Stability And Synchronization Of Coupled Systems On Networks With Reaction-diffusion Terms

In the real world, coupled systems on networks are extensively applied in the fieldsof physics, biology, and engineering. Thus they have aroused high attention of domesticand international scholars. However, the analysis for dynamic properties of coupledsystems generally is a complex and formidable task, for the reason that the dynamics ofthe coupled systems depend on not only the individual vertex dynamics but also thecoupling topology. This will increase the difficulty of the research.On the basis of the reality, when considering finite transmission of interaction,time delay is always unavoidable. In fact, except the changes in time, the state of thenodes also depends on the influence of the space. Therefore, to describe the dynamicalchanges more accurately, both the existence of time delay and the impression ofreaction should be taken into consideration, which will make the systems more complex.Consequently, dynamic properties of delayed coupled systems on networks withreaction-diffusion terms have been the focus of the research.As two of the most important dynamics of coupled systems on networks, thestability and synchronization have attracted increasing attention. It is well-known thatthe most effective method is Lyapunov method, which is not perfect. For some complexsystems, how to construct proper Lyapunov functions is still an open problem.As is known to all, coupled systems on networks are composed of plentifulinterconnected dynamical nodes. To solve the above problems, a digraph will be used todescribe a coupled system in this paper. Graph theory will be brought in. Combiningwith graph theory and Lyapunov method, this paper will study the stability andsynchronization for coupled systems on networks with reaction-diffusion termsrespectively. By making use of vertex-Lyapunov functions and the topological structure,a systematic method will be proposed to construct Lyapunov functions for givencoupled systems on networks. Sequentially, judgment theorems will be provided.Firstly, this paper will study the exponential stability for a coupled system withmixed delays and rection-diffusion terms. The first theorem is presented in the form ofLyapunov functions. Furthermore, on the basis of the first theorem, a second theoremwill be presented for more convenient to judgement by utilizing the coefficients ofcoupled systems. Meanwhile, a numerical example will be illustrated to show theeffectiveness of the proposed criteria.Secondly, the exponential synchronization for delayed coupled systems withrection-diffusion terms will be studied. Two different kinds of judgment theorems willbe given, keeping from finding the Lyapunov function of a coupled system directly.Finally, a numerical example will be illustrated to show the effectiveness of the obtainedresults.