Fluctuational transitions across locally-disconnected and locally-connected fractal basin boundaries

We study fluctuational transitions in a discrete dynamical system that has two co-existing attractors in phase space, separated by a fractal basin boundary which may be either locally-disconnected or locally-connected. It is shown that, in each case, transitions occur via an accessible point on the boundary. The complicated structure of paths inside the locally-disconnected fractal boundary is determined by a hierarchy of homoclinic original saddles. The most probable escape path from a regular attractor to the fractal boundary is found for the each type of boundary using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.