Fractal boundaries for exit in Hamiltonian dynamics

Fractal boundaries can occur for certain situations involving chaotic Hamiltonian systems. In particular, situations are considered in which an orbit can exit from the system in one of several different ways, and the question is asked which of these ways applies for a given initial condition. As an illustration, specific examples are considered for which there are two possible ways in which a particle can exit from the system. We examine the space of initial conditions to see which of the two exit possibilities applies for each initial condition. It is found that the regions of initial-condition state space corresponding to the two exit modes are separated by a boundary that has both fractal and smooth (nonfractal) regions, for one example, and by a fractal boundary for the other example. Furthermore, it is found for the example where the boundary has fractal and smooth regions that these regions are intertwined on arbitrarily fine scale. The existence of fractal boundaries is conjectured to be a typical property of chaotic Hamiltonian dynamics with multiple exit modes. Two situations in space physics where our results may be relevant are discussed.