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 ${\lambda }_{12}$ (or ${\lambda }_{21}$) 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.

随机马尔可夫跳跃系统通常用于描述具有开关结构的复杂实际系统，例如发电厂和通信网络。本文介绍了在组合谐波和噪声激励下的非线性马尔可夫跳跃系统的分析研究。结合加权平均法、随机平均法和有限差分法，研究了谐波和噪声组合激励下非线性马尔可夫跳跃系统的平稳响应和分岔。在确定性情况下，马尔可夫跳跃过程的存在可以将系统的平稳响应从极限环转变为扩散极限环。在随机情况下，我们分析振幅和联合 SPDF 的平稳概率密度函数 (SPDF)，发现马尔可夫跳跃过程可以诱导随机 P 分叉的出现。不断增加的过渡率${\lambda }_{12}$ （要么 ${\lambda }_{21}$) 将幅度的 SPDF 从一个峰值转移到两个峰值，然后再转移到一个峰值并保持不变。数值模拟显示与我们的理论预测基本一致。