【作者】 许晶；

【导师】 苏欢；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2019， 硕士

【摘要】 许多自然和人工的系统都可以抽象为延迟微分系统,并且很多系统都会发生分支现象。通常情况下,人们希望产生的分支是有利的,但在现实生活中,许多分支是有害的,这时就需要对它进行控制。因此,延迟微分系统的分支存在性和分支控制问题近年来已成为一个热门研究领域。半离散化是一种众所周知的技术,其基本思想是将系统中的部分项进行离散处理,使其仅需在有限个离散点处进行观测,与连续系统的观测相比,更经济,更实用。因此,将半离散方法应用于研究延迟微分系统的分支和分支控制问题是很有意义的。本文系统地对两类半离散延迟微分系统的Hopf分支问题进行了研究,具体安排如下:第二章中,讨论了一类半离散中立型延迟微分系统的Hopf分支问题。基于已有的连续时间中立型延迟微分系统Hopf分支的研究结果,对半离散中立型延迟微分系统的Hopf分支进行探究。证明了在采样周期充分小的情况下,半离散中立型延迟微分系统与连续时间中立型延迟微分系统的稳定性和Hopf分支结构近似。给出了半离散系统稳定和经历Hopf分支的条件。最后,给出一个数值算例来说明结果的正确性。第三章中,研究一类具有离散时间延迟反馈控制的延迟微分系统的Hopf分支问题。首先,对连续时间延迟反馈控制系统的稳定性进行分析,得到系统经历Hopf分支的条件和使系统渐近稳定的控制参数的有效范围。接着,基于所得到结果对离散时间延迟反馈控制系统进行研究,证明了当采样周期充分小时,离散时间延迟反馈控制系统的有效控制范围与连续时间系统的近似。同时,对采样周期的界限进行估计。最后,通过数值仿真说明结果的正确性和有效性。更多还原

【Abstract】 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.更多还原