Stochastic P-bifurcation analysis of a class of nonlinear Markov jump systems under combined harmonic and random excitations

Stochastic Markov jump systems are commonly used to describe complicate practical systems with switching structures such as power plants and communication networks. This paper presents analytical studies of a nonlinear Markov jump system under combined harmonic and noise excitations. Combining the weighted-average method, stochastic averaging, and finite difference method, the stationary responses and bifurcations of a nonlinear Markov jump system under combined harmonic and noise excitations are investigated. In deterministic case, the existence of Markov jump process can transform the stationary responses of system from limit cycle to diffusion limit cycle. In the stochastic case, we analyze the stationary probability density functions (SPDFs) of the amplitude and the joint SPDF, finding that the Markov jump process can induce the appearance of stochastic P-bifurcation. An increasing transition rate λ₁₂ (or λ₂₁) transfers SPDFs of amplitude from one-peak to two-peak and then to one-peak and remaining unchanged. Numerical simulations show basic agreement with our theoretical predictions.