Search

The coexistence of stable limit cycles and chaotic attractors has already been observed in some 3D dynamical systems. In this paper we show, using the T-system, that unstable limit cycles and chaotic attractors can also coexist. Moreover, by completing the characterization of the existence of invariant algebraic surfaces and their associated global dynamics, we give a better understanding on the disappearance of the strange attractor and the limit cycles of the studied system.

In this paper we analyze the motion of charged particles in a non-standard magnetic field that is of a Clebsch type representation where a magnetic perturbation is considered. The safety factor of ASDEX-Upgrade is used. Some aspects of the solutions of the guiding center system of equations are featured.

In our model the diffusion produced by a two-dimensional electrostatic stochastic field combined with a sheared magnetic field is analyzed numerically. We assumed, as in a lot of other papers, an isotropic spectrum of electrostatic fluctuations as a combination of a Gaussian one (for small wave-vectors) and a power-law k-3 (for large wave-vectors). The running and asymptotic diffusion coefficients of electrons are obtained for physically relevant parameter values.

We analyze statistically the energization of particles in a large scale environment of strong turbulence that is fragmented into a large number of distributed current filaments. The turbulent environment is generated through strongly perturbed, 3D, resistive MHD simulations, and it emerges naturally from the nonlinear evolution, without a specific reconnection geometry being set up. Based on test-particle simulations, we estimate the transport coefficients in energy space for use in the classical Fokker-Planck (FP) equation, and we show that the latter fails to reproduce the simulation results. The reason is that transport in energy space is highly anomalous (strange), the particles perform Levy flights, and the energy distributions show extended power-law tails. Newly then, we motivate the use and derive the specific form of a fractional transport equation (FTE), we determine its parameters and the order of the fractional derivatives from the simulation data, and we show that the FTE is able to reproduce the high energy part of the simulation data very well. The procedure for determining the FTE parameters also makes clear that it is the analysis of the simulation data that allows to make the decision whether a classical FP or a FTE is appropriate., in press at Physical Review Letters, 2017 (6 pages, 7 figures)

The formation of internal transport barriers (ITB) in tokamaks has been experimentally associated with the region of low magnetic shear and the presence of main rational magnetic surfaces. We study the ITB from a mathematical point of view in a Hamiltonian model (the rev-tokamap model) that describes the magnetic field lines dynamics in reversed shear tokamaks. The non-twist properties of the map that generates the system are exploited in order to explain analytically the existence of ITB surrounding the shearless curve (in the low magnetic shear zone) and the robustness of invariant circles (corresponding to closed magnetic surfaces) situated in its proximity. We describe the location of the ITB for various q-profiles and perturbations of the (ideal) integrable system. We use an analytical method to estimate the maximum radius of magnetic field lines confinement, hence the position of the barrier, and we explain the destruction of invariant circles when the amplitude of the perturbation increases. The reconnection phenomena are also related with ITB and an explanation for the (experimentally observed) enlargement of ITB when the minimal value of the q-profile becomes close but less than a main rational is obtained.

Magnetic field lines behaviour in a reversed shear tokamak can be described by a one and a half degree of freedom Hamiltonian system. In order to get insights into its dynamics we study numerically a global model for a Poincare map associated to such a system. Mainly we investigate the scenario of reconnection of the invariant manifolds of two hyperbolic orbits of the same type n/m and show that it is a generic one. When the two Poincare–Birkhoff chains involved in this process are aligned in phase, i.e. they are in a nongeneric position, a sequence of two saddle–center bifurcations occur in one of the chains, interfering with the former elliptic orbit of that chain, such that at the reconnection threshold the two chains are in a generic position. Dynamics around the new created configuration at the reconnection appears to vary from a regular motion to a chaotic one. In connection with the study of this global bifurcation we give the first example of region of instability in the dynamics of a nontwist map.

In hot laboratory plasmas, internal transport barriers (ITBs) have recently been observed, localized in the radial profile `around' rational values of the winding number ω(r) = 1/q(r). Such barriers are obviously related to the perturbed magnetic structure, described by a 1+1/2 Hamiltonian in presence of a perturbation. From the point of view of non-linear Hamiltonian dynamical systems [1], this experimental result appears highly paradoxical since rational q-values generally correspond to the less robust tori. We have studied the appearance of chaos of toroidal magnetic lines by a discrete area-preserving map named `tokamap' [2]. By increasing the perturbation, we have observed in a wide chaotic sea the destruction of the last confining Kolmogorov–Arnold–Moser surfaces, broken and transformed into permeable Cantor sets (Cantori). The flux across a Cantorus has been computed by using refined mathematical techniques due to MacKay, Mather and Aubry. We have proved that the ITB observed in the tokamap is actually composed [3] of two permeable Cantori with `noble' values of ω (in the definition of Percival), each Cantorus forming a partial (permeable) barrier inhibiting the magnetic line motion. More generally, between the dominant chains of rational islands q = m/(m−1), the most resistant barriers between q = (m+2)/(m+1) and (m+1)/m have been checked (Greene, MacKay and Stark) to be localized on the `most irrational' numbers in these Farey intervals, i.e. on the noble numbers N(1,m)≡1+[1/(m+1/G)] (where G is the Golden number), defined by their continuous fraction expansion N(i,m) = [i,m,(1)∞]. In conclusion, the study of the tokamap mapping allowed us to predict on mathematical basis that ITB can occur in tokamak plasmas not only `around' rational magnetic surfaces but more precisely on noble q−values of irrational surfaces, and to localize them by the Fibonacci series of their convergents.

International audience; The existence of transport barriers in 3/2 degrees of freedom degenerate Hamiltonian systems is studied using the associated stroboscopic maps. For small enough amplitudes of the perturbations, a transport barrier (formed by infinitely many invariant rotational circles) is proven to exist and to form in the degenerate annulus. These results are applied to Hamiltonian models which describe some magnetic configurations in tokamak plasmas. Read More: http://www.worldscientific.com/doi/abs/10.1142/S021812741350034X

International audience; The issue of magnetic confinement in magnetic fusion devices is addressed within a purely magnetic approach. Using some Hamiltonian models for the magnetic field lines, the dual impact of low magnetic shear is shown in a unified way. Away from resonances, it induces a drastic enhancement of magnetic confinement that favors robust internal transport barriers (ITBs) and stochastic transport reduction. When low-shear occurs for values of the winding of the magnetic field lines close to low-order rationals, the amplitude thresholds of the resonant modes that break internal transport barriers by allowing a radial stochastic transport of the magnetic field lines may be quite low. The approach can be applied to assess the robustness versus magnetic perturbations of general (almost) integrable magnetic steady states, including non-axisymmetric ones such as the important single helicity steady states. This analysis puts a constraint on the tolerable mode amplitudes compatible with ITBs and may be proposed as a possible explanation of diverse experimental and numerical signatures of their collapses.

The dynamics of magnetic field lines and of charged particles in toroidal chambers are commonly analysed by numerically solving the dynamical equations. They may also be analysed using deterministic reduced models, i.e. low-dimensional discrete time approximations (maps) of the Hamiltonian continuous time models. We report on the accuracy of the latter method by considering the mapping technique derived from the Hamilton–Jacobi equation. The optimum time stepping in some models for the study of the magnetic field in tokamaks is determined by using local criteria. Special attention is given to the analysis of stochasticity produced by time discretization.

Single trajectories of magnetic line motion indicate the persistence of a central protected plasma core, surrounded by a chaotic shell enclosed in a double-sided transport barrier : the latter is identified as being composed of two Cantori located on two successive "most-noble" numbers values of the perturbed safety factor, and forming an internal transport barrier (ITB). Magnetic lines which succeed to escape across this barrier begin to wander in a wide chaotic sea extending up to a very robust barrier (as long as L, TEX file, 84 pages including 32 color figures. Higher quality figures can be seen on the PDF file at http://membres.lycos.fr/fusionbfr/JHM/Tokamap/JSP.pdf

Using Hamiltonian models for the magnetic field lines, we propose a methodology to improve their confinement through the creation of transport barriers. A local modification of the safety profile creating a low-shear zone is shown to be sufficient to locally enhance drastically the regularity of the magnetic field lines without requesting a reversed shear. The optimal benefits of low-shear are obtained when the value q 0 of the safety profile in the low-shear zone is sufficiently far from the main resonance values m/n with low m and n, in the case of large enough values of those (m, n) mode amplitudes. A practical implementation in tokamak plasmas should involve electron cyclotron current drive to locally modify the magnetic shear.

Larger scale plasma instabilities not leading to an immediate termination of a discharge often result in periodic nonlinear perturbations of the plasma. A minimal possible model is formulated for description of the system with drive and relaxation processes which have different time scales. The model is based on two equations: the first being responsible for the relaxation dynamics and the second for the drive. The model can be generalized to describe the pellet injection.

The role of stochastization of magnetic field lines in fast reconnection phenomena occurring in magnetized fusion plasma is analyzed. A mapping technique is applied to trace the field lines of toroidally confined plasma where the perturbation parameter is expressed in terms of experimental perturbation amplitudes determined from the Axially Symmetric Divertor Experiment (ASDEX) Upgrade tokamak [S. Gunter, C. Angioni, M. Apostoliceanu et al., Nucl. Fusion 45, S98 (2005)]. It is found that fast reconnection observed during amplitude drops of the neoclassical tearing mode instability in the frequently interrupted regime can be related to stochastization.