Asymptotic boundedness for stochastic coupled systems on networks with Markovian switching

• White noise and color noise are considered in the stochastic coupled networks.
• A new approach combining Lyapunov method and some graph theory is presented.
• The sufficient principle obtained by using it M-matrix technique is easy to be verified.
• The impact of Markovian switching and coupled structure is showed by two examples.

In this paper, a novel class of stochastic coupled systems on networks with Markovian switching is presented. In such model, the white noise, the color noise and the coupling between different vertices of the network are taken into account. Focusing on the boundedness problem, this paper employs the Lyapunov method, some graph theory and the method of M-matrix to establish some simple and easy-verified boundedness criteria. These criteria can directly show the link between the graph structure of the network and the dynamics of coupled systems. Finally, stochastic coupled van der Pol׳s equations with Markovian switching are used to demonstrate our findings. Meanwhile, two numerical examples are also provided to clearly show the influence of coupled structure on the boundedness of coupled systems.