Fluctuational escape from a chaotic attractor

Noise-induced escape from a non-hyperbolic attractor, and from a quasihyperbolic attractor with nonfractal boundaries, is investigated by means of analogue experiments and numerical simulations. It is found that there exists a most probable (optimal) escape trajectory, the prehistory of the escape being de ned by the structure of the chaotic attractor. The corresponding optimal uctuational force is found. The possibility of achieving analytic estimates of the escape probability within the framework of Hamiltonian formalism is discussed. 1 The escape problem for chaotic systems There have been recently a number of interesting studies of the interplay of noise with chaotic behaviour [1, 2] including e.g. a demonstration of chaotic features in a purely stochastic Kramers oscillator [3], the phenomena of both noise-induced instability [4] and noise-induced order [5], noise-induced chaos [6] and quantum noise-induced chaotic oscillations [7]. However, the analytic estimation of the probability of noise-induced escape from the basin of attraction of a chaotic attractor remains an unsolved fundamental problem in the theory of uctuations [8, 9]. It is of broad interdisciplinary interest in view of a host of important applications including e.g. stabilisation of the voltage standard [10], neuron dynamics [11], and laser systems [12]. It has now been established, however, that the uctuational dynamics of escape can be investigated directly through measurements of the prehistory probability distribution [13{16]. The underlying idea is based on the concept of large uctuations [17], in which the system uctuates to the remote state along an optimal path. Mathematical equivalents of this physical concept are asymptotic equations for the solution of the Fokker-Plank equation written in the form of rays (Hamilton equations), or wavefronts (HamiltonJacobi equation) [18]. In this method the dynamical variables of the system, and sometimes also the external force, are recorded simultaneously, and the statistics of all actual trajectories along which system evolves to a given state are then analyzed [13]. The advantages of this technique were demonstrated Published in Stochastic Processes in Physics, Chemistry and Biology, eds. J. A. Freund and T. Poeschel, Springer, Berlin, 2000, pp 378-389.