbits occur with any nonzero perturbation, and a great deal of information has been obtained from a thorough analysis of the ͑a , b͒ parameter plane. 5,6,23,24 A physically meaningful question is to what extent do the peculiar characteristics of the SNM affect the diffusion ͑in action͒ that occurs in the chaotic orbits. In the applications of SNM listed above, the formation of a transport barrier located in the neighborhood of the shearless curve has been observed.…”

Section: ͑11͒mentioning

confidence: 99%

“…I, the SNM plays a role for nontwist systems similar to that played by the SM for twist systems. 5,6,23 However, proofs of many important results of Hamiltonian nonlinear dynamics, such as the KolmogorovArnold-Moser ͑KAM͒ theorem, the Poincaré-Birkhoff theorem, and Aubry-Mather theory, may require the twist condition. 2,25,26 Hence, many phenomena typical of twist maps do not occur in nontwist maps, and new features may also occur.…”

Section: Reconnection and Barrier Destructionmentioning

confidence: 99%

“…Such non-KAM tori, also called meanders, are not graphs over the x-axis. 23,27 Accordingly, we call the region occupied by the collection of meanders the meandering region, which appears in the neighborhood of the shearless curve.…”

Section: Reconnection and Barrier Destructionmentioning

confidence: 99%

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Transport properties in nontwist area-preserving maps

Szezech

Caldas

Lopes

et al. 2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds. © 2009 American Institute of Physics. ͓doi:10.1063/1.3247349͔Nonmonotonic flows with reverse shear are observed in many physical systems. Much previous research indicates that transport properties of such systems can be well described by area preserving maps. Many examples exist, e.g., in the fields of fluid mechanics and plasma physics; in particular, in models that describe zonal flows that occur in geophysics, atmospheric science, and fusion plasma physics. The standard nontwist map (SNM) is a well-known paradigm for investigating transport in reverse shear systems. The nonmonotonicity property of the SNM gives rise to transport barriers due to robust tori (invariant curves) that occur in zonal flows. These tori separate regions of the two-dimensional phase space. Moreover, the influence of these barriers on the transport remains even after the breakup of the tori. This phenomenon, i.e., the difficulty encountered in crossing broken barriers, is explained by examining the stickiness of orbits that occur in some regions of the map phase space. For a certain range of control parameters, these regions emerge near resonances. The presence of stickiness is closely related to the structure of the stable and unstable manifolds of hyperbolic orbits, becoming prevalent when the manifolds reconnect and change from a dominant homoclinic tangle to a combination that includes both homoclinic and heteroclinic tangles.

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Section: Reconnection and Barrier Destructionmentioning

confidence: 98%

Section: ͑11͒mentioning

confidence: 99%

Section: Reconnection and Barrier Destructionmentioning

confidence: 99%

Section: Reconnection and Barrier Destructionmentioning

confidence: 99%

See 2 more Smart Citations

Transport properties in nontwist area-preserving maps

Szezech

Caldas

Lopes

et al. 2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…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]. In our study we will derive a generalization of this map for the four dimensional case.…”

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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