Systems whose specifications change abruptly and statistically, referred to as Markovianjump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional Itô equation governing the energy envelope. The associated Fokker-Planck-Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demonstrated by using a nonlinear two-degree oscillator that is stochastically driven and capable of Markovian jumps.

Systems subjected to switching random excitations are practically significant because they include many safety-critical systems such as power plants and communication networks. In this paper, the reliability of multi-degree, nonlinear, non-integrable Hamiltonian systems subjected to switching random excitations is investigated. Such a system is formulated as a continuous–discrete hybrid based upon the Markov jump theory. Stochastic averaging is applied to suppress the rapidly varying parameters of the Markov jump process in order to generate a probability-weighted diffusion equation. The associated backward Kolmogorov equation is then set up, from which the approximate reliability function and probability density of first passage time are obtained. The utility and accuracy of this approximate procedure are demonstrated by two examples.