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Barriers have been identified in magnetically confined plasmas reducing the particle transport and improving the confinement. One of them, the primary shearless barriers are associated to extrema of non-monotonic plasma profiles. Previously, we identified these barriers in a model described by a map that allows the integration of charged particles motion in drift waves for a long time scale. In this work, we show how the existence of these robust barriers depends on the fluctuation amplitude and on the electric shear. Moreover, we also find control parameter intervals for which these primary barriers onset and break-up are recurrent. Another noticeable feature, in these transitions, is the appearance of a layer of particle trajectory stickiness after the shearless barrier break-up or before its onset. Besides the mentioned primary barriers, we also observe sequences of secondary shearless barriers, not reported before, created and destroyed by a sequence of bifurcations as the main control parameters, the fluctuation amplitude and electric shear, are varied. Furthermore, in these bifurcations, we also find hitherto unknown double and triple secondary shearless barriers which constitute a noticeable obstacle to the chaotic transport., 12 pages, 28 figures. This article has been accepted by the Journal Physics of Plasmas. It can be found at: https://doi.org/10.1063/5.0056428

Intrinsically nonlinear coupled systems present different oscillating components that exchange energy among themselves. We present a new approach to deal with such energy exchanges and to investigate how it depends on the system control parameters. The method consists in writing the total energy of the system, and properly identifying the energy terms for each component and, especially, their coupling. To illustrate the proposed approach, we work with the bi-dimensional spring pendulum, which is a paradigm to study nonlinear coupled systems, and is used as a model for several systems. For the spring pendulum, we identify three energy components, resembling the spring and pendulum like motions, and the coupling between them. With these analytical expressions, we analyze the energy exchange for individual trajectories, and we also obtain global characteristics of the spring pendulum energy distribution by calculating spatial and time average energy components for a great number of trajectories (periodic, quasi-periodic and chaotic) throughout the phase space. Considering an energy term due to the nonlinear coupling, we identify regions in the parameter space that correspond to strong and weak coupling. The presented procedure can be applied to nonlinear coupled systems to reveal how the coupling mediates internal energy exchanges, and how the energy distribution varies according to the system parameters., Submitted for publication

The turbulence of magnetically confined plasmas usually presents high-density pulses with short duration known as bursts. In the Texas Helimak, it is possible to suppress bursts in a broader region by applying a negative electrostatic bias. However, an almost unchanged burst rate persists in a region far from the location where bias is applied. We investigate the turbulence transition that occurs from the burst suppressed region to the burst dominated region by analyzing data from Langmuir probes in the whole radial extension of the machine. We find that such turbulence transition can be understood as an alternation of two different turbulent regimes, with the probability of being in each regime depending on the radial position. One regime, named as burst-free regime, consists of only Gaussian fluctuations and the other, named as bursty turbulent regime, is a train of pulses with double exponential temporal profiles, exponential amplitude distribution, and random occurring instants. This succession between burst-free and bursty turbulent regimes influences the equilibrium parameters.

The cerebral cortex plays a key role in complex cortical functions. It can be divided into areas according to their function (motor, sensory and association areas). In this paper, the cerebral cortex is described as a network of networks (cortex network), we consider that each cortical area is composed of a network with small-world property (cortical network). The neurons are assumed to have bursting properties with the dynamics described by the Rulkov model. We study the phase synchronization of the cortex network and the cortical networks. In our simulations, we verify that synchronization in cortex network is not homogeneous. Besides, we focus on the suppression of neural phase synchronization. Synchronization can be related to undesired and pathological abnormal rhythms in the brain. For this reason, we consider the delayed feedback control to suppress the synchronization. We show that delayed feedback control is efficient to suppress synchronous behavior in our network model when an appropriate signal intensity and time delay are defined.

In this work, we study the dynamic range in a neural network modelled by cellular automaton. We consider deterministic and non-deterministic rules to simulate electrical and chemical synapses. Chemical synapses have an intrinsic time-delay and are susceptible to parameter variations guided by learning Hebbian rules of behaviour. The learning rules are related to neuroplasticity that describes change to the neural connections in the brain. Our results show that chemical synapses can abruptly enhance sensibility of the neural network, a manifestation that can become even more predominant if learning rules of evolution are applied to the chemical synapses.

We construct an area preserving and integrable map to represent magnetic surfaces with triangularity in single-null divertor tokamaks. The magnetic surfaces obtained by the map can assume different asymmetric geometries and the position of X-point, through the choices of values for some free parameters. The safety factor profile is independent of the geometric parameters and can also be chosen arbitrarily. We combine the divertor integrable map with a nonintegrable map that simulates the effect of external magnetostatic perturbations. The application of this methodology permits to obtain the escape patterns of magnetic field lines on the divertor plates.

Extreme events are by definition rare and exhibit unusual values of relevant observables. In literature, it is possible to find many studies about the predictability and suppression of extreme events. In this work, we show the existence of dragon-kings extreme events in nonlinear three-wave interactions. Dragon-king extreme events, identified by phase transitions, tipping points and catastrophes, affects fluctuating systems. We show that these events can be avoided by adding a perturbing small amplitude wave to the system.

Chimera states in coupled oscillator systems present both spatially coherent and incoherent domains. The number and size of these domains depend on many factors like the system parameters and initial conditions. Systematic investigations of these dependences require a quantification of the degree of coherence present in a given snapshot spatial pattern. We propose the use of a local order parameter magnitude combined with the counting of the corresponding plateaus so as to provide such quantification. We use this technique in non-locally coupled lattices of chaotic logistic maps and chaotic R\"ossler systems to investigate the dependence of the degree of coherence on the coupling strength., Comment: 6 figures

Nowadays, divertors are used in the main tokamaks to control the magnetic field and to improve the plasma confinement. In this article, we present analytical symplectic maps describing Poincar\'e maps of the magnetic field lines in confined plasmas with a single null poloidal divertor. Initially, we present a divertor map and the tokamap for a diverted configuration. We also introduce the Ullmann map for a diverted plasma, whose control parameters are determined from tokamak experiments. Finally, an explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent toroidal equilibrium magnetic surfaces. In this method, we also give examples of onset of chaotic field lines at the plasma edge due to resonant perturbations.

According to Stevens’ law the relationship between stimulus and response is a power-law within an interval called the dynamic range. The dynamic range of sensory organs is found to be larger than that of a single neuron, suggesting that the network structure plays a key role in the behavior of both the scaling exponent and the dynamic range of neuron assemblies. In order to verify computationally the relationships between stimulus and response for spiking neurons, we investigate small-world networks of neurons described by the Hodgkin–Huxley equations connected by chemical synapses. We found that the dynamic range increases with the network size, suggesting that the enhancement of the dynamic range observed in sensory organs, with respect to single neurons, is an emergent property of complex network dynamics.

Using a method developed by Clemente, it is possible to obtain anisotropic magnetohydrodynamic equilibrium in axially symmetric systems, from a previously known solution of the Grad-Schluter-Shafranov equation. We generalize this method to symmetric systems described by orthogonal as well as nonorthogonal systems of coordinates. Two examples are presented in cylindrical and spherical geometries, for which we give an exact analytic solution of the anisotropic MHD equilibrium.

a b s t r a c t We study the firing rate properties of a cellular automaton model for a neuronal network with chemical synapses. We propose a simple mechanism in which the nonlocal connections are included, through electrical and chemical synapses. In the latter case, we introduce a time delay which produces self-sustained activity. Nonlocal connections, or shortcuts, are randomly introduced according to a specified connection probability. There is a range of connection probabilities for which neuron firing occurs, as well as a critical probability for which the firing ceases in the absence of time delay. The critical probability for nonlocal shortcuts depends on the network size according to a power-law. We also compute the firing rate amplification factor by varying both the connection probability and the time delay for different network sizes.

Oscilações de relaxação são mantidas por uma influência externa constante e apresentam duas escalas de tempo diferentes (lenta e rápida). Elas têm propriedades diferentes das oscilações pendulares que, para pequenas amplitudes, reduzem-se às oscilações harmônicas. Neste primeiro trabalho apresentamos as diferenças entre oscilações pendulares e de relaxação, mostrando exemplos ilustrativos de aplicações de oscilações de relaxação em um sistema mecânico (vaso de Tântalo) e num circuito elétrico. Relaxation oscillations are maintained by a constant external influence and present two diferent timescales (fast and slow). They have properties diferent from the pendular oscillations which, for small amplitudes, reduce to harmonic oscillations. In this first paper we present the diferences between relaxation and pendular oscillations, showing illustrative examples of application of relaxation oscillations in a mechanical system (Tantal vase) and an electric circuit.

The transition to turbulence (spatio-temporal chaos) in a wide class of spatially extended dynamical system is due to the loss of transversal stability of a chaotic attractor lying on a homogeneous manifold (in the Fourier phase space of the system), causing spatial mode excitation. Since the latter manifests as intermittent spikes this has been called a bubbling transition. We present numerical evidences that this transition occurs due to the so-called blowout bifurcation, whereby the attractor as a whole loses transversal stability and becomes a chaotic saddle. We used a nonlinear three-wave interacting model with spatial diffusion as an example of this transition.

We investigated the transition to spatio-temporal chaos in spatially extended nonlinear dynamical systems possessing an invariant subspace with a low-dimensional attractor. When the latter is chaotic and the subspace is transversely stable we have a spatially homogeneous state only. The onset of spatio-temporal chaos, i.e. the excitation of spatially inhomogeneous modes, occur through the loss of transversal stability of some unstable periodic orbit embedded in the chaotic attractor lying in the invariant subspace. This is a bubbling transition, since there is a switching between spatially homogeneous and nonhomogeneous states with statistical properties of on–off intermittency. Hence the onset of spatio-temporal chaos depends critically both on the existence of a chaotic attractor in the invariant subspace and its being transversely stable or unstable.

Neuron activity presents two timescales, a fast one related to action-potential spiking, and a slow timescale in which bursting takes place. Bursting activity in neuron ensembles can be synchronized, meaning the adjustment of the bursting phases due to coupling. We investigated bursting synchronization in a non-locally coupled lattice using a two-dimensional map to describe neuron activity. The coupling involves all sites in a lattice, the corresponding strength decreasing with the lattice distance in a power-law fashion. We observed bursting synchronization for wide intervals of the coupling parameters. We also investigated the bursting synchronization of the ensemble with an external time-periodic signal applied to one or more selected neurons.

Numerical simulations of coupled map lattices with non-local interactions (i.e., the coupling of a given map occurs with all lattice sites) often involve a large computer time if the lattice size is too large. In order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution. This emulates a “thermal bath”, where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude. We demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied. We considered one-dimensional lattices of chaotic logistic maps at outer crisis x → 4 x ( 1 − x ) .

We considered a coupled chaotic logistic map lattice exhibiting the scale-free property: the outreach connectivity of each node obeys a power-law distribution. We analyzed a weak form of coherent spatio-temporal behavior (direction coherence) which presents features common to completely synchronized states, like a transitional behavior with intermittent bursting. We studied such phenomena and their dependence on the parameters of the coupled scale-free lattice. Prospective applications in neuronal networks are emphasized.

The magnetic field line structure of tokamaks with reversed magnetic shear is analyzed by means of a nontwist map model that takes into account non-integrable perturbations that describe ergodic magnetic limiters. The map studied possess behavior expected of the standard nontwist map, a well-studied map, despite the different symmetries and the existence of coupled perturbations. A distinguising feature of nontwist maps is the presence of good surfaces in the reveresed shear region, and consequently the appearance of a transport barrier inside the plasma. Such barriers are observed in the present model and are seen to be very robust. Very strong perturbations are required to destroy them, and even after breaking, the transport turns out to be diffusive. Poloidal diffusion is found to be two orders of magnitude higher than radial diffusion.

We proposed a simple feedback control method to suppress chaotic behavior in oscillators with limited power supply. The small-amplitude controlling signal is applied directly to the power supply system, so as to alter the characteristic curve of the driving motor. Numerical results are presented showing the method efficiency for a wide range of control parameters. Moreover, we have found that, for some parameters, this kind of control may introduce coexisting periodic attractors with complex basins of attraction and, therefore, serious problems with predictability of the final state the system will asymptote to.

We consider regular lattices of coupled chaotic maps whose local dynamics is ruled by the logistic map x ↦ 4 x ( 1 − x ) , a usual paradigm of chaotic systems. Through finite-time exponents, we scrutinize the lattice dynamics in the vicinity of the threshold of complete synchronization. We connect dynamical features such as relaxation to the coherent state and intermittency with the statistics of finite-time exponents, focusing on the implications of the particular statistics related to the logistic map. Although numerical examples are given for lattice couplings decaying with distance as a power law, our results are expected to be valid for a wider class of schemes coupling logistic maps.

Spatially extended dynamical systems subjected to a periodic external input are able to store short-term memories as a result of self-organization. This phenomenon has been used to interpret sliding charge density wave experiments, and has been found in lattices of coupled maps and oscillators. In this paper, we aim to describe the influence of a stochastic perturbation in the self-organized memories in a lattice of coupled linear and weakly nonlinear maps. We consider the case in which the coupling between maps depends on the lattice distance as a power law.

We investigate aspects of the spatio-temporal dynamics exhibited by a one-dimensional lattice of chaotic piecewise linear maps in a coupling prescription which includes both regular (nearest and next-to-nearest neighbors) and randomly chosen couplings. We discuss the conditions for the existence of chaotic synchronized states, and relate them to the coupling parameters. The transition to synchronized behavior is described, and we explore some statistical properties of the time it takes to achieve this regime.

Spatially extended dynamical systems may exhibit intermittent behavior in both spatial and temporal scales, characterized by repeated conversions from spatially localized transient chaos into global laminar patterns. A simple model, yet retaining some features of more complex systems, consists of a lattice of a class of tent maps with an escaping region. The coupling prescription we adopt in this work considers the interaction of a site with all its neighbors, the corresponding strength decaying with the lattice distance as a power-law. This makes possible to pass continuously from a local (nearest-neighbor) to a global kind of coupling. We investigate statistical properties of both the chaotic transient bursts and the periodic laminar states, with respect to the coupling parameters.

Analytical tokamak plasma equilibria, with non-monotonic plasma current profile and perturbed by ergodic magnetic limiters, are described by non-twist conservative maps. Examples are given of concentration of magnetic field lines escaping to the tokamak wall. The robustness of invariant curves on the shearless region is also observed.

Tuned liquid column dampers are U-tubes filled with some liquid, acting as an active vibration damper in structures of engineering interest like buildings and bridges. We study the effect of a tuned liquid column damper in a vibrating system consisting of a cart which vibrates under driving by a source with limited power supply (non-ideal excitation). The effect of a liquid damper is studied in some dynamical regimes characterized by coexistence of both periodic and chaotic motion.

The ergodic magnetic limiter is a device designed to generate a cold boundary layer of chaotic magnetic field lines at the peripheral region of a tokamak, with the main purpose of reducing the deleterious effects of the plasma–wall interaction. In the TCABR tokamak an ergodic limiter was constructed and recently installed inside the vacuum chamber. We developed a theoretical model for the action of an ergodic magnetic limiter in a large aspect-ratio tokamak taking into account the finite width of the limiter. The theoretical results are in good agreement with measurements of the vacuum magnetic field created by the limiter. Poincare maps of field line flow are computed to reveal the resulting magnetic field line structure due to the ergodic limiter and show that the operation of the ergodic limiter in the TCABR tokamak is feasible and results in a chaotic boundary layer for limiter currents of about 6% of the plasma current.

Complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high-dimensional chaos. This paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly non-hyperbolic behavior, the system undergoes on–off intermittency with respect to the synchronization manifold. This has been confirmed by numerical diagnostics of synchronization and non-hyperbolic behavior, the latter using the statistical properties of finite-time Lyapunov exponents.

We investigate numerically the dynamical behavior of a non-ideal mechanical system consisting of a vibrating cart containing a particle which can oscillate back and forth colliding with walls carved in the cart. This system represents an impact damper for controlling high-amplitude vibrations and chaotic motion. The motion of the cart is induced by an in-board non-ideal motor driving an unbalanced rotor. We study the phase space of the cart and the bouncing particle, in particular the intertwined smooth and fractal basin boundary structure. The control of the chaotic motion of the cart due to the particle impacts is also investigated. Our numerical results suggests that impact dampers of small masses are effective to suppress chaos, but they also increase the final-state sensitivity of the system in its phase space.

We used numerical diagnostics to quantify spatial disorder, and its relation with temporal chaos, for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice distance in a power-law fashion. The main tool is spatial return plots, whose properties are used to obtain information about the chaotic synchronized states of the system. A spatial correlation integral is introduced to characterize the clustering of points in the spatial return plots.

We examine the existence of an invariant manifold in the phase space structure presented by a dissipative four-wave coupling. We show that, once in such manifold, the system preserves two quantities as time evolves: a quantity related to the energy of the system and another, related to the exchange of energy between the waves. We observe that the presence of this manifold allows the system to manifest stable solutions for a specific interval of the dissipation parameters. Out of this range the system brings up only uncoupled states (no nonlinear saturated states). The dissipation parameters affect only the transversal dynamics to the manifold, changing how quickly the invariant manifold is reached. In the invariant manifold, the system exhibits a large number of coexisting periodic attractors—limit cycles (coupled states), with an interwoven structure of basins of attraction. The time series of conserved quantities and the properties of the Lyapunov spectra are used to imply the existence of a lower-dimensional invariant manifold. Since the dynamics in such invariant manifold preserves an energy-like function, we identify it as being conservative.

We investigated the structure of magnetic field lines in a tokamak with reversed magnetic shear and non-symmetric perturbations by means of analytically derived area-preserving non-twist Poincare maps. The maps are used to study magnetic island dimerization and the formation of a transport barrier in the chaotic layer of field lines.

We investigate a phenomenological model for the heartbeat consisting of two coupled Van der Pol oscillators. The coupling between these nodes can be both unidirectional or bidirectional, and an external driving produced by a pacemaker is also included in this model. In order to warrant a robust operation, it is desirable that both units oscillate in a synchronized way, even though in the presence of external in0uences or parameter mismatches which are unavoidable in a physiological setting. We study the synchronization properties of such an association with respect to the nature andintensity of coupling. We analyze in particular the (generalized ) synchronization of rhythms characterizedby a chaotic mod ulation of the oscillator frequencies. We also investigate the shadowing breakdown of numerically generated chaotic trajectories of the coupled oscillator system via unstable dimension variability in its chaotic invariant set. c

Reversed magnetic shear configurations in tokamaks reduce substantially particle diffusivity and improve plasma confinement due to the formation of a transport barrier, as a result of magnetic field line reconnection and bifurcation. The latter are caused by the resonant perturbation of an ergodic limiter on a tokamak with reversed magnetic shear, which creates a region with chaotic field lines in the vicinity of the dimerized island chains. An analytically derived nontwist map for field lines is used to describe the formation of a transport barrier, manifested in the decrease of field line diffusion rate. This barrier appears due to the chaotic field line trapping near the reconnection layer that survives the bifurcation.

We investigate the emergence of mode locking, or frequency synchronization, in a chain of coupled sine-circle maps with randomly distributed parameters, and exhibiting the small-world property. The coupling prescription we adopt considers the nearest and next-to-the-nearest neighbors of a given site, as well as randomly chosen non-local shortcuts, according to a given probability. A transition between synchronized and non-synchronized patterns is observed as this probability is varied. We also study the statistics of the synchronization plateaus, evidencing a Poisson-type distribution.

We consider a symplectic map for magnetic field lines in a tokamak with an ergodic limiter, whose function is to generate a perturbing magnetic field which causes the formation of chaotic field lines in the vicinity of the tokamak inner wall. We analyze numerically the onset of global chaos due to the interaction of adjacent magnetic islands, the size of the chaotic region, and investigate anomalous diffusion and loss of field lines due to their collisions with the tokamak wall.

The action of a set of ergodic magnetic limiters in tokamaks is investigated from the Hamiltonian chaotic scattering point of view. Special attention is paid to the influence of invariant sets, such as stable and unstable manifolds, as well as the strange saddle, on the formation of the chaotic layer at the plasma edge. The nonuniform escape process associated to chaotic field lines is also analyzed. It is shown that the ergodic layer produced by the limiters has not only a fractal structure, but it possesses the even more restrictive Wada property.

We present analytical results for the Kolmogorov–Sinai entropy of a one-dimensional lattice of locally coupled piecewise linear maps, for some particular values of the coupling strength. Our results explain the numerically observed fact that the entropy of a lattice of chaotic maps increases for strong coupling.

Short-term memory formation is a collective e-ect displayed by driven spatiotemporal dynamical systems, in which the injectedsignal is echoedback in many d ways d to the coupling e-ect. The existence of multiple short-term memories enables us to use these systems to encode signals. We present numerical evidence of short-term memory generation in a lattice of AC-driven circuits with inductive coupling. The multiple con4gurations exhibited by this oscillator chain can be usedto encod e graphical characters. c 2002 Elsevier Science B.V. All rights reserved.

A numerical study of chaotic field line diffusion in a tokamak with an ergodic magnetic limiter is described. The equilibrium model field is analytically obtained by solving a Grad–Schluter–Shafranov equation in toroidal polar coordinates, and the limiter field is determined by supposing its action as a sequence of delta-function pulses. A symplectic twist mapping is introduced to analyze the mean square radial deviation of a bunch of field lines in a predominantly chaotic region. The formation of a stochastic layer and field diffusivity at the plasma edge are investigated. Field line transport is initially subdiffusive and becomes superdiffusive after a few iterations. The field lines are lost when they collide with the tokamak inner wall; their decay rate is exponential with Poisson statistics.

We consider the stationary magnetohydrodynamical (MHD) equilibrium equation for an axisymmetric plasma undergoing azimuthal rotations. The case of cylindrical symmetry is treated, and we present two semi-analytical solutions for the stationary MHD equilibrium equations, from which a number of physical properties of the magnetically confined plasma are derived.

We consider the effects of external nonsymmetric magnetostatic perturbations caused by resonant helical windings and a chaotic magnetic limiter on the plasma confined in a tokamak. The main purpose of both types of perturbation is to create a region in which field lines are chaotic in the Lagrangian sense: two initially nearby field lines diverge exponentially through many turns around the tokamak. The equilibrium field is obtained from the equations of magneto-hydrodynamic equilibrium written down in a polar toroidal coordinate system. The magnetic fields generated by the resonant helical windings and the chaotic magnetic limiter are obtained through an analytical solution of Laplace equation. The magnetic field line equations are integrated to give a Hamiltonian mapping of field lines that we use to characterize the structure of chaotic field lines. In the case of resonant windings, we obtained the map by both numerical integration and a Hamiltonian formulation. For a chaotic limiter, we analytically derived a symplectic map by using a Hamiltonian formulation.

The stationary equilibrium of an axisymmetric plasma characterized by toroidal and poloidal flows is considered within the framework of ideal double adiabatic magnetohydrodynamic equations. The problem is reduced to a nonlinear partial differential equation for the poloidal magnetic flux function, containing six surface functions, plus a nonlinear algebraic Bernoulli equation defining the plasma density. Ellipticity conditions and bifurcations of its solutions are discussed in the limit of small beta, appropriated for tokamak-like equilibria. Possible connections with the L-H transition are suggested.

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.

An equation for stationary MHD equilibrium with azimuthal rotation in spherical coordinates is derived. Following a procedure introduced by Maschke and Perrin, by supposing that the plasma temperature is a surface quantity, we describe a two-parameter family of axisymmetric toroidal configurations. The equilibrium equation is analytically solved for given hypotheses in such a way as to investigate the effect of plasma rotation on the magnetic field structure.

An explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent equilibrium magnetic surfaces. The magnetic surfaces obtained from the map are capable of fitting different geometries with freely specified position of the X-point, by varying free model parameters. The safety factor profile of the map is independent of the geometric parameters and can also be chosen arbitrarily. The divertor integrable map is composed of a nonintegrable map that simulates the effect of external symmetry-breaking resonances, so as to generate a chaotic region near the separatrix passing through the X-point. The composed field line map is used to analyze escape patterns (the connection length distribution and magnetic footprints on the divertor plate) for two equilibrium configurations with different magnetic shear profiles at the plasma edge. (Some figures may appear in colour only in the online journal)

Chaotic magnetic field lines play an important role in plasma confinement by tokamaks. They can either be generated in the plasma as a result of natural instabilities or artifficially produced by external conductors, like resonant helical windings and ergodic magnetic limiters. This is a review of works carried out at the Universidade de Sao Paulo and Universidade Federal do Parana on theoretical and experimental aspects of generation and control of chaotic magnetic field lines in tokamaks.

We have studied numerically the chaotic field line diffusion in a Tokamak with ergodic magnetic limiters. The model field is analytically obtained by solving a Grad-Shafranov equation in toroidal polar coordinates, and the limiter field is obtained by supposing its action as a sequence of delta-function pulses. The mean square radial deviation of a bunch of field lines in a predominantly chaotic region is analyzed. Our results show an initially superdiffusive behavior, followed by a subdiffusive regime, with subsequent loss of field lines due to collisions with the inner wall.

We explore a method for constructing two-dimensional area-preserving, integrable maps associated with Hamiltonian systems, with a given set of fixed points and given invariant curves. The method is used to find an integrable Poincaré map for the field lines in a large aspect ratio tokamak with a poloidal single-null divertor. The divertor field is a superposition of a magnetohydrodynamic equilibrium with an arbitrarily chosen safety factor profile, with a wire carrying an electric current to create an X-point. This integrable map is perturbed by an impulsive perturbation that describes non-axisymmetric magnetic resonances at the plasma edge. The non-integrable perturbed map is applied to study the structure of the open field lines in the scrape-off layer, reproducing the main transport features obtained by integrating numerically the magnetic field line equations, such as the connection lengths and magnetic footprints on the divertor plate.

In Tokamak Chauffage Alfven Bresilien [R. M. O. Galvao et al., Plasma Phys. Controlled Fusion 43, 1181 (2001)], high magnetohydrodynamic (MHD) activity may appear spontaneously or during discharges with a voltage biased electrode inserted at the plasma edge. The turbulent electrostatic fluctuations, measured by Langmuir probes, are modulated by Mirnov oscillations presenting a dominant peak with a common frequency around 10kHz. We report the occurrence of phase locking of the turbulent potential fluctuations driven by MHD activity at this frequency. Using wavelet cross-spectral analysis, we characterized the phase and frequency synchronization in the plasma edge region. We introduced an order parameter to characterize the radial dependence of the phase-locking intensity.