Energy-optimal steering of transitions through a fractal basin boundary

We study fluctuational transitions in a discrete dynamical system having two co-existing attractors in phase space, separated by a fractal basin boundary. It is shown that transitions occur via a unique accessible point on the boundary. The complicated structure of the paths inside the fractal boundary is determined by a hierarchy of homoclinic original saddles. By exploiting an analogy between the control problem and the concept of an optimal fluctuational path, we identify the optimal deterministic control function as being equivalent to the optimal fluctuational force obtained from a numerical analysis of the fluctuational transitions between two states.