Hopf Bifurcation For Two Kinds Of Semi-discrete Delay Differential Systems

Many natural and artificial systems can be abstracted into delay differential systems.And many systems will occur bifurcation.In many cases,it is hoped that the generated bifurcation is beneficial.However,in real life,the effect of some bifurcation is negative,which requires us to control bifurcation of such systems.Therefore,the existence of bifurcation and bifurcation control for delay differential systems have become a hot research field in recent years.Semi-discretization is a well-known technique used.The basic idea is to discretize some items in the system so that only finite number of discrete points need to be observed.It is more economical and practical than the continuous-time observation of systems.Therefore,it is significant to apply the semi-discrete method to study the problem of bifurcation and bifurcation control for delay differential systems.This paper systematically discusses the Hopf bifurcation problem of two types of semidiscrete delay differential systems.Specific arrangements are as follows:In the second chapter,the Hopf bifurcation problem for a class of semi-discrete neutral delay differential systems is discussed.Based on the research results of the Hopf bifurcation of the continuous-time neutral delay differential system,the Hopf bifurcation of the semi-discrete system is explored.It is proved that the stability of the semi-discrete system and the continuous-time system have similar Hopf bifurcation structure when the sampling period is sufficiently small.And the conditions of stability and existence of Hopf bifurcation for discrete-time system are obtained.Finally,a numerical example is given to illustrate the validity of the results.In the third chapter,the Hopf bifurcation for a class of delay differential systems with discrete-time delayed feedback control is studied.Firstly,the stability of the continuous-time delayed feedback controlled system is analyzed.The Hopf bifurcation existence conditions and an effective interval of control parameter that make the system asymptotically stable are obtained.Then,based on the above results,the discrete-time controlled system is studied.It is proved that the effective control interval of the discrete-time delayed feedback controlled system is approximate to the continuous-time controlled system when the sampling period is sufficiently small.At the same time,the bound of the sampling period is estimated.Finally,numerical examples are given to verify the correctness and validity of the results.