The Analysis Of Stability And Synchronization Of Coupled Systems Based On Graph-theoretic Approach

The study of complex networks has become the common focus of many fields such as mathematics, physics, biology and social sciences. Dynamical behaviors of many complex networks can be described by coupled systems on networks. Hence, researchers in many fields are interested in the research about coupled systems on networks. However,few researchers studied the case about stochastic coupled systems driven by many kinds of environmental noise and coupled systems with multiple dispersal. Additionally, stability and synchronization are very important properties of coupled systems. Therefore, the researches about stability and synchronization of stochastic coupled systems driven by white noise, telegraph noise or L′evy noise and coupled systems with multiple dispersal are very novel. Based on graph-theoretic approach, this thesis aims to study the stability and synchronization of stochastic coupled systems driven by many kinds of noise and stability of coupled systems with multiple dispersal by combining stability theory and synchronization theory. The contents of this thesis are as follows.First topic is to study the boundedness of stochastic coupled Van der Pol oscillators based on graph-theoretic approach. The global Lyapunov function of stochastic coupled Van der Pol oscillators is constructed by using graph-theoretic approach and Lyapunov functions of vertex systems. With the help of boundedness theory, criteria about boundedness of stochastic coupled Van der Pol oscillators are presented. These results can show that stochastic coupled Van der Pol oscillators can be bounded by adjusting the network of oscillators under appropriate perturbation.Second topic is to study the stability of neutral stochastic coupled oscillators with time-varying delays based on graph-theoretic approach. By using graph-theoretic approach, stability theory of stochastic differential equations and theory of functional differential equation, neutral stochastic coupled oscillators with white noise are studied. Criteria are deduced to ensure moment exponential stability and almost surely exponential stability of the addressed system. These results show that the stability of neutral stochastic coupled oscillators can be effectively investigated based on graph-theoretic approach.In addition, the approach for stability in this part can also be used to study the stability of other neutral stochastic coupled systems.Third topic is to study the stability of stochastic coupled systems with L′evy noise based on graph-theoretic approach. By utilizing graph-theoretic approach and stability theory of stochastic differential equations, stochastic coupled systems with L′evy noise are investigated and then criteria about moment exponential stability and stability in probability are showed. These theoretical results are applied to study the stability of stochastic coupled oscillators with L′evy noise and stochastic Volterra predator–prey system with L′evy noise. Some corresponding stability criteria are obtained for them. There results can show that stochastic coupled systems with L′evy noise can be studied based on graph-theoretic approach. Stochastic coupled systems with L′evy noise can be stable under appropriate disturbance and coupling.Fourth topic is to study the synchronization of stochastic coupled systems with white noise and telegraph noise based on graph-theoretic approach. By combining graphtheoretic approach, synchronization theory, M-matrix theory and state feedback control technique, several su?cient conditions ensuring moment exponential synchronization and almost surely exponential synchronization between stochastic drive-response coupled systems are given. These theoretical results are applied to study the synchronization of stochastic Cohen-Grossberg neural networks with white noise and telegraph noise. The corresponding synchronization criterion is obtained. These results show that the synchronization of coupled systems with white noise and telegraph noise can be investigated by the graph-theoretic approach and the synchronization criteria are closely related with topological properties of corresponding network and generator of the Markov chain.Fifth topic is to study the stability of coupled systems with multiple dispersal based on graph-theoretic approach. The graph-theoretic approach based on a single graph is generalized to the case on multi-graph. By using the graph-theoretic approach based on multiple graphs, stability of general coupled systems with multiple dispersal is obtained.These theoretical results are applied to study the stability of multi-group predator-prey models with predator and prey dispersal and coupled oscillators. Some corresponding stability criteria are obtained for them. These results show that graph-theoretic approach is much more effective to study the coupled systems with multiple dispersal networks.