cpt proposals for master internships · this phenomena some control strategies have been devised,...

CPT proposals for Master internshipsAcademic year 2018-2019

Warning(!!): Due to a limited funding, the CPT director will fix the numberof financial rewardings up to a max of 6 internships.

(This warning doesn’t hold for students coming from French ENS).

The application file will be made with

• The internship topic title;

• A motivation letter;

• A 1-page short CV;

• Transcripts of the 1st Masters year;

and sent to the CPT director as a unique pdf file.

In addition: a recommendation letter can also be addressed to the CPT director.

mailto:[email protected]

Master 2 Internship

Centre de Physique Théorique, UMR 7332 and IRFM CEA

Address: Centre de Physique théorique, Campus de Luminy, Case 907, 13288 Marseille Cedex 09 &IRFM, CEA Cadarache 13108 St Paul lez Durance Cedex,

Name of advisers: Xavier Leoncini and Guilhem Dif-Pradalieremail: [email protected], [email protected]

Title : Chaotic motion of charged particles in magnetic �elds, from single trajectories to collective

equilibrium: applications to fusion plasmas.

General Context:In the last few years, the impact of low dimensional chaos in the motion of charged particles in ideal plasma

con�gurations has been shown to be able to destroy quasi-invariants in some regions of the phase space. Most notablythe existence of an adiabatic constant, namely the magnetic moment μ, which is at the heart of the gyrokineticsreduction appeared to be questionable in these regions, or when it exists, this was shown not to imply the integrabilityof the dynamics, even in axisymmetric magnetic �elds [1�3]. It appeared therefore useful to compute exact equilibriumsolutions of the Vlasov equation [4, 5]. Surprisingly these solutions can display a bifurcation from a plasma that is wellcon�ned with steep density gradients to one that is less con�ned. This is reminiscent of the so called L-H transitionobserved in fusion machine, and is supposedly linked to the presence of a transport barrier [3]. In the present case,the di�erent pro�les appear to emerge depending on whether or not unstable points with an associated separatrixexist in the microscopic Hamiltonian dynamics of the particles. During this internship the student will familiarizehimself with the problems, and develop new tools both conceptual and numerical. One of the problems being ableto analyze the stability of the proposed exact Valsov solution, and analyze the Hamiltonian chaotic behavior of themicroscopic trajectories, while measuring the e�ects on the particle density function. Extracting the �uid propertiesfrom the obtained kinetic ones and comparing with magneto-hydrodynamic equilibrium will be another challenge.This internship could naturally lead to a PhD.Scienti�c Environment: This internship work will be part of a long standing collaboration of the CPT with the

IRFM of the CEA Cadarache within the French national research federation of magnetic con�nement fusion (FR-FCM) and the TOP project from the A*MIDEX initiative. Good knowledge of dynamical systems and classicalstatistical physics will be a plus, mixing both numerical and analytical work will be expected from the interestedcandidate.

[1] B. Cambon, X. Leoncini, M. Vittot, R. Dumont, and X. X. Garbet. Chaotic motion of charged particles in toroidal magneticcon�gurations. Chaos, 24:033101, 2014.

[2] S. Ogawa, B. Cambon, X. Leoncini, M. Vittot, D. Del Castillo-Negrete, G. Dif-Pradalier, and X. Garbet. Full particle orbite�ects in regular and stochastic magnetic �elds. Phys. Plasmas, 23:072506, 2016.

[3] S. Ogawa, X. Leoncini, G. Dif-Pradalier, and X. Garbet. Study on creation and destruction of transport barriers via e�ectivesafety factors for energetic particles. Phys. Plasmas, 23:122510, 2016.

[4] Shun Ogawa, Xavier Leoncini, Alexei Vasiliev, and Xavier Garbet. Tailoring steep density pro�le with unstable points. Tobe published in Phys. Lett. A, 2018.

[5] Elias Laribi. Solutions stationnaires de vlasov-maxwell en geometrie cylindrique. Master's thesis, 2018.

Master Research Internship proposal

Research Center of Nonlinear Theory and Applications

Research tutors :

AMU : Xavier Leoncini, Laurent Raymond : [email protected], [email protected]

SAU : Wei Guan, Hao Liu, Yiwen Qi : [email protected], [email protected]

Title of the project : From dynamical control to network topology

General Context : The research work will fall in the general context of dynamical systems and complex networks.The study of complex networks is currently in a booming progress. In fact, applications of complex networks canrange from biology to social sciences. These systems can be considered as systems composed of many agents, mostoften in interaction through non-linear processes on complex networks. Recent developments in this area have focusedon the creation of the network, and currently consist of consider dynamics on these networks or the dynamics of thenetwork himself. One important aspect is when the network models some critical communications between agents,since we expect nonlinearities and time delays a chaotic behavior arise naturally in these systems. This chaoticbehavior is usually not desired, and needs to be controlled. Indeed when considering actual practical situations withtechnological applications, the non-predictive behavior is not always, if not almost never a suitable regime. To avoidthis phenomena some control strategies have been devised, however these are network dependent, they cover somepartial system failure, but usually depend on the speci�c network topology [3].In the last few years the notion of lace networks emerged, these networks can be constructed and monitored through

their fractal dimension[2, 4]. Depending on this parameter, the dynamics of a coupled rotators system could changedrastically from displaying regular behavior to much more complex and chaotic dynamics[1]. The goal of this internshipis to test this kind of topology on another type of dynamical system, namely some model systems with applicationsto control theory. How will the automated control strategy de�ned on such network fare ? Based on the results anda few observables can we devise an optimal topology such that for instance the controlled system will not only scalewell with the number of nodes, but as well minimize some quantity : power consumption, data transmission etc...In this internship, the goal is to set up the basis for a thorough investigation of the response and behavior of

some speci�c control strategy when de�ning them on lace type of networks. Moreover, the since this is part of aburgeoning international collaboration between Aix-Marseille University and Shenyang Aerospace University, theinterested candidate must be willing to step in an be part of an of �adventure�.Keys words : non-linear dynamics and chaos, complex networks, switched system, privacy safety of control systems,

control theory

If at the master 2 level, this internship could naturally lead to a PhD.

Scienti�c Environment : This internship work will be part of a collaboration of the CPT with the ShenyangAerospace University within the Center For Nonlinear Theory and Applications.

Location : The internship will be localized in the Center of Nonlinear Theory and Applications in ShenyangAerospace University, Shenyang, China. Mobility is thus required.

Required Knowledge : Good knowledge of dynamical systems and classical statistical physics will be a plus, themastering of numerical tools in the spirit of being able to perform numerical simulations (programming language of thetype fortran90, C,C++) and data analysis (octave, matlab, python) is necessary. Knowledge of english is mandatory,

2

as well as good communications skill in order to work in this international environment.

[1] Martin Belger, Sarah De Nigris, and Xavier Leoncini. Slowing down of chaotic states : Freezing the initial state. Discontinuity,Nonlinearity and Complexity, 5(4) :427�435, 2016.

[2] Sarah De Nigris and X. Leoncini. Crafting networks to achieve, or not achieve, chaotic states. Phys. Rev. E, 91 :042809,2015.

[3] Hao Liu. Sampled-data-based consensus of multi-agent systems under asynchronous denial-of-service attacks. IEEE Trans.

Cybernetics, submitted, 2018.[4] Sarah De Nigris and Xavier Leoncini. Hidden dimensions in an Hamiltonian system on networks, volume 15 of Nonlinear

Systems and Complexity. Springer, 2016.

Master-2 internship proposal

Lab: Centre de Physique Theorique (CPT), UMR 7332, Aix-Marseille Universite - LuminyResearch team: E7: " Systèmes dynamiques: théorie et applications"

Supervisor: Michel VITTOTTel: 04 91 26 95 24e-mail: [email protected]

Level (M1, M2 or both): M2Group Size: This is part 1 of a group of 3 separated subjects (somewhat related), for 1 student.

Project title: A Lie-Jordan Framework for Classical and Quantum Mechanics.

------------------------------------------------------------------------------------Abstract: The Lie-Jordan Algebraic structure is a non-associative generalization of the Poisson structure. It is well adapted to describe the Classical or Quantum Hamiltonian Dynamics (via its Lie structure), as well as the associated "States" (via its Jordan structure), giving rise to an"Heisenberg Uncertainty Principle".

This Master2 project consists in a short introduction to this framework. And it can eventually be continued into a PhD thesis.

References:Doering & Isham: "A Topos Foundation for Theories of Physics: I. Formal Languages for Physics", ArXiv: quant-ph/0703060------------------------------------------------------------------------------------





Project title:Hamiltonian Description of the Electrodynamics, via the Poisson Algebra of Maxwell-Vlasov.Application to the Physics of Magnetically Confined Plasmas, in Tokamaks (like ITER).------------------------------------------------------------------------------------Abstract:

This project will study the Hamiltonian approach of classical electrodynamics, via (non-canonical) Poisson structures. This relativistic Hamiltonian framework (introduced by Morrison, Marsden, Weinstein) is independent of the gauge potentials, and is well suited for a perturbation theory, in a strong inhomogeneous magnetic field (expansion in 1/|B|, with all thecurvature terms...). This algebraic and geometric description of the Maxwell-Vlasov kinetics yields some very concrete applications. For instance the reduced dynamics of the "gyrocenter", or "guiding-center" dynamics, in order to improve the efficiency of the computation and the confinement of the magnetically confined plasmas.

This is important in view of the thermonuclear fusion, as in Tokamaks (international project ITER, in CEA-Cadarache) or Stellarators. The geometric approach may be implemented in any coordinates, for instance adapted to the Tokamak (toroidal coordinates or even more adapted...).


References: P. J. Morrison: "Poisson brackets for fluids and plasmas"AIP Conference Proceedings 88, 13 (1982)------------------------------------------------------------------------------------





Project title:A Lie-Algebraic Framework for Hamiltonian Control Theory, and HamiltonianPerturbation Theory.------------------------------------------------------------------------------------Abstract:The "Hamiltonian Control" problem is to implement a small modification of the hamiltonian of a system in order to restore some properties, such as an "Internal Transport Barrier" or other structures in the phase space. The mathematical problem is to detail which properties are useful to restore, and to compute the associated modification of the hamiltonian (the "control term"). An experimental realization was already made via some external antennas, in a simple machine. A new route is to extend this method to the general (dissipative) case.The "Hamiltonian Perturbation" problem is a point of view dual to the preceding one, wherewe don't want (or we can't) modify the system. Instead we search for the "optimal canonical transformation" of the perturbed system which builds a deformation of the simple properties of the unperturbed system.


References:Michel Vittot: "Perturbation theory and control in classical or quantum mechanics by an inversion formula", J. Phys. A: Math. Gen. 37 (2004) p 6337-6357------------------------------------------------------------------------------------

cpt proposals for master internships · this phenomena some control strategies have been devised,...

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