Iberê L. Caldas

Calculation of Lyapunov exponents in systems with impacts

We apply a model based algorithm for the calculation of the spectrum of the Lyapunov exponents of attractors of mechanical systems with impacts. For that, we introduce the transcendental maps that describe solutions of integrable differential equations, between impacts, supplemented by transition conditions at the instants of impacts. We apply this procedure to an impact oscillator and to an impact-pair system (with periodic and chaotic driving). In order to show the method precision, for large parameters range, we calculate Lyapunov exponents to classify attractors observed in bifurcation diagrams. In addition, we characterize the system dynamics by the largest Lyapunov exponent diagram in the parameter space.

Controlling chaotic orbits in mechanical systems with impacts

Abstract We stabilize desired unstable periodic orbits, embedded in the chaotic invariant sets of mechanical systems with impacts, by applying a small and precise perturbation on an available control parameter. To obtain such perturbation numerically, we introduce a transcendental map (impact map) for the dynamical variables computed just after the impacts. To show how to implement the method, we apply it to an impact oscillator and to an impact-pair system.

Escape patterns, magnetic footprints, and homoclinic tangles due to ergodic magnetic limiters

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.

SCRAPE-OFF LAYER INTERMITTENCY IN THE CASTOR TOKAMAK

Spatial–temporal intermittency of floating potential and ion saturation current fluctuations is analyzed by using data obtained from two probes arrays in the scrape-off layer of the CASTOR tokamak [Proceedings of the 1996 International Conference on Plasma Physics (Nagoya) (International Atomic Energy Agency, Vienna, 1997), Vol. I, p. 322]. For these ion saturation current fluctuations with non-Gaussian probability density functions, a conditional averaging analysis shows coherent structures with correlation lengths and lifetimes larger for larger amplitude conditions. Nevertheless, there is no evidence of such large structures in the potential fluctuations. Furthermore, wavelet transforms are used to analyze these nonstationary fluctuations and obtain details not observed with the Fourier technique. So, examining wavelet power and coherence spectra, strong intermittency is found for both kinds of fluctuations, in a time scale two orders of magnitude higher than that observed in the conditional analysis. ...

Basins of Attraction and Transient Chaos in a Gear-Rattling Model

The authors numerically investigate basins of attraction of coexisting periodic and chaotic attrac tors in a gear-rattling impact model. These attractors are strongly dependent on small changes of the initial conditions. Gradually varying a control parameter, the size of these basins of attraction is modified by global bifurcations of their boundaries. Moreover, the topology of these basins is also modified by appearance or disappearance of coexisting attractors. Furthermore, for the considered control parameter range, the frac tal basin boundaries are so interleaved that trajectories are practically unpredictable in some regions of phase space. The authors also examine an example of a crisis on which a chaotic attractor is converted into a chaotic transient that goes to a periodic attractor. For this crisis, the authors show the evolution of transient lifetime dependence of the initial conditions as the control parameter is varied.

Transport properties in nontwist area-preserving maps.

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.

Recurrence time statistics for finite size intervals

We investigate the statistics of recurrences to finite size intervals for chaotic dynamical systems. We find that the typical distribution presents an exponential decay for almost all recurrence times except for a few short times affected by a kind of memory effect. We interpret this effect as being related to the unstable periodic orbits inside the interval. Although it is restricted to a few short times it changes the whole distribution of recurrences. We show that for systems with strong mixing properties the exponential decay converges to the Poissonian statistics when the width of the interval goes to zero. However, we alert that special attention to the size of the interval is required in order to guarantee that the short time memory effect is negligible when one is interested in numerically or experimentally calculated Poincare recurrence time statistics.

The structure of chaotic magnetic field lines in a tokamak with external nonsymmetric magnetic perturbations

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.

Field line diffusion and loss in a tokamak with an ergodic magnetic limiter

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.

Reduction of chaotic particle transport driven by drift waves in sheared flows

Investigations of chaotic particle transport by drift waves propagating in the edge plasma of tokamaks with poloidal zonal flow are described. For large aspect ratio tokamaks, the influence of radial electric field profiles on convective cells and transport barriers, created by the nonlinear interaction between the poloidal flow and resonant waves, is investigated. For equilibria with edge shear flow, particle transport is seen to be reduced when the electric field shear is reversed. The transport reduction is attributed to the robust invariant tori that occur in nontwist Hamiltonian systems. This mechanism is proposed as an explanation for the transport reduction in Tokamak Chauffage Alfven Bresilien [R. M. O. Galvao et al., Plasma Phys. Controlled Fusion 43, 1181 (2001)] for discharges with a biased electrode at the plasma edge.