Boundedness Analysis For Stochastic Coupled Systems On Networks With MARKOV Switching

Recently, coupled systems have wide range of applications in many fields. But lessattention was paid on the stochastic coupled systems under the perturbation of many sortsof noise. In addition, a lot of researchers focus on the stability of coupled systems, butrelatively few scholars study the boundedness. Therefore, the investigation on the bound-edness of stochastic coupled systems, subjected to various types of environmental noise,including white noise and color noise, is a hot and significant topic. In this paper, theboundedness of stochastic coupled systems with Markov switching is mainly consideredbased on the Lyapunov method, the graph theory and M-matrix.Firstly, by combining graph theory and Lyapunov function of vertex systems, theLyapunov function of stochastic coupled systems is constructed. Some su?cient criteriaon the boundedness of systems are given by using Lyapunov method. Additionally, bycombining the theory of M-matrix, other su?cient principles are given in the form ofcoeficients of a system to judge the boundedness of the system. The conditions in theseprinciples are easily verified in practice.Secondly, the boundedness of every vertex system is obtained, respectively. Andthe relationship between the boundedness of vertex systems and the boundedness of largesystem after coupling is established, that is, under some conditions established in thepaper, the coupled system is bounded whenever the vertex systems are bounded or not.Finally, two particular examples are given to demonstrate the applicability of thetheory established in this paper. One is stochastic coupled Van der Pol's equations withMarkov switching. The other is stochastic Cohen-Grossberg neural networks with Markovswitching. The su?cient conditions for the boundedness of two systems above are provid-ed, respectively. The detailed discussion for coupled Van der Pol's equations under twocoupled ways are presented. Numerical simulation are also considered, which reveals thegreat significance of the coupling structure on the boundedness of system.