Periodic Solutions Of Several Coupled Systems

Complex systems have been one of the important topics in the 21 th century.Complex systems in reality can be described by coupled systems,and their main applications depend on the dynamics of coupled systems.Therefore,many scholars pay attention to the dynamics properties of coupled systems,and many results have been reported.However,the results are mainly related to the synchronization and stability of coupled systems.As we all know,periodicity is also one of the important dynamics properties.In this paper,we study the existence of periodic solutions for several coupled systems.The results can enrich the existing theoretical results,and provide theoretical basis for applications of many fields.The results mainly include five points.1.The existence of periodic solutions for neutral coupled oscillators with feedback and time-varying delays.By making use of the coincidence degree theory,graph theory,and Lyapunov method,we obtain two kinds of sufficient criteria about the existence of periodic solutions for neutral coupled oscillators with feedback and time-varying delay,i.e.a Lyapunov-type criterion and a coefficients-type criterion.The results imply that when the network topology property of coupled oscillators,feedback control,and timevarying delays satisfy certain conditions,the periodic solutions exist.2.The existence and global exponential stability of periodic solutions for coupled control systems with feedback and time delays.By introducing feedback and time delays into coupled control systems,and the existence and global exponential stability of periodic solutions of the systems are analyzed.We obtain sufficient criteria about the existence and global exponential stability of periodic solutions of the systems.Finally,applying our results to a class of Lurie coupled control systems,we obtain sufficient conditions of the existence and global exponential stability of periodic solution.The results imply that when feedback control,time delays,coupling form and intensity satisfy certain conditions,the global exponentially stable periodic solutions exist.3.The existence of periodic solutions for discrete time coupled systems with timevarying delays.By introducing time-varying delays into discrete time coupled systems,we obtain a Lyapunov-type sufficient criterion and a coefficients-type criterion about the existence of periodic solutions.Finally,by applying our results to discrete time coupled oscillators systems with time-varying delays,we obtain sufficient conditions for the existence of periodic solutions.The results imply that when the form and intensity of coupling,and time-varying delays satisfy certain conditions,the periodic solutions exist.4.The existence of Periodic solutions for discrete time periodic time-varying coupled systems.Generalizing intensity of constants coupling to intensity of periodic timevarying coupling about discrete time coupled systems.By making use of the coincidence degree theory,graph theory,and Lyapunov method,we obtain sufficient criteria about the existence of periodic solution of the systems.Moreover,we apply our results to discrete time Cohen-Grossberg Neural Networks and obtain the existence of periodic solutions for it.The results imply that when the form and intensity of time-varying coupling,satisfy certain conditions,the periodic solutions exist.5.The existence of periodic solutions for stochastic multi-group models with multidispersal.Generalizing single digraph to multi-digraph and considering noise perturbation,we establish stochastic multi-group models with multi-dispersal.By making use of stochastic analysis skills,graph theory,and Lyapunov method,we obtain sufficient criteria about the existence of periodic solutions for stochastic multi-group models with multi-dispersal.Moreover,we apply our results to stochastic coupled oscillators on two networks and obtain the existence of periodic solutions for them.The results imply that when multi-dispersal influences among different groups and intensity of noise pertubation satisfy certain conditions,the periodic solutions exist.