Computing Complex Horseshoes by Means of Piecewise Maps

A systematic procedure to numerically compute a horseshoe map is presented. This new method uses piecewise functions and expresses the required operations by means of elementary transformations, such as translations, scalings, projections and rotations. By repeatedly combining such transformations, arbitrarily complex folding structures can be created. We show the potential of these horseshoe piecewise maps to illustrate several central concepts of nonlinear dynamical systems, as for example, the Wada property.

FRACTAL AND WADA EXIT BASIN BOUNDARIES IN TOKAMAKS

The creation of an outer layer of chaotic magnetic field lines in a tokamak is useful to control plasma-wall interactions. Chaotic field lines (in the Lagrangian sense) in this region eventually hit the tokamak wall and are considered lost. Due to the underlying dynamical structure of this chaotic region, namely a chaotic saddle formed by intersections of invariant stable and unstable manifolds, the exit patterns are far from being uniform, rather presenting an involved fractal structure. If three or more exit basins are considered, the respective basins exhibit an even stronger Wada property, for which a boundary point is arbitrarily close to points belonging to all exit basins. We describe such a structure for a tokamak with an ergodic limiter by means of an analytical Poincaré field line mapping.

WADA BASIN BOUNDARIES IN CHAOTIC SCATTERING

Chaotic scattering systems with multiple exit modes typically have fractal phase space boundaries separating the sets of initial conditions (basins) going to the various exits. If the exits number more than two, we show that the system may possess the stronger property that any initial condition which is on the boundary of one exit basin is also simultaneously on the boundary of all the other exit basins. This interesting property is known as the Wada property and basin boundaries having this property are called Wada basin boundaries.