Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

Dullin, Holger R.; Meiss, James D.

doi:10.1137/080728160

Cited by 27 publications

(35 citation statements)

References 46 publications

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“…However, the overall behaviour bears some similarity to a bifurcation documented by Dullin & Meiss (2013, figure 20) in a three-dimensional generalization of the Hénon map. There it is also confined to a narrow parameter range.…”

Section: Stability Of the Central Orbitsupporting

confidence: 62%

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Chaotic advection in a steady, three-dimensional, Ekman-driven eddy

et al. 2013

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We investigate and quantify stirring due to chaotic advection within a steady, three-dimensional, Ekman-driven, rotating cylinder flow. The flow field has vertical overturning and horizontal swirling motion, and is an idealization of motion observed in some ocean eddies. The flow is characterized by strong background rotation, and we explore variations in Ekman and Rossby numbers, E and R o , over ranges appropriate for the ocean mesoscale and submesoscale. A high-resolution spectral element model is used in conjunction with linear analytical theory, weakly nonlinear resonance analysis and a kinematic model in order to map out the barriers, manifolds, resonance layers and other objects that provide a template for chaotic stirring. As expected, chaos arises when a radially symmetric background state is perturbed by a symmetrybreaking disturbance. In the background state, each trajectory lives on a torus and some of the latter survive the perturbation and act as barriers to chaotic transport, a result consistent with an extension of the KAM theorem for three-dimensional, volume-preserving flow. For shallow eddies, where E is O(1), the flow is dominated by thin resonant layers sandwiched between KAM-type barriers, and the stirring rate is weak. On the other hand, eddies with moderately small E experience thicker resonant layers, wider-spread chaos and much more rapid stirring. This trend reverses for sufficiently small E, corresponding to deep eddies, where the vertical rigidity imposed by strong rotation limits the stirring. The bulk stirring rate, estimated from a passive tracer release, confirms the non-monotonic variation in stirring rate with E. This result is shown to be consistent with linear Ekman layer theory in conjunction with a resonant width calculation and the Taylor-Proudman theorem. The theory is able to roughly predict the value of E at which stirring is maximum. For large disturbances, the stirring rate becomes monotonic over the range of Ekman numbers explored. We also explore variation in the eddy aspect ratio.

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Section: Stability Of the Central Orbitsupporting

confidence: 62%

“…Further inquiry is hindered by the fact that the action-angle-angle variable set is singular there. Dullin & Meiss (2013) Note that for the boundary condition (2.4), v T (r) = 4r(1 − r).…”

Section: Discussionmentioning

confidence: 99%

Chaotic advection in a steady, three-dimensional, Ekman-driven eddy

et al. 2013

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“…We observe that local 2d projections of the bifurcations in 3d phase-space slices remarkably resemble phase-space plots of bifurcations of periodic orbits in 2d maps. This is consistent with normal form results for quasi-periodically forced oscillators 36 and investigations in 3d volume-preserving maps 47 . Moreover, the families of 1d-tori arising from a bifurcation due to a crossing resonance are also the skeleton of this resonance channel.…”

Section: Introductionsupporting

confidence: 91%

Bifurcations of families of 1D-tori in 4D symplectic maps

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Lange

Ketzmerick

et al. 2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The regular structures of a generic 4d symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1d-tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations, no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example, we study two coupled standard maps by visualizing the elliptic and hyperbolic 1d-tori in a 3d phase-space slice, local 2d projections, and frequency space. The observed bifurcations are consistent with the analytical predictions previously obtained for quasi-periodically forced oscillators. Moreover, the new families emerging from such a bifurcation form the skeleton of the corresponding resonance channel.

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“…For these values of the parameters some of the bubble structure of the flow is preserved. Namely, the 2D invariant manifolds of Q l and Q r (which do not coincide), bound a Cantor family of invariant tori that enclose, for most values of the parameters ϕ and a, an elliptic invariant circle [18].…”

Section: The Hopf-one Bifurcation In Volume-preserving Mapsmentioning

confidence: 99%

Accelerator modes and anomalous diffusion in 3D volume-preserving maps

et al. 2018

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Angle-action maps that are periodic in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a volume-preserving family of maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map.

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Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

Cited by 27 publications

References 46 publications

Chaotic advection in a steady, three-dimensional, Ekman-driven eddy

Chaotic advection in a steady, three-dimensional, Ekman-driven eddy

Bifurcations of families of 1D-tori in 4D symplectic maps

Accelerator modes and anomalous diffusion in 3D volume-preserving maps

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