Synchronized stationary distribution criteria for stochastic delayed coupled systems with Markovian switching

This paper investigates the existence of synchronized stationary distributions for stochastic coupled systems with time delays and Markovian switching (SCSDM). A novel approach, which integrates graph theory, the convergence and uniform boundedness theorem, and the Lyapunov method based on a feedback control strategy, is employed to analyze stationary distributions. Sufficient conditions are established to guarantee the existence of such distributions, including a Semi-Lyapunov type theorem and a coefficients-type theorem. Furthermore, the theoretical results are applied to a specific class of stochastic coupled oscillators with time delays and Markovian switching. Numerical simulations and an illustrative example are provided to validate the theoretical findings and demonstrate their applicability.