Michel Vittot

Control of stochasticity in magnetic field lines

We present a method of control which is able to create barriers to magnetic field line diffusion by a small modification of the magnetic perturbation. This method of control is based on a localized control of chaos in Hamiltonian systems. The aim is to modify the perturbation (of order e) locally by a small control term (of order e2) which creates invariant tori acting as barriers to diffusion for Hamiltonian systems with two degrees of freedom. The location of the invariant torus is enforced in the vicinity of the chosen target (at a distance of order e due to the angle dependence). Given the importance of confinement in magnetic fusion devices, the method is applied to two examples with a loss of magnetic confinement. In the case of locked tearing modes, an invariant torus can be restored that aims at showing the current quench and therefore the generation of runaway electrons. In the second case, the method is applied to the control of stochastic boundaries allowing one to define a transport barrier within the stochastic boundary and therefore to monitor the volume of closed field lines.

Controlling chaotic transport in a Hamiltonian model of interest to magnetized plasmas

We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a model that reproduces turbulent E × B drift and show numerically that the control is able to drastically reduce chaotic transport.

Localized control for non-resonant Hamiltonian systems

We present a method of localized control of chaos in Hamiltonian systems. The aim is to modify the perturbation locally using a small control term which makes the controlled Hamiltonian more regular. We provide an explicit expression for the control term which is able to recreate invariant (KAM) tori without modifying other parts of phase space. We apply this method of localized control to a forced pendulum model, to the delta-kicked rotor (standard map) and to a non-twist Hamiltonian.

EFFICIENT CONTROL OF ACCELERATOR MAPS

Recently, the Hamiltonian Control Theory was used in [Boreux et al., 2012] to increase the dynamic aperture of a ring particle accelerator having a localized thin sextupole magnet. In this paper, these results are extended by proving that a simplified version of the obtained general control term leads to significant improvements of the dynamic aperture of the uncontrolled model. In addition, the dynamics of flat beams based on the same accelerator model can be significantly improved by a reduced controlled term applied in only one degree of freedom.

Perturbation of an eigenvalue from a dense point spectrum: an example

We study an example of a perturbed Floquet Hamiltonian depending on a coupling constant . The spectrum is pure point and dense. We pick up an eigenvalue, namely , and show the existence of a function defined on such that for all , 0 is a point of density for the set I, and the Rayleigh - Schrodinger perturbation series represents an asymptotic series for the function . All ideas are developed and demonstrated when treating the explicit example, but some of them are expected to have an essentially wider range of application.

Stabilizing the intensity for a Hamiltonian model of the FEL

Abstract The intensity of an electromagnetic wave interacting self-consistently with a beam of charged particles, as in a Free Electron Laser, displays large oscillations due to an aggregate of particles, called the macro-particle. In this article, we propose a strategy to stabilize the intensity by destabilizing the macro-particle. This strategy involves the study of the linear stability of a specific periodic orbit of a mean-field model. As a control parameter—the amplitude of an external wave—is varied, a bifurcation occurs in the system which has drastic effects on the self-consistent dynamics, and in particular, on the macro-particle. We show how to obtain an appropriate tuning of the control parameter which is able to strongly decrease the oscillations of the intensity without reducing its mean-value.

Control of test particle transport in a turbulent electrostatic model of the Scrape-Off-Layer

The Ex B drift motion of charged test particle dynamics in the Scrape-Off-Layer (SOL) is analyzed to investigate a transport control strategy based on Hamiltonian dynamics. We model SOL turbulence using a 2D non-linear fluid code based on interchange instability which was found to exhibit intermittent dynamics of the particle flux. The effect of a small and appropriate modification of the turbulent electric potential is studied with respect to the chaotic diffusion of test particle dynamics. Over a significant range in the magnitude of the turbulent electrostatic field, a three-fold reduction of the test particle diffusion coefficient is achieved.

A Simple and Compact Presentation of Birkhoff Series

A new, and more algebraic method for the construction of the perturbation series of the classical mechanics is given. The use of Poisson bracket operators yields a compact expression, which actually is a formal summation of the recurrence formulas usually obtained for the normal form of a quasi-integrable hamiltonian.

Hamiltonian description of a self-consistent interaction between charged particles and electromagnetic waves

The Hamiltonian description of the self-consistent interaction between an electromagnetic plane wave and a copropagating beam of charged particles is considered. We show how the motion can be reduced to a one-dimensional Hamiltonian model (in a canonical setting) from the Vlasov-Maxwell Poisson brackets. The reduction to this paradigmatic Hamiltonian model is performed using a Lie algebraic formalism which allows us to preserve the Hamiltonian character at each step of the derivation.

A Hamiltonian system for interacting Benjamin–Feir resonances

In this paper, we present a model describing the time evolution of two-dimensional surface waves in gravity and infinite depth. The model of six interacting modes derives from the normal form of the system describing the dynamics of surface waves and is governed by a Hamiltonian system of equations of cubic order in the amplitudes of the waves. We derive a Hamiltonian system with two degrees of freedom from this Hamiltonian using conserved quantities. The interactions are those of two coupled Benjamin–Feir resonances. The temporal evolution of the amplitude of the different modes is described according to the parameters of the system. In particular, we study the energy exchange produced by the modulations of the amplitudes of the modes. The evolution of the modes reveals a chaotic dynamics.