Solution techniques for pulse problems in non-linear stochastic dynamics

Abstract

Advantages and disadvantages of available solution techniques for pulse problems in non-linear stochastic dynamics are discussed. First, random pulse problems, both, those which do and do not lead to Markov theory, are presented. Next, the analytical and analytically numerical techniques suitable for Markov response problems such as moment equations, Petrov–Galerkin and cell-to-cell mapping techniques are briefly discussed. Usefulness of these techniques is limited by the fact that effectiveness of each of them depends on the mean rate of impulses. Another limitation is the size of the problem, i.e. the number of state variables of the dynamical system. In contrast, the applicability of the simulation techniques is not limited to Markov problems, nor is it dependent on the mean rate of impulses. Moreover their use is straightforward for a large class of point processes, at least for renewal processes.