Imperfect fractal repellers and irregular families of periodic orbits in a 3-D model potential

Abstract

A model, plane symmetric, 3-D potential, which preserves some features of galactic problems,is used in order to examine the phase space structure through the study of the properties of orbits crossing perpendicularly the plane of symmetry. It is found that the lines formed by periodic orbits, belonging to Farey sequences, are not smooth neither continuous. Instead they are deformed and broken in regions characterised by high Lyapunov Characteristic Numbers (LCN's). It is suggested that these lines are an incomplete form of a fractal repeller, as discussed by Gaspard and Baras (1995), and are thus closely associated to the ``quasi-barriers'' discussed by Varvoglis et al. (1997). There are numerical indications that the contour lines of constant LCN's possess fractal properties. Finally it is shown numerically that some of the periodic orbits -members of the lines- belong to true irregular families. It is argued that the fractal properties of the phase space should affect the transport of trajectories in phase or action space and,therefore, play a certain role in the chaotic motion of stars in more realistic galactic potentials.