The Stability Of Stochastic Coupled Systems With Time Delays And Time-varying Coupling Structure

In this paper, stochastic coupled systems with time delays and time-varying coupling structure(SCTC) are investigated. Since systems in many domains can be modeled by coupled systems, such as physics, mechanics and biology and so on, many scholars have investigated coupled systems extensively. The stability of coupled systems is necessary in practical applications, thus it is important to investigate the stability of coupled systems. The stability of coupled systems is affected by some factors, such as environmental noises, time delays and coupling structure. Generally, the coupling structure of coupled systems is not constant, and coupled systems with time-varying coupling structure can model systems in practice more reasonably. Therefore, the investigation of SCTC is important and significant in practice.Lyapunov method has been used by more and more scholars in investigating the stability of coupled systems so far. However, it is not easy to construct a global Lyapunov function directly. Motivated by Michael. Li et al., in this paper, we use graph theory and Lyapunov method to construct an appropriate Lyapunov function for SCTC, which avoids the difficulty of constructing Lyapunov function directly.In Chapter 2 of this paper, we establish SCTC on strongly connected digraphs toinvestigate the stability of the trivial solution. Based on graph theory and Lyapunovmethod, a Lyapunov-type theorem and coefficients-type criteria are presented.Correspondingly, theoretical results are applied to stochastic coupled oscillators withtime-varying coupling structure(SCOT), and a stability criterion is obtained. Finally,two numerical examples are presented to check the feasibility of theoretical results.In Chapter 3 of this paper, we establish SCTC on a digraph without strong connectivity(DWSC) to investigate the stability of the trivial solution. Firstly, a hierarchical method and algorithm for a condensed digraph are given to layer a DWSC into some strongly connected components. A Lyapunov-type theorem and a coefficients-type criterion are presented by the theory of asymptotically autonomous systems, graph theory and Lyapunov method. Theoretical results are applied to SCOTCS, and a stability criterion is obtained. Finally, we present a numerical example to show the effectiveness of results obtained.