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Plasma turbulence at the edge of tokamaks is an issue of major importance in the study of the anomalous transport of particles and energy. Although the behavior of a turbulent plasma seems intractable, it turns out that many of its aspects can be described by low-dimensional non-integrable dynamical models. In this paper, we consider a number of dynamical effects occurring in tokamak plasma edge—in particular the role of internal transport barriers. Furthermore, we present experimental results on turbulent-driven transport for two machines—the Brazilian TCABR tokamak and University of Texas’ Helimak—that can be explained by those theoretical models.

Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos.

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 study the standard nontwist map that describes the dynamic behaviour of magnetic field lines near a local minimum or maximum of frequency. The standard nontwist map has a shearless invariant curve that acts like a barrier in phase space. Critical parameters for the breakup of the shearless curve have been determined by procedures based on the indicator points and bifurcations of periodical orbits, a methodology that demands high computational cost. To determine the breakup critical parameters, we propose a new simpler and general procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition. We also show that the coexistence of islands and chaotic sea in phase space can be analysed by using the recurrence plot. In particular, the measurement of determinism from the recurrence plot provides us with a simple procedure to distinguish periodic from chaotic structures in the parameter space. We identify an invariant shearless breakup scenario, and we also show that recurrence plots are useful tools to determine the presence of periodic orbit collisions and bifurcation curves.

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.

In this paper we study how hyperbolic and nonhyperbolic regions in the neighborhood of a resonant island perform an important role allowing or forbidding stickiness phenomenon around islands in conservative systems. The vicinity of the island is composed of nonhyperbolic areas that almost prevent the trajectory to visit the island edge. For some specific parameters tiny channels are embedded in the nonhyperbolic area that are associated to hyperbolic fixed points localized in the neighborhood of the islands. Such channels allow the trajectory to be injected in the inner portion of the vicinity. When the trajectory crosses the barrier imposed by the nonhyperbolic regions, it spends a long time abandoning the vicinity of the island, since the barrier also prevents the trajectory from escaping from the neighborhood of the island. In this scenario the nonhyperbolic structures are responsible for the stickiness phenomena and, more than that, the strength of the sticky effect. We show that those properties of the phase space allow us to manipulate the existence of extreme events (and the transport associated to it) responsible for the nonequilibrium fluctuation of the system. In fact we demonstrate that by monitoring very small portions of the phase space (namely, $\ensuremath{\approx}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}%$ of it) it is possible to generate a completely diffusive system eliminating long-time recurrences that result from the stickiness phenomenon.

We claim that dynamical traps displayed by chaotic orbits of non-integrable Hamiltonian systems can be characterized using properties of the finite-time Lyapunov exponent. We show that, for the case where the phase space presents stickiness regions, the distribution of the finite-time Lyapunov exponent is bimodal, while, for the case where no such regions exist, the distribution is a Gaussian-like one.

Master–slave systems have been intensively investigated for modelling the application of chaos in communications. We considered Colpitts oscillators coupled according to a master–slave configuration in order to study chaos synchronization. We revealed the existence of superpersistent transients in this coupled system. The transient is a ubiquitous phenomenon in nonlinear dynamical systems, and it is responsible for important physical phenomena. We characterized superpersistent transients through a scaling law for their average lifetime. Unstable–unstable pair bifurcation has been identified as the generic mechanism for these transients. Moreover, we showed that an additive noise, added to the slave system, may suppress the chaos synchronization. Our results show that synchronization can be achieved for higher coupling strength when there is noise. However, the noise may induce a longer transient if the synchronization is not suppressed.

Lagrangian coherent structures are effective barriers, sticky regions, that separate phase space regions of different dynamical behavior. The usual way to detect such structures is via finite-time Lyapunov exponents. We show that similar results can be obtained for single-frequency systems from finite-time rotation numbers, which are much faster to compute. We illustrate our claim by considering examples of continuous and discrete-time dynamical systems of physical interest., Comment: 4 pages, 3 figures

We review symplectic nontwist maps that we have introduced to describe Lagrangian transport properties in magnetically confined plasmas in tokamaks. These nontwist maps are suitable to describe the formation and destruction of transport barriers in the shearless region (i.e., near the curve where the twist condition does not hold). The maps can be used to investigate two kinds of problems in plasmas with non-monotonic field profiles: the first is the chaotic magnetic field line transport in plasmas with external resonant perturbations. The second problem is the chaotic particle drift motion caused by electrostatic drift waves. The presented analytical maps, derived from plasma models with equilibrium field profiles and control parameters that are commonly measured in plasma discharges, can be used to investigate long-term transport properties.

In this letter we study how deterministic features presented by a system can be used to perform direct transport in a {\it quasi}-symmetric potential and weak dissipative system. We show that the presence of nonhyperbolic regions around acceleration areas of the phase space plays an important role in the acceleration of particles giving rise to direct transport in the system. Such effect can be observed for a large interval of the weak asymmetric potential parameter allowing the possibility to obtain useful work from unbiased nonequilibrium fluctuation in real systems even in a presence of a {\it quasi}-symmetric potential., Comment: 4 pages, 3 figures

We describe the onset of spatiotemporal chaos in a spatially extended nonlinear dynamical system as a result of the loss of transversal stability of an invariant manifold representing a spatially homogeneous and temporally chaotic state. The onset of spatiotemporal chaos is characterized by the switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency.

We analyze alterations on the electrostatic turbulence in experiments with imposed bias to control the plasma radial electric field in Texas Helimak (K. W. Gentle and H. He, Plasma Sci. Technol. 10, 284 (2008)), a toroidal plasma device with a one-dimensional equilibrium, magnetic curvature, and shear. Comparing discharges from different biased potentials, we identify, in a roughly uniform gradient region, a continuous variation from low turbulence level and narrower frequency spectra, for negative bias, to high turbulence level and broadband spectra for positive bias. Overall, we distinguish two kinds of perturbed turbulence, classified according to their intensity, spectral, statistical, and recurrence properties. When the bias is positive, the turbulence shows enhanced and broadband spectra with non Gaussian probability distribution functions having noticeable long tails (extreme events) similar to the turbulence in tokamak scrape-off layer. On the other hand, negative bias reduces the turbulence level and decreases the spectrum widths. Also for negative bias, we found large frequency widths whenever the coupling between drift waves and the sheared plasma flow is fast enough to allow the enhancement of sidebands modes.

Shearless transport barriers appear in confined plasmas due to non-monotonic radial profiles and cause localized reduction of transport even after they have been broken. In this paper we summarize our recent theoretical and experimental research on shearless transport barriers in plasmas confined in toroidal devices. In particular, we discuss shearless barriers in Lagrangian magnetic field line transport caused by non-monotonic safety factor profiles. We also discuss evidence of particle transport barriers found in the TCABR Tokamak (University of S˜ ao Paulo) and the Texas Helimak (University of Texas at Austin) in biased discharges with non-monotonic plasma flows. (Some figures may appear in colour only in the online journal)

We investigated the transition to wave turbulence in a spatially extended three-wave interacting model, where a spatially homogeneous state undergoing chaotic dynamics undergoes spatial mode excitation. The transition to this weakly turbulent state can be regarded as the loss of synchronization of chaos of mode oscillators describing the spatial dynamics.

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds.