时滞微分方程的Hopf分支的时域与频域分析
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摘要
本篇博士学位论文由六章组成.
第一章,简述时滞微分方程的历史背景,时滞微分方程的Hopf分支的历史发展与研究现状,阐述问题产生的背景和本文的主要工作.
第二章,简单介绍时滞微分方程的稳定性理论,时域中的Hopf分支理论、中心流形和规范型理论、全局Hopf分支理论、频域中的Hopf分支理论及一些与本论文研究相关的重要背景知识.
第三章,我们研究了一类具有双时滞的5维BAM神经网络模型,得到了该系统平衡点渐近稳定的充分条件和Hopf分支产生的的充分条件；用时域中的Hopf分支理论及中心流形和规范型理论,给出了确定Hopf分支方向和Hopf分支周期解稳定性及周期的具体计算表达式,并给出实例进行数值模拟验证我们所得结论的正确性.
第四章,我们用时域中的Hopf分支理论及中心流形和规范型理论研究了一类具有多时滞的6维BAM神经网络模型,得到了系统平衡点渐近稳定的充分条件和Hopf分支产生的充分条件,同时给出了确定Hopf分支方向和Hopf分支周期解稳定性及周期的具体计算表达式,并给出实例进行数值模拟验证我们所得结论的正确性.
第五章,我们研究了一类具有时滞和依赖时滞的变系数的2维捕食模型.通过分析其相应的特征超越方程,研究了系统的线性稳定性,用时域中的Hopf分支理论研究了Hopf分支产生的的条件,同时运用中心流形和规范型理论,给出了确定Hopf分支方向和Hopf分支周期解稳定性及周期的具体计算表达式.并给出实例进行数值模拟验证我们所得结论的正确性.
第六章,我们用频域法研究了一类双时滞的3阶BAM神经网络模型,确定了Hopf分支点的存在性,以时滞为参数,研究Hopf分支现象,当分支参数通过某一临界值时,Hopf分支产生；利用图示Hopf分支定理给出了频域法中的确定Hopf分支方向及Hopf分支周期解的稳定性的方向指标和稳定性指标.
最后,对本论文工作进行全面的总结,提出一些期待解决的问题,并对未来的研究方向进行展望.
This Ph.D.thesis is divided into six chapters and main contents are as follows:
In Chapter 1, we give a survey to the developments of the theory of Hopf bifurcation for delayed differential equations. Then we introduce the background of problems, the main results of this dissertation.
In Chapter 2, we briefly introduce the stability theory of delayed differential equation, Hopf bifurcation theory in the time domain, the normal form theory and center manifold theory, global Hopf bifurcation theory and Hopf bifurcation theory in the frequency domain. Some important preliminaries are also summarized.
In Chapter 3, we consider a five dimensional BAM neural network model with two discrete delays. Some sufficient conditions to ensure that the equilibrium of system is asymptotically stable and the Hopf bifurcation exists near the equilibrium are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations and periods are obtained by using the normal form theory and center manifold theory in the time domain. Finally, numerical simulations supporting the theoretical analysis are given.
In Chapter 4, we consider a six dimensional BAM neural network model with three discrete delays. Some sufficient conditions to ensure that the equilibrium of system is asymptotically stable and the Hopf bifurcation exists near the equilibrium are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations and periods are obtained by using the normal form theory and center manifold theory in the time domain. Using the global Hopf bifurcation theorem for functional differential equation (FDE) and Bendixson' criterion for high-dimensional ordinary differential equation(ODE), we obtain the global existence of periodic solutions. Finally, numerical simulations supporting the theoretical analysis are given.
In Chapter 5, we consider a class of stage-structure predator-prey model with time delay and delays dependent parameters. By analyzing the associated char-acteristic transcendental equation, its linear stability is investigated. Using the Hopf bifurcation theorem in the time domain, we investigate the existence of Hopf bifurcation of the model. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bi-furcations and periods are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.
In Chapter 6, a class of simplified tri-neuron BAM network model with two delays is considered. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. If the sumτof delayτ1 andτ2 is chosen as a bifurcation parameter, it is found that Hopf bifurcation occurs when the sumτpass through a series of critical values. The direction and the stability of Hopf bifurcation periodic solu-tions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulation for justifying the theoretical analysis are also provided. Finally, main conclusions are given.
Finally, the research work of this paper is summarized. Some problems which are expected to resolve are put forward and the directions in the near future are included.
第一章,简述时滞微分方程的历史背景,时滞微分方程的Hopf分支的历史发展与研究现状,阐述问题产生的背景和本文的主要工作.
第二章,简单介绍时滞微分方程的稳定性理论,时域中的Hopf分支理论、中心流形和规范型理论、全局Hopf分支理论、频域中的Hopf分支理论及一些与本论文研究相关的重要背景知识.
第三章,我们研究了一类具有双时滞的5维BAM神经网络模型,得到了该系统平衡点渐近稳定的充分条件和Hopf分支产生的的充分条件；用时域中的Hopf分支理论及中心流形和规范型理论,给出了确定Hopf分支方向和Hopf分支周期解稳定性及周期的具体计算表达式,并给出实例进行数值模拟验证我们所得结论的正确性.
第四章,我们用时域中的Hopf分支理论及中心流形和规范型理论研究了一类具有多时滞的6维BAM神经网络模型,得到了系统平衡点渐近稳定的充分条件和Hopf分支产生的充分条件,同时给出了确定Hopf分支方向和Hopf分支周期解稳定性及周期的具体计算表达式,并给出实例进行数值模拟验证我们所得结论的正确性.
第五章,我们研究了一类具有时滞和依赖时滞的变系数的2维捕食模型.通过分析其相应的特征超越方程,研究了系统的线性稳定性,用时域中的Hopf分支理论研究了Hopf分支产生的的条件,同时运用中心流形和规范型理论,给出了确定Hopf分支方向和Hopf分支周期解稳定性及周期的具体计算表达式.并给出实例进行数值模拟验证我们所得结论的正确性.
第六章,我们用频域法研究了一类双时滞的3阶BAM神经网络模型,确定了Hopf分支点的存在性,以时滞为参数,研究Hopf分支现象,当分支参数通过某一临界值时,Hopf分支产生；利用图示Hopf分支定理给出了频域法中的确定Hopf分支方向及Hopf分支周期解的稳定性的方向指标和稳定性指标.
最后,对本论文工作进行全面的总结,提出一些期待解决的问题,并对未来的研究方向进行展望.
This Ph.D.thesis is divided into six chapters and main contents are as follows:
In Chapter 1, we give a survey to the developments of the theory of Hopf bifurcation for delayed differential equations. Then we introduce the background of problems, the main results of this dissertation.
In Chapter 2, we briefly introduce the stability theory of delayed differential equation, Hopf bifurcation theory in the time domain, the normal form theory and center manifold theory, global Hopf bifurcation theory and Hopf bifurcation theory in the frequency domain. Some important preliminaries are also summarized.
In Chapter 3, we consider a five dimensional BAM neural network model with two discrete delays. Some sufficient conditions to ensure that the equilibrium of system is asymptotically stable and the Hopf bifurcation exists near the equilibrium are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations and periods are obtained by using the normal form theory and center manifold theory in the time domain. Finally, numerical simulations supporting the theoretical analysis are given.
In Chapter 4, we consider a six dimensional BAM neural network model with three discrete delays. Some sufficient conditions to ensure that the equilibrium of system is asymptotically stable and the Hopf bifurcation exists near the equilibrium are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations and periods are obtained by using the normal form theory and center manifold theory in the time domain. Using the global Hopf bifurcation theorem for functional differential equation (FDE) and Bendixson' criterion for high-dimensional ordinary differential equation(ODE), we obtain the global existence of periodic solutions. Finally, numerical simulations supporting the theoretical analysis are given.
In Chapter 5, we consider a class of stage-structure predator-prey model with time delay and delays dependent parameters. By analyzing the associated char-acteristic transcendental equation, its linear stability is investigated. Using the Hopf bifurcation theorem in the time domain, we investigate the existence of Hopf bifurcation of the model. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bi-furcations and periods are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.
In Chapter 6, a class of simplified tri-neuron BAM network model with two delays is considered. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. If the sumτof delayτ1 andτ2 is chosen as a bifurcation parameter, it is found that Hopf bifurcation occurs when the sumτpass through a series of critical values. The direction and the stability of Hopf bifurcation periodic solu-tions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulation for justifying the theoretical analysis are also provided. Finally, main conclusions are given.
Finally, the research work of this paper is summarized. Some problems which are expected to resolve are put forward and the directions in the near future are included.
引文
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