Double Transient Chaotic Behaviour of a Rolling Ball

Abstract

The attractor of a dynamical system is a subset of the state space to which orbits originating from typical initial conditions tend as time increases. An attractor is called chaotic or strange if it has a fractal structure in the phase space. While in case of permanent chaos the phase points of the system do never leave the chaotic attractor, if the chaos is transient the trajectory is staying only a finite time at the chaotic attractor and after it runs asymptotically into a simple attractor. If two or more attractors coexist, trajectories may hesitate for a long time before getting captured by one of the attractors. The basin of attraction of an attractor is the set of initial conditions which produces trajectory approaching the attractor. Fractal basin boundaries are common properties of dynamical systems [1]. Trajectories starting from a fractal boundary show often a transient chaos. In dissipative systems without any driving all motion must eventually cease because of the continuous decay of the energy. In this case, sequential magnifications of the phase space indicate that the set of long lifetimes becomes increasingly sparse at sufficiently small scales, so we can find that the fractality of the basin boundary is scale-dependent (the fractaldimension of the basin boundaries is found to decrease with magnification 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 driving is added. For example in the case of external excitation unstable periodic orbits immediately appear, and the long term dynamics tend to permanent chaos [2].