Generic twistless bifurcations

Dullin, Holger R.; Meiss, James D.; Sterling, David

doi:10.1088/0951-7715/13/1/310

Cited by 51 publications

(96 citation statements)

References 19 publications

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“…4͑d͒, where we depict a magnification of the vicinity of a selected island belonging to the lower chain, in which a second-order "period-3 chain" has been formed ͑actually with period 3 ϫ 11͒. For such period-3 phenomena, it has been shown that generically there exist shearless curves, 28 This strong dependence on initial conditions and associated lingering adjacent to island chains, also called stickiness, has been extensively studied in twist systems ͑see, for example, Ref. 30͒.…”

Section: Diffusion and Escape Of Trajectoriesmentioning

confidence: 99%

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: Diffusion and Escape Of Trajectoriesmentioning

confidence: 99%

Transport properties in nontwist area-preserving maps

Szezech

Caldas

Lopes

et al. 2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Note that the considered pairs of parameters ( , α) are located above the curve α * ( ), which is the curve of triplication of the elliptic fixed point. According to [43,44] the corresponding Poincaré map is in these cases a nontwist map. The sequence of detected local and global bifurcations is typical for such a map [45][46][47][48].…”

Section: The Role Of the Total Driving Amplitudementioning

confidence: 99%

Complex dynamics in a simple model of pulsations for super-asymptotic giant branch stars

Munteanu

Garcı́a–Berro

José

et al. 2002

Chaos: An Interdisciplinary Journal of Nonlinear Science

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When intermediate mass stars reach their last stages of evolution they show pronounced oscillations. This phenomenon happens when these stars reach the so-called Asymptotic Giant Branch (AGB), which is a region of the Hertzsprung-Russell diagram located at about the same region of effective temperatures but at larger luminosities than those of regular giant stars. The period of these oscillations depends on the mass of the star. There is growing evidence that these oscillations are highly correlated with mass loss and that, as the mass loss increases, the pulsations become more chaotic. In this paper we study a simple oscillator which accounts for the observed properties of this kind of stars. This oscillator was first proposed and studied in [1] and we extend their study to the region of more massive and luminous stars -the region of Super-AGB stars. The oscillator consists of a periodic nonlinear perturbation of a linear Hamiltonian system. The formalism of dynamical systems theory has been used to explore the associated Poincaré map for the range of parameters typical of those stars. We have studied and characterized the dynamical behaviour of the oscillator as the parameters of the model are varied, leading us to explore a sequence of local and global bifurcations. Among these, a tripling bifurcation is remarkable, which allows us to show that the Poincaré map is a nontwist area preserving map. Meandering curves, hierarchical-islands traps and sticky orbits also show up. We discuss the implications of the stickiness phenomenon in the evolution and stability of the Super-AGB stars.

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“…Twist singularities are are not unusual; for example, the frequency is not a monotone function of action in any system that has a pair of nested separatrices [MB99,Mor02]. This "fold" singularity also generically occurs in any system near tripling resonances [Moe90,DMS99,DM03]. In this paper we will discuss both the fold and cusp singularities and some of the dynamical consequences of the breakdown of twist.…”

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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Generic twistless bifurcations

Cited by 51 publications

References 19 publications

Transport properties in nontwist area-preserving maps

Transport properties in nontwist area-preserving maps

Complex dynamics in a simple model of pulsations for super-asymptotic giant branch stars

Normal forms for 4D symplectic maps with twist singularities

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