The non-twist standard map with robust tori

Martins, Caroline G. L.; Carvalho, R. Egydio de; Caldas, Iberê L.; Roberto, Marisa

doi:10.1088/1751-8113/43/17/175501

Cited by 12 publications

(11 citation statements)

References 35 publications

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“…Note that this shows that nontwist tori are more stable under non stationary perturbations, which gives a rigorous meaning to the observation that non-twist KAM tori are "more robust". See [MEdCCR10,dCNGM96,Mor00,Eas91]. We also note that the models in [FW14], satisfy (22) since the unperturbed systems is a harmonic oscillator.…”

Section: 2mentioning

confidence: 80%

KAM tori and whiskered invariant tori for non-autonomous systems

Canadell

Llave

2015

Physica D: Nonlinear Phenomena

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We consider non-autonomous dynamical systems which converge to autonomous (or periodic) systems exponentially fast in time. Such systems appear naturally as models of many physical processes affected by external pulses. We introduce definitions of non-autonomous invariant tori and nonautonomous whiskered tori and their invariant manifolds and we prove their persistence under small perturbations, smooth dependence on parameters and several geometric properties (if the systems are Hamiltonian, the tori are Lagrangian manifolds). We note that such definitions are problematic for general time-dependent systems, but we show that they are unambiguous for systems converging exponentially fast to autonomous. The proof of persistence relies only on a standard implicit function theorem in Banach spaces and it does not require that the rotations in the tori are Diophantine nor that the systems we consider preserve any geometric structure. We only require that the autonomous system preserves these objects. In particular, when the autonomous system is integrable, we obtain the persistence of tori with rational rotational. We also discuss fast and efficient algorithms for their computation. The method also applies to infinite dimensional systems which define a good evolution, e.g. PDE's. When the systems considered are Hamiltonian, we show that the time dependent invariant tori are isotropic. Hence, the invariant tori of maximal dimension are Lagrangian manifolds. We also obtain that the (un)stable manifolds of whiskered tori are Lagrangian manifolds. We also include a comparison with the more global theory developed in [BdlL11].

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Section: 2mentioning

confidence: 80%

KAM tori and whiskered invariant tori for non-autonomous systems

Canadell

Llave

2015

Physica D: Nonlinear Phenomena

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“…Magnetic field lines in toroidal plasma devices can be described by area preserving nontwist maps. 3 Martins and collaborators 4 used the SNM to study transport barriers in plasmas confined in tokamaks. Studies concerning internal transport barriers are important to improve plasma a) Electronic mail: antoniomarcosbatista@gmail.com confinement and stability properties.…”

Section: Introductionmentioning

confidence: 99%

Recurrence-based analysis of barrier breakup in the standard nontwist map

Santos

Mugnaine

Szezech

et al. 2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study the standard nontwist map that describes the dynamic behaviour of magnetic field lines near a local minimum or maximum of frequency. The standard nontwist map has a shearless invariant curve that acts like a barrier in phase space. Critical parameters for the breakup of the shearless curve have been determined by procedures based on the indicator points and bifurcations of periodical orbits, a methodology that demands high computational cost. To determine the breakup critical parameters, we propose a new simpler and general procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition. We also show that the coexistence of islands and chaotic sea in phase space can be analysed by using the recurrence plot. In particular, the measurement of determinism from the recurrence plot provides us with a simple procedure to distinguish periodic from chaotic structures in the parameter space. We identify an invariant shearless breakup scenario, and we also show that recurrence plots are useful tools to determine the presence of periodic orbit collisions and bifurcation curves.

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“…In some cases, the barrier acts only locally, not affecting regions far from it in phase space 21,[33][34][35] . In other methods, the barrier is created in such a way that it is able to alter the structures of phase space in regions far from it [36][37][38][39] . Some methods of control of chaos have already been used to control chaotic transport in plasmas confined by a) meirielenso@gmail.com b) ibere@if.usp.br magnetic fields 21,30,31 , to control chaotic transport in turbulent electric fields 34 , and to regularize particle dynamics in wave-particle interactions 9,32,35 .…”

Section: Introductionmentioning

confidence: 99%

Improving particle beam acceleration in plasmas

Sousa

Caldas

2018

Physics of Plasmas

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The dynamics of wave-particle interactions in magnetized plasmas restricts the wave amplitude to moderate values for particle beam acceleration from rest energy. We analyze how a perturbing invariant robust barrier modifies the phase space of the system and enlarges the wave amplitude interval for particle acceleration.For low values of the wave amplitude, the acceleration becomes effective for particles with initial energy close to the rest energy. For higher values of the wave amplitude, the robust barrier controls chaos in the system and restores the acceleration process. We also determine the best position for the perturbing barrier in phase space in order to increase the final energy of the particles.

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The non-twist standard map with robust tori

Cited by 12 publications

References 35 publications

KAM tori and whiskered invariant tori for non-autonomous systems

KAM tori and whiskered invariant tori for non-autonomous systems

Recurrence-based analysis of barrier breakup in the standard nontwist map

Improving particle beam acceleration in plasmas

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