广义Logistic模型的Hopf分支与计算机仿真

研究了一类含时滞且含干扰和收获率的广义Logistic模型的Hopf分支周期解。得到该模型正平衡态存在唯一的充要条件,利用特征值理论得到该模型产生Hopf分支的条件;利用周期函数正交性方法得到其近似周期解的表达式;运用计算机仿真,给出了参数取不同数值时的曲线拟合图,讨论了参数对周期解的周期、振幅及正平衡态的影响。     

具时滞的非线性纵向飞行模型稳定性和分支分析

研究一类具有时滞的非线性飞行模型的稳定性和分支问题。首先考虑数据测量的时间延迟,给出了含时滞的大迎角纵向多项式飞行模型;然后应用泛函微分方程Hopf分支理论和中心流形等非线性方法给出了该模型稳定性和分支的解析分析,得到了由时滞引起的Hopf分支存在条件、分支点计算公式以及分支周期解的稳定性判别准则;最后利用所得结论进行了飞行实例分析,分析结果表明,数据测量延时可能会引起飞行稳定性的改变,而且延时超过一定临界值时将产生Hopf分支,出现纵向周期振荡,其结论具有实际参考意义。     

具有扩散的三种群食物链模型的Hopf分支

研究了一类在Neumann边界条件下,具有扩散现象和群体防卫能力的三种群食物链模型的Hopf分支,以捕食者的死亡率为分支参数,利用Hurwitz判据讨论了系统正常数平衡解的稳定性,并通过理论分析给出了Hopf分支产生的条件,又利用规范形理论和中心流形定理得到了空间非齐次情形下Hopf分支的方向和分支周期解的稳定性。     

色氨酸操纵子系统的数值分析

研究色氨酸操纵子系统的分支问题。首先介绍该模型的建立过程,然后通过分析系统关于正平衡点的线性变分方程的特征根来研究系统在正平衡点的稳定性,应用泛函微分方程的局部Hopf分支理论给出了该模型出现周期解的条件,并利用正规型方法和中心流形理论得到了描述系统在第一个分支点出现的Hopf分支的分支方向和分支周期解稳定性的计算公式,最后通过数值模拟进行验证。     

具免疫时滞的HBV传染病模型的稳定性和Hopf分支

考虑了溶解性和非溶解性机制下的一类具有免疫时滞的HBV感染动力学模型。分析了无感染平衡点及感染无免疫平衡点的全局稳定性,讨论了感染免疫平衡点的局部稳定性和Hopf分支的存在条件。数值模拟结果表明：当易感细胞生成率的取值使得基本再生数满足平衡存在条件且低于某一临界值时,时滞对平衡点的稳定性没有影响;当大于该临界值时,随着时滞增大,平衡点不稳定,出现一系列Hopf分支,最终表现为周期波动模式。     

二维离散抛物映射的分支

本文研究一个二维离散抛物映射的动力学行为.首先,引用文献[1]中关于映射不动点的存在性和稳定性的结果:映射有三个不动点,及当参数b变化时,每个不动点稳定性的充分条件;接着,把b作为分支参数,利用中心流形定理和分支理论,分别导出Fold分支、Flip分支、Hopf分支存在的充分条件;最后通过数值模拟,验证Fold分支、Flip分支、Hopf分支存在条件的理论结果,同时,也发现映射存在复杂的对称性破缺分支.     

具有时滞和饱和接触率的SIRS模型的Hopf分支

应用微分方程分支理论,研究了具有时滞和饱和接触率的SIRS模型,以时滞[τ]为分支参数,运用Hopf分支理论,得到当时滞[τ]充分小时正平衡点是局部渐近稳定的,当[τ]经过一系列临界值时模型出现Hopf分支。用Matlab软件进行数值仿真验证了结论的正确性。     

二维离散Duffing-Holmes系统的分支与混沌研究

通过欧拉方法可将Duffing-Holmes方程变换为离散非线性动力学系统,得到标准Holmes映射.研究该映射不动点的存在性与稳定性条件,并运用中心流形定理分析映射的Pitchfork分支,Flip分支和Hopf分支的存在性,具体给出了发生相应分支所满足的参数条件.此外,证明了映射存在Marotto意义下的混沌,最后用数值模拟验证了所得理论结果.     

具时滞和功能反应模型的Hopf分支与Matlab计算

研究具有离散和连续时滞的Host-Parasitoid模型的Hopf分支问题。以时滞为参数,利用特征值理论给出系统惟一正平衡态的稳定性和Hopf分支存在的充分条件;根据规范型理论和中心流形定理分析了在临界值处出现分支周期解的方向及稳定性,并给出计算其方向和稳定性的计算公式;最后利用Matlab对实例进行验证得到理论分析和数值计算的一致性。     

具有修正的Lelie-Gower项Holling-III类时滞捕食系统的Hopf分支

研究了一类具有修正的Leslie-Gower项与Holling-III类功能性反应函数的时滞捕食系统. 以时滞为分支参数, 讨论系统正平衡点的局部稳定性, 给出系统产生Hopf分支的时滞关键值. 进一步, 确定系统Hopf分支的方向与分支周期解稳定性, 并对系统全局分支周期解的存在性进行讨论. 最后, 利用仿真实例验证理论分析结果的正确性.     

Bifurcation solutions in the diffusive minimal sediment

This paper is concerned with the diffusive minimal sediment model with no-flux boundary conditions. The steady-state bifurcation with two-dimensional kernel is firstly obtained with respect to this model by the space decomposition and the implicit function theorem. The existence of the Hopf bifurcation is next attained, and the direction and the stability of the Hopf bifurcation are analyzed in detail. Numerical simulations are done to support and complement the theoretical results.     

传输时延环境下信息物理融合系统中恶意病毒传播的稳定性与分岔分析

本文考虑到恶意病毒在信息物理融合系统中的传播具有时延性,基于非线性动力学理论建立了一类更具一般性的含有时滞的恶意病毒传播模型.通过选取时滞作为分岔参数,并讨论相关的特征方程,研究了时滞对系统局部稳定性和Hopf分岔的影响.研究发现,系统的动力学行为依赖于分岔的临界值.此外,给出了保证系统稳定性和产生Hopf分岔的条件....     

Bifurcation control of small‐world networks with delays VIA PID controller

In this paper, the problem of bifurcation control for a small‐world network model with time delay is studied. We first put forward a Proportional‐Integral‐Derivative (PID) feedback scheme to control the Hopf bifurcation of the network. The time delay is selected as the bifurcation parameter. The conditions of the stability and Hopf bifurcation are given for the controlled network. By using the center manifold theorem and the normal form theory, the direction and stability of bifurcating periodic solutions are confirmed. The feasible region of the parameters of the controller is determined. It is found that the bifurcation dynamics of the small‐world network are optimized by adjusting the parameters of the PID controller. Finally, a numerical example verifies the effectiveness of the designed PID controller, and the relationships between the onset of the Hopf bifurcation and the control parameters are obtained.     

Stability and bifurcation analysis of a six-neuron BAM neural network model with discrete delays

In this paper, a six-neuron BAM neural network model with discrete delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the model is investigated and Hopf bifurcation is demonstrated. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form method and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are given.     

不可压缩流中二元机翼运动的Hopf分岔

机翼的颤振是一种典型的自激振动,它是由气动力、弹性力和惯性力的相互作用引起的一种气动弹性现象.本文研究了具有结构非线性刚度恢复力的机翼颤振的Hopf分岔问题.首先,利用连续时间的Hopf分岔显式临界准则分析了机翼颤振Hopf分岔的存在性,推导了第一李雅普诺夫系数的通项公式,为判定机翼Hopf分岔的稳定性提供了依据.其次,分析了机翼颤振退化的余维二Hopf分岔的存在性条件,得到了满足条件的双参数分岔区域.然后,推导了第二李雅普诺夫系数的通项公式并结合中心流形降阶原理和同构变换进一步分析了余维二Hopf分岔的稳定性以及其局部开折问题.最后,通过推导第三李雅普诺夫系数分析了余维三Hopf分岔中心的稳定性.     

Stability and Hopf bifurcation of a general delayed recurrent neural network.

In this paper, stability and bifurcation of a general recurrent neural network with multiple time delays is considered, where all the variables of the network can be regarded as bifurcation parameters. It is found that Hopf bifurcation occurs when these parameters pass through some critical values where the conditions for local asymptotical stability of the equilibrium are not satisfied. By analyzing the characteristic equation and using the frequency domain method, the existence of Hopf bifurcation is proved. The stability of bifurcating periodic solutions is determined by the harmonic balance approach, Nyquist criterion, and graphic Hopf bifurcation theorem. Moreover, a critical condition is derived under which the stability is not guaranteed, thus a necessary and sufficient condition for ensuring the local asymptotical stability is well understood, and from which the essential dynamics of the delayed neural network are revealed. Finally, numerical results are given to verify the theoretical analysis, and some interesting phenomena are observed and reported.     

Local Bifurcation Analysis of a Delayed Fractional-order Dynamic Model of Dual Congestion Control Algorithms

In this paper, we propose a delayed fractional-order congestion control model which is more accurate than the original integer-order model when depicting the dual congestion control algorithms. The presence of fractional orders requires the use of suitable criteria which usually make the analytical work so harder. Based on the stability theorems on delayed fractionalorder differential equations, we study the issue of the stability and bifurcations for such a model by choosing the communication delay as the bifurcation parameter. By analyzing the associated characteristic equation, some explicit conditions for the local stability of the equilibrium are given for the delayed fractionalorder model of congestion control algorithms. Moreover, the Hopf bifurcation conditions for general delayed fractional-order systems are proposed. The existence of Hopf bifurcations at the equilibrium is established. The critical values of the delay are identified, where the Hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium. Same as the delay, the fractional order normally plays an important role in the dynamics of delayed fractional-order systems. It is found that the critical value of Hopf bifurcations is crucially dependent on the fractional order. Finally, numerical simulations are carried out to illustrate the main results.      

分数阶时滞传染病模型的传播动力学

本文研究了一个具有时滞的分数阶SEIR传染病模型,并且着重研究了时滞的引入对模型的动力学行为的影响.首先,建立了分数阶SEIR传染病模型并给出了无时滞情况下地方病平衡点稳定的充分条件,以此来确保时滞的引入具有实际意义.其次,结合分岔理论求得了Hopf分岔发生的条件以及分岔阈值的表达式.研究发现,系统的动力学行为依赖于分岔的临界值.在此基础上,研究了分数阶阶次的变化对分岔阈值的影响,发现随着阶次的增加系统的Hopf分岔将会提前.最后用数值仿真结果来验证理论推导的正确性.     

一类具有饱和发生率的时滞恶意病毒传播模型的分岔控制策略

考虑非线性的饱和发生率,建立一种刻画信息物理融合系统(cyber-physical systems, CPS)中恶意病毒传播的SIRS(susceptible-infected-recovered-susceptible)模型.为了避免因Hopf分岔的产生致使恶意病毒传播扩散,采用参数调节法和状态反馈法相结合的混合分岔控制策略,研究信息物理融合系统的Hopf分岔控制问题,建立受控系统的稳定性条件和分岔判据,探明控制增益参数对Hopf分岔点和分岔极限环幅值的影响规律,并给出分岔阈值与增益参数间的关系图.数值仿真结果表明,所提出的混合分岔控制策略不仅能够改变Hopf分岔点的位置,而且可以有效调节极限环幅值的大小,使得信息物理融合系统产生预期的动力学行为,有效降低恶意病毒传播的危害.