Abstract
The stability of coupled systems with time-varying coupling structure (CSTCS) is considered in this paper. The graph-theoretic method on a digraph with constant weight has been successfully generalized into a digraph with time-varying weight. In addition, we construct a global Lyapunov function for CSTCS. By using the graph theory and the Lyapunov method, a Lyapunov-type theorem and some sufficient criteria are obtained. Furthermore, the theoretical conclusions on CSTCS can successfully be applied to the predator-prey model with time-varying dispersal. Finally, a numerical example of CSTCS is given to illustrate the effectiveness and feasibility of our results.
Highlights
During the past few decades, coupled systems (CSs) have been used to model a wide variety of systems in many fields, such as physics [ – ], biology [ – ] and social science [ ]
The method has been extended to stochastic systems [ – ], discrete time systems [ ], and time delay systems [ ]
When some regions discontinue with each other due to some reasons, the transmission rate of infectious diseases will be lower. All these facts can illustrate that time-varying coupling structure should not be neglected
Summary
During the past few decades, coupled systems (CSs) have been used to model a wide variety of systems in many fields, such as physics [ – ], biology [ – ] and social science [ ]. In [ , ], Li et al used graph theory to explore the global stability for general coupled systems of ordinary differential equations Following this pioneering work in [ , ], many scholars have studied the dynamics of CSs by this technique and obtained a number of conclusions [ – ]. When some regions discontinue with each other due to some reasons, the transmission rate of infectious diseases will be lower All these facts can illustrate that time-varying coupling structure should not be neglected. Many researchers, including us, have given some results of global stability of CSs. in our previous work we do not consider the time-varying coupling structure.
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