Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

Systems whose specifications change abruptly and statistically, referred to as Markovian-jump 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 Ito 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.