CPT proposals for Master internships

Transcription

1 CPT proposals for Master internships Academic year Warning(!!): Due to a limited funding, the CPT director will fix the number of 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.

2 Master 2 Internship Centre de Physique Théorique, UMR 7332 and IRFM CEA Address: Centre de Physique théorique, Campus de Luminy, Case 907, Marseille Cedex 09 & IRFM, CEA Cadarache St Paul lez Durance Cedex, Name of advisers: Xavier Leoncini and Guilhem Dif-Pradalier Xavier.Leoncini@cpt.univ-mrs.fr, Guilhem.Dif-Pradalier@cea.fr 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 congurations has been shown to be able to destroy quasi-invariants in some regions of the phase space. Most notably the existence of an adiabatic constant, namely the magnetic moment μ, which is at the heart of the gyrokinetics reduction appeared to be questionable in these regions, or when it exists, this was shown not to imply the integrability of the dynamics, even in axisymmetric magnetic elds [13]. It appeared therefore useful to compute exact equilibrium solutions of the Vlasov equation [4, 5]. Surprisingly these solutions can display a bifurcation from a plasma that is well conned with steep density gradients to one that is less conned. This is reminiscent of the so called L-H transition observed in fusion machine, and is supposedly linked to the presence of a transport barrier [3]. In the present case, the dierent proles appear to emerge depending on whether or not unstable points with an associated separatrix exist in the microscopic Hamiltonian dynamics of the particles. During this internship the student will familiarize himself with the problems, and develop new tools both conceptual and numerical. One of the problems being able to analyze the stability of the proposed exact Valsov solution, and analyze the Hamiltonian chaotic behavior of the microscopic trajectories, while measuring the eects on the particle density function. Extracting the uid properties from the obtained kinetic ones and comparing with magneto-hydrodynamic equilibrium will be another challenge. This internship could naturally lead to a PhD. Scientic 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 connement fusion (FR- FCM) and the TOP project from the A*MIDEX initiative. Good knowledge of dynamical systems and classical statistical physics will be a plus, mixing both numerical and analytical work will be expected from the interested candidate. [1] B. Cambon, X. Leoncini, M. Vittot, R. Dumont, and X. X. Garbet. Chaotic motion of charged particles in toroidal magnetic congurations. Chaos, 24:033101, [2] S. Ogawa, B. Cambon, X. Leoncini, M. Vittot, D. Del Castillo-Negrete, G. Dif-Pradalier, and X. Garbet. Full particle orbit eects in regular and stochastic magnetic elds. Phys. Plasmas, 23:072506, [3] S. Ogawa, X. Leoncini, G. Dif-Pradalier, and X. Garbet. Study on creation and destruction of transport barriers via eective safety factors for energetic particles. Phys. Plasmas, 23:122510, [4] Shun Ogawa, Xavier Leoncini, Alexei Vasiliev, and Xavier Garbet. Tailoring steep density prole with unstable points. To be published in Phys. Lett. A, [5] Elias Laribi. Solutions stationnaires de vlasov-maxwell en geometrie cylindrique. Master's thesis, 2018.

3 Master Research Internship proposal Research Center of Nonlinear Theory and Applications Research tutors : AMU : Xavier Leoncini, Laurent Raymond : Xavier.Leoncini@cpt.univ-mrs.fr, Laurent.Raymond@cpt.univ-mrs.fr SAU : Wei Guan, Hao Liu, Yiwen Qi : guanweihaha@163.com, qiyiwen@sau.edu.cn 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 can range from biology to social sciences. These systems can be considered as systems composed of many agents, most often in interaction through non-linear processes on complex networks. Recent developments in this area have focused on the creation of the network, and currently consist of consider dynamics on these networks or the dynamics of the network 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 chaotic behavior is usually not desired, and needs to be controlled. Indeed when considering actual practical situations with technological applications, the non-predictive behavior is not always, if not almost never a suitable regime. To avoid this phenomena some control strategies have been devised, however these are network dependent, they cover some partial system failure, but usually depend on the specic 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 change drastically from displaying regular behavior to much more complex and chaotic dynamics[1]. The goal of this internship is to test this kind of topology on another type of dynamical system, namely some model systems with applications to control theory. How will the automated control strategy dened on such network fare? Based on the results and a few observables can we devise an optimal topology such that for instance the controlled system will not only scale well 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 specic control strategy when dening them on lace type of networks. Moreover, the since this is part of a burgeoning international collaboration between Aix-Marseille University and Shenyang Aerospace University, the interested 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. Scientic Environment : This internship work will be part of a collaboration of the CPT with the Shenyang Aerospace University within the Center For Nonlinear Theory and Applications. Location : The internship will be localized in the Center of Nonlinear Theory and Applications in Shenyang Aerospace University, Shenyang, China. Mobility is thus required. Required Knowledge : Good knowledge of dynamical systems and classical statistical physics will be a plus, the mastering of numerical tools in the spirit of being able to perform numerical simulations (programming language of the type fortran90, C,C++) and data analysis (octave, matlab, python) is necessary. Knowledge of english is mandatory,

4 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) :427435, [2] Sarah De Nigris and X. Leoncini. Crafting networks to achieve, or not achieve, chaotic states. Phys. Rev. E, 91 :042809, [3] Hao Liu. Sampled-data-based consensus of multi-agent systems under asynchronous denial-of-service attacks. IEEE Trans. Cybernetics, submitted, [4] Sarah De Nigris and Xavier Leoncini. Hidden dimensions in an Hamiltonian system on networks, volume 15 of Nonlinear Systems and Complexity. Springer, 2016.

5 Master-2 internship proposal Lab: Centre de Physique Theorique (CPT), UMR 7332, Aix-Marseille Universite - Luminy Research team: E7: " Systèmes dynamiques: théorie et applications" Supervisor: Michel VITTOT Tel: vittot@cpt.univ-mrs.fr Level (M1, M2 or both): M2 Group 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/

6 Master-2 internship proposal Lab: Centre de Physique Theorique (CPT), UMR 7332, Aix-Marseille Universite - Luminy Research team: E7: " Systèmes dynamiques: théorie et applications" Supervisor: Michel VITTOT Tel: vittot@cpt.univ-mrs.fr Level (M1, M2 or both): M2 Group Size: This is part 2 of a group of 3 separated subjects (somewhat related), for 1 student. 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 (noncanonical) 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 the curvature 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...). This Master2 project consists in a short introduction to this framework. And it can eventually be continued into a PhD thesis. References: P. J. Morrison: "Poisson brackets for fluids and plasmas" AIP Conference Proceedings 88, 13 (1982)

7 Master-2 internship proposal Lab: Centre de Physique Theorique (CPT), UMR 7332, Aix-Marseille Universite - Luminy Research team: E7: " Systèmes dynamiques: théorie et applications" Supervisor: Michel VITTOT Tel: vittot@cpt.univ-mrs.fr Level (M1, M2 or both): M2 Group Size: This is part 3 of a group of 3 separated subjects (somewhat related), for 1 student. Project title: A Lie-Algebraic Framework for Hamiltonian Control Theory, and Hamiltonian Perturbation 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, where we 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. This Master2 project consists in a short introduction to this framework. And it can eventually be continued into a PhD thesis. References: Michel Vittot: "Perturbation theory and control in classical or quantum mechanics by an inversion formula", J. Phys. A: Math. Gen. 37 (2004) p