Study On Dynamic Properties Of Two Types Of Coupled Systems

Since the 21 st century,the research of complex networks has become a common focus in many fields such as mathematics,physics,biology and social sciences.Many complex networks in real life can be described by coupled systems on the network.However,the actual coupled system is always disturbed by various factors.For example,in real life,the time delay is everywhere.In biology,if there is no interaction between patches,some species will be on the verge of extinction,so it is very important to consider the effect of diffusion on the coupled systems.Because the actual coupled systems will be disturbed by various kinds of environmental noise in nature,and the environmental noises will affect the dynamic properties of the system.Therefore,we introduce the environmental noises into coupled systems,and establish a stochastic coupled system which is based on the needs of reality.The application of coupled systems mainly depends their dynamic properties,in which stability,synchronization and the existence of stationary distribution are very important properties of coupled systems.Therefore,in this paper,diffusion,delay and environmental noises are introduced into coupled systems,and the mathematical models of two kinds of coupled systems are established,which are the coupled delay systems based on memristor,and the stochastic multi-group models with dispersal.Their dynamic properties are studied by using graph theory and Lyapunov method.The research contents of this paper are as follows:Firstly,the global exponential stability of coupled delay systems based on memristor is studied.When studying the coupled systems,we consider two factors: memristor and time delay,and then we establish the coupled delay systems based on memristor.Through graph theory and Lyapunov method,the Lyapunov type theorem and the coefficient type theorem which make the system exponentially stable are obtained.Next,the theoretical results are applied to the stability analysis of nonlinear coupled oscillator based on memristor.Finally,a numerical example is given to illustrate the effectiveness of the theoretical results.Secondly,combining Lyapunov method and graph theory,the existence of stationary distribution of stochastic multi-group models with dispersal is studied.Two sufficient conditions to guarantee the existence of stationary distribution are obtained,which are showed in Lyapunov function and system coefficient form.The sufficient conditions obtained are less conservative,which shows that the stationary distribution is closely related to the environmental noises and topological structure.The theoretical results are applied to the stochastic coupled oscillator.Finally,the effectiveness of the theoretical results is illustrated by numerical simulation.Thirdly,by using aperiodic intermittent control,Lyapunov method and graph theory,the existence of synchronized stationary distribution for stochastic multi-group models is analyzed,and two theorems are obtained to guarantee the existence of synchronized stationary distribution,in which the sufficient conditions are less restrictive.The theoretical results show that the synchronized stationary distribution is closely related to the environmental noise and topological structure.Finally,the numerical simulation is to illustrate the availability of the theoretical results.