First-Passage Problems in Randomly Perturbed Dynamical Systems

Countless problems faced by engineers and scientists deal with the reliability of dynamical systems excited by random forces. Such excitations or disturbances may cause large random deviations of the system's response above critical threshold levels. After the first crossing of that level the system may immediately fail or it may switch to another unwanted regime. Examples include the prediction of the probability of capsizing of ships in rough seas, of the toppling of free-standing structures due to ground motions during earthquake, of the ``snap-through'' of shell-type structures due to acoustic noise, of the extinction or explosion of biological populations due to random environmental fluctuations, etc. This research project addresses the prediction of probabilities of such y``first-passage'' events, hence providing a measure of system performance or reliability.

This project consists of three parts. In the first part, first-passage probabilities are determined analytically by nonlocal, asymptotic methods based on Markov process theory. These nonlocal methods, based on singular perturbation expansion of boundary value problems, take into account the strongly nonlinear character of the governing equations inherent to large excursion problems. Both single and multiple degree-of-freedom systems are considered with wide-band or narrow-band, Gaussian or non-Gaussian random excitation. One seeks to obtain first-passage probabilities, mean residence times, most probable trajectories, and expected number of crossings.

In the second part, we conduct digital simulation to assess the range of validity of the asymptotic method. Since the probability of first-passage events of interest are generally very small, digital simulation of these events can be prohibitively long. To alleviate this problem, we seek to ``accelerate'' these simulations, that is, to increase the probability of rare events by changing the measure of the original probability space. The idea is to follow the optimal, most probable paths which lead to exits from a safe region of state space. The determination of these paths are given asymptotically by the theory of large deviations which hence gives the form of the transformation to be used. A form of importance sampling is obtained. The resulting numerical method offers a powerful alternative for the prediction of failure probabilities.

In the third part, we conduct analog electronic experiments of the mathematical models of interest to provide real physical systems which can yield in ``real-time'' a large flow of results with speed and flexibility. They provide a useful hands-on complement to digital simulations of nonlinear stochastic dynamics.