Fluctuational Escape from Chaotic Attractors

Fluctuational transitions between two coexisting attractors are investigated. Two different systems are considered: the periodically driven nonlinear oscillator and the two-dimensional map introduced by Holmes. These two systems have smooth and fractal boundaries, respectively, separating their coexisting attractors. It is shown that, starting from a cycle embedded in the chaotic attractor, the periodically-driven oscillator escapes to a saddle cycle at the boundary of the basin of attraction, and does so through sequential transitions between saddles cycles embedded in the attractor. In the case of discrete dynamics with locally disconnected fractal boundaries, it is shown that escape from an attractor always seems to occur through an accessible boundary orbit and further through the specific homoclinic points forming a fractal structure of the boundary. It is shown that analysis of fluctuational transitions between attractors can be used to solve a problem of the energy-optimal migration of a chaotic system. The deterministic optimal control functions are identified with the corresponding optimal fluctuational forces in the limit of small noise intensity. We discuss possible applications and related unsolved problems of stochastic dynamics.