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.      

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.     

Delayed feedback on the 3-D chaotic system only with two stable node-foci

In this paper, we investigate the effect of delayed feedbacks on the 3-D chaotic system only with two stable node-foci by Yang et al. The stability of equilibria and the existence of Hopf bifurcations are considered. The explicit formulas determining the direction, stability and period of the bifurcating periodic solutions are obtained by employing the normal form theory and the center manifold theorem. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodic orbit in the chaotic system with direct time delay feedback. We also find that the control law can be applied to the chaotic system only with two stable node-foci for the purpose of control and anti-control of chaos. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.     

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。     

速度时滞反馈控制下磁浮系统的稳定性与Hopf分岔

The problem of time delay speed feedback in the control loop is considered here. Its effects on the linear stability and dynamic behavior of the maglev system are investigated. It is found that a Hopf bifurcation can take place when the time delay exceeds certain values. The stability condition of the maglev system with the time delay is acquired. The direction and stability of the Hopf bifurcation are determined by constructing a center manifold and by applying the normal form method. Finally, numerical simulations are performed to verify the analytical result.     

速度时滞反馈控制下磁浮系统的稳定性与Hopf分岔

The problem of time delay speed feedback in the control loop is considered here.Its effects on the linear stability and dynamic behavior of the maglev system are investigated.It is found that a Hopf bifurcation can take place when the time delay exceeds certain values.The stability condition of the maglev system with the time delay is acquired.The direction and stability of the Hopf bifurcation are determined by constructing a center manifold and by applying the normal form method.Finally,numerical simulations are performed to verify the analytical result.     

Controlling the spreading in small-world evolving networks: stability, oscillation, and topology

The spreading of viruses, diseases, and even disasters (such as power blackouts and financial crises) in many large-scale and small-world networks is one of the mostly concerned issues today. In this note, we study general spreading dynamical behaviors in small-world evolving networks when control strategies are applied to suppress the propagation of diseases, viruses, and disasters. After proposing a novel Watts-Strogatz (W-S) spreading model to capture the general spreading mechanism in small-world networks, we investigate the stability and Hopf bifurcations of delay-controlled spreading models with linear and nonlinear feedback controllers, where parameters of small-world rewiring probability, feedback control gain, and time delay are analyzed for the oscillating behaviors. We conclude that the oscillatory spreading phenomena in delay-controlled small-world networks are topologically inherent.     

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 Three-Neuron Network with Multiple Discrete and Distributed Delays

In this paper, a class of three-neuron network with discrete and distributed delays is introduced. We first give a detailed Hopf bifurcation analysis for the proposed network. Choosing the discrete time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Moreover, by using the normal form theory and center manifold theorem, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, numerical simulations are presented to demonstrate the effectiveness of our theoretical results.     

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。     

Bifurcation Control Of A Fractional‐Order Van Der Pol Oscillator Based On The State Feedback

In this paper, a dynamic state feedback is applied to control Hopf bifurcations arising from a fractional‐order Van Der Pol oscillator. The degree parameter indicating the strength of the nonlinear damping is chosen as the bifurcation parameter. It is shown that in the absences of the dynamic state feedback controller, the fractional‐order Van Der Pol oscillator loses the stability via the Hopf bifurcation early, and can maintain the stability only in a certain domain of the degree parameter. When applying the state feedback controller to the fractional‐order Van Der Pol oscillator, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the system possesses the stability in a larger parameter range. Numerical simulations are given to justify the validity of the dynamic state feedback controller in bifurcation controls.     

Chaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback

In this paper, the problem of controlling chaos in a Sprott E system with distributed delay feedback is considered. By analyzing the associated characteristic transcendental equation, we focus on the local stability and Hopf bifurcation nature of the Sprott E system with distributed delay feedback. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form theory and center manifold theory. Numerical simulations for justifying the theoretical analysis are provided.     

Hopf Bifurcation Analysis of a Reaction-Diffusion Neural Network with Time Delay in Leakage Terms and Distributed Delays

In this paper, a reaction-diffusion neural network with time delay in leakage terms and distributed synaptic transmission delays under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation are established. By using the normal form theory and the center manifold reduction of partial functional differential equations, explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.     

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.     

Spatiotemporal dynamics induced by delay and diffusion in a predator–prey model with mutual interference among the predator

In this work, we study a reaction–diffusion predator–prey model with mutual interference among the predators while searching for food. We prove that the model exhibits bistability, which indicates that there are no patterns for our model. When time delay is incorporated into the model, multiple stability switches phenomenon of positive constant steady state emerged. By taking delay as a bifurcation parameter, the Hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay. The algorithm for determining the direction and the stability of the bifurcating periodic solutions is also derived. The delay–diffusion driven Turing instability of the positive constant steady state is investigated. Our results show that delay and diffusion can create periodic oscillatory patterns of spatially homogeneous and inhomogeneous and Turing patterns.     

Delay-induced Hopf bifurcation in a noise-driven excitable neuron model

A FitzHugh–Nagumo neuron model with cubic nonlinearity and discrete delay is considered, in which the time delay is regarded as a parameter. The effect of time delay on the linear stability and Hopf bifurcation of the model is studied. The existence, stability and direction of the local and global Hopf bifurcation are derived. Some numerical simulations are employed to validate the main results of this work.     

Stability analysis in a second-order differential equation with delays

A second-order differential equation with finite discrete delays is considered. Local stability of the zero equilibrium is investigated, and we obtain some sufficient conditions for the zero equilibrium is stable or unstable. Moreover, it is found that there exist the local Hopf bifurcations of the system when the delay varies.     

基于分岔理论的永磁同步电机有限时间混沌控制

永磁同步电机运行系统具有不稳定的分岔特性,随着系统参数的变化,系统会在平衡点处发生分岔行为.首先,基于分岔理论构建了永磁同步电机的混沌模型.其次,通过研究系统的分岔参数,分析了系统在平衡点处的分岔特性,发现系统在零平衡点处会产生静态分岔并出现新的平衡点,随着参数的继续变化,系统在新的平衡点处发生连续的Hopf分岔,而连...     

A method for stability and bifurcation control

In this note, we use the Ro/spl uml/ssler system as an illustrative example to present a method for controlling stability and bifurcations of nonlinear dynamical systems. This approach employs the idea used in computing the transition variety sets of constrained bifurcations to find the stability boundaries of equilibrium points in parameter space. With this method and feedback control, one can obtain appropriate parameter values to delay either static, dynamic, or both bifurcations. A feedback controller is designed to stabilize the Ro/spl uml/ssler system using all feasible control parameters. A numerical example is given to demonstrate the theoretical results.     

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.