Abstract This paper is concerned with the existence of a synchronized stationary distribution for stochastic multi-links systems with Markov jump (SMMJs). By employing Lyapunov method, Kirchhoff’s Matrix Tree Theorem in graph theory as well as M-matrix method, several criteria are given to guarantee the existence of a synchronized stationary distribution of SMMJs, including the Lyapunov-type theorem and a coefficients-type theorem. As a subsequent, the theoretical results are applied to a class of stochastic Markov jump oscillators with multi-links and stochastic multi-links Chua’s circuits with Markov jump, which indicates the results present widely applied prospect in various physical systems. Eventually, two examples together with numerical simulations are provided to validate the effectiveness of the theoretical results.

中文翻译：

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具有马尔可夫跳跃的随机多链接系统的同步平稳分布

摘要 本文关注具有马尔可夫跳跃 (SMMJ) 的随机多链接系统的同步平稳分布的存在。利用Lyapunov方法、图论中的Kirchhoff矩阵树定理以及M矩阵方法，给出了保证SMMJ同步平稳分布存在的几个准则，包括Lyapunov型定理和系数型定理。随后，将理论结果应用于一类具有多链接的随机马尔可夫跳跃振荡器和具有马尔可夫跳跃的随机多链接蔡氏电路，表明该结果在各种物理系统中具有广泛的应用前景。最后，通过两个例子和数值模拟来验证理论结果的有效性。