Double Transient Chaotic Behaviour of a Rolling Ball

The study of relatively simple chaotic systems can provide a deep insight into the deterministic and probabilistic behaviour of the natural processes. The joint presentation of a real system and its mathematical model helps effectively to understand the intricate concepts and ideas used for the description of the physics of chaotic motion. In the present paper the dynamic behaviour of a ball moving in a complexshaped bowl will be studied. It is shown, that this motion exhibit characteristically transient chaotic behaviour, and the boundaries of its attraction basins have typical fractal structure. Sequential magnifications of the phase space have revealed that the fractality of the basin boundaries are scaledependent (the fractal-dimension of the basin boundaries is found to decrease and tend to unit). This behaviour has been termed the doubly transient chaos. It is an interesting fact that the character of chaos changes when a driving force is added. For example in the case of external excitation the unstable periodic orbits immediately appear, and the long term dynamics tend to permanent chaos.

Keywords: Chaotic motion, permanent and transient chaos, fractal basinsntroduction.