An application of the path-integral approach to an analysis of the
fluctuations in complex dynamical systems is discussed. It is
shown that essentially the same ideas underly recent progress in
the solutions of a number of long-standing problems in complex
dynamics. In particular, we consider the problems of prediction,
control and inference of chaotic dynamics perturbed by noise in
the framework of path-integral approach.
We study fluctuational transitions in a discrete dynamical system
between two co-existing chaotic attractors separated by a
fractal basin boundary. It is shown that there is a generic
mechanism of fluctuational transition through a fractal boundary
determined by a hierarchy of homoclinic original saddles. The most
probable escape path from a chaotic attractors to the fractal boundary is found using both statistical analysis of fluctuational trajectories and Hamiltonian theory of fluctuations.
Fluctuational escape via an unstable limit cycle is investigated
in stochastic flows and maps. A new topological method is
suggested for analysis of the corresponding boundary value
problems when the action functional has multiple local minima
along the escape trajectories and the search for the global
minimum is otherwise impossible. The method is applied to the
analysis of the escape problem in the inverted Van der Pol
oscillator and in the Henon map. An application of this technique
to solution of the escape problem in chaotic maps with fractal
boundaries, and in maps with chaotic saddles embedded within the
basin of attraction, is discussed.