Reliability assessment of dynamical systems with random excitation

During recent years, a variety of important interconnections between the theory of nonlinear dynamical systems, nonlinear stochastically excited systems, and nonlinear control systems have been obtained by mathematicians, physicists, and engineers. This paper is the beginning of a systematic study of reliability aspects within the emerging unified view of the different classes of nonlinear dynamical systems. We study the response behavior of dynamical systems with random excitation depending on two parameters: bifurcation parameter αεR, and a parameter ρ⩽0 that measures the strength of the disturbance. The goal of this paper is to obtain precise stability and reliability diagrams in α-ρ-space. In particular, we characterize for each parameter combination the levers that the (maximal) system response will reach with probability 1, with positive probability, or with probability 0. Emphasis is placed on determining the parameter combinations for which the system behavior changes drastically, i.e., on stochastic bifurcation phenomena, to determine crucial system parameters and their critical values