Bifurcation in two-dimensional neural network model with delay

A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined by using the normal form method and centre manifold theory. Foundation item: the National Natural Science, Foundation of China (19831030) Biography: WEI Jun-jie, Professor, Doctor, E-mail: weijj@hit.edu.cn     

Diffusion-driven instability and Hopf bifurcation in Brusselator system

The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.     

Hopf Bifurcation on a Two-Neuron System with Distributed Delays: A Frequency Domain Approach

In this paper, a more general two-neuron model with distributed delays and weak kernel is investigated. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. Furthermore, we found that if the mean delay is used as a bifurcation parameter, Hopf bifurcation occurs for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determine by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also given.     

Stability and bifurcation of a human respiratory system model with time delay

The stability and bifurcation of the trivial solution in the two-dimensional differential equation of a model describing human respiratory system with time delay were investigated. Formulas about the stability of bifurcating periodic solution and the directionof Hopf bifurcation were exhibited by applying the normal form theory and the center manifold theorem.Furthermore, numerical simulation was carried out.     

van der Pol型时滞系统的两参数余维一Hopf分岔及其稳定性

研究具有三次非线性时滞项的ｖａｎ ｄｅｒ Ｐｏｌ型时滞系统随两参数（时滞量和增益系数）余维一Ｈｏｐｆ分岔,说明了线性化特性方程随两参数变化时的根的分布和Ｈｏｐｆ分岔存在性;通过构造中心流形并且使用范式方法确定出Ｈｏｐｆ分岔的方向以及周期解的稳定性;分析了时滞量对所论系统发生Ｈｏｐｆ分岔的影响。     

轴承-转子系统Hopf分叉及其后动力学行为研究

采用长轴承解析模型研究滑动轴承支承的平衡单盘柔性转子-轴承系统的自激振动,把结合打靶法的延续算法应用于柔性平衡转子-轴承系统Hopf分叉后周期解的追踪和求解上,基于Floquet理论对周期解的稳定性加以分析.通过持续追踪周期解频率变化并与失稳固有频率进行对比,分析了自激锁相现象,研究了非线性油膜力自激源对系统的作用机理.运用Poincare映射、分叉图、及Lyapnov指数对周期解分叉、混沌及进入和脱离混沌的过程进行了分析.     

Hopf分岔的代数判据及其在车辆动力学中的应用

利用Hurwitz行列式,给出平衡点失稳而发生Hopf分岔的代数判定准则和计算方法,这一方法将Hopf分岔点的求解转化为一个非线性方程的求解问题,从而克服了以前方法在计算Hopf分岔点时,对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量。应用这一方法,我们进行了非线性车辆系统蛇行运动稳定性的研究,得到了轮对系统发生蛇行运动的临界速度的解析表达式。     

Hopf Bifurcation of the Generalized Lorenz Canonical Form

Hopf bifurcation of a unified chaotic system – the generalized Lorenz canonical form (GLCF) – is investigated. Based on rigorous mathematical analysis and symbolic computations, some conditions for stability and direction of the periodic obits from the Hopf bifurcation are derived.     

高超声速飞行器横侧向失稳非线性分岔分析

针对滑翔式高超声速飞行器大攻角横侧向失稳问题,采用延拓算法和分岔理论,求解并分析了以俯仰舵偏为连续参数的稳态平衡分岔图和以副翼舵偏为连续参数的横侧向机动稳态平衡分岔图,对平衡分支的稳定性和突变点进行了分析,并给出了特征根拓扑结构变化.研究表明,高超声速飞行器存在极限分岔点、Hopf分岔点以及叉型分岔点,且从叉型分岔点延伸出多个平衡分支,引起横侧向的自滚转失稳;从Hopf分岔点延伸出极限环分支,该分支对应较为复杂的极限环运动,其中还包含倍周期分岔、花环分岔、极限环极限点分岔等复杂的分岔现象;在横侧向机动飞行情况下,模型存在横向操作偏离失稳问题,且存在多个不稳定的平衡点.研究结果为实现高超声速飞行器的稳定飞行和控制器的设计提供了极其重要的动力学信息.      

Nonlinear dynamics of hardware-in-the-loop experiments on stick–slip phenomena

A single degree-of-freedom nonlinear mechanical model of the stick–slip phenomenon is studied when the Stribeck-type friction force is emulated by means of a digitally controlled actuator. The relative velocity of the slipping contact surfaces is considered as bifurcation parameter. The original physical system presents subcritical Hopf bifurcation with a wide bistable parameter region where stick–slip and steady-state slipping are both stable locally. Hardware-in-the-loop experiments are performed with a physical oscillatory system subjected to the emulated Stribeck forces. The effect of sampling time is studied with respect to the stability and nonlinear behavior of this experimental system. The existence of subcritical Neimark–Sacker bifurcations are proven in the digital system, the stability and bifurcation characteristics of the continuous and the digital systems are compared, and the counter-intuitive stabilizing effect of sampling time is shown both analytically and experimentally. The conclusions draw the attention to the limitations of hardware-in-the-loop experiments when the corresponding systems are strongly nonlinear.     

Periodic Delay Effects on Cutting Dynamics

We consider a delay equation whose delay is perturbed by a small periodic fluctuation. In particular, it is assumed that the delay equation exhibits a Hopf bifurcation when its delay is unperturbed. The periodically perturbed system exhibits more delicate bifurcations than a Hopf bifurcation. We show that these bifurcations are well explained by the Bogdanov-Takens bifurcation when the ratio between the frequencies of the periodic solution of the unperturbed system (Hopf bifurcation) and the external periodic perturbation is 1:2. Our analysis is based on center manifold reduction theory.     

Double Hopf Bifurcation Analysis Using Frequency Domain Methods

The dynamic behavior close to a non-resonant double Hopf bifurcation is analyzed via a frequency-domain technique. Approximate expressions of the periodic solutions are computed using the higher order harmonic balance method while their accuracy and stability have been evaluated through the calculation of the multipliers of the monodromy matrix. Furthermore, the detection of secondary Hopf or torus bifurcations (Neimark–Sacker bifurcation for maps) close to the analyzed singularity has been obtained for a coupled electrical oscillatory circuit. Then, quasi-periodic solutions are likely to exist in certain regions of the parameter space. Extending this analysis to the unfolding of the 1:1 resonant double Hopf bifurcation, cyclic fold and torus bifurcations have also been detected in a controlled oscillatory coupled electrical circuit. The comparison of the results obtained with the suggested technique, and with continuation software packages, has been included.     

挤压油膜阻尼器－滑动轴承－刚性转子系统的稳定性及分岔行为

对挤压油膜阻尼器－滑动轴承－转子系统的稳定性及分岔行为进行了研究，由于该动力系统为一强非线性系统，具有复杂的非线性现象。本文采用Ｆｌｏｑｕｅｔ理论对其周期解的稳定性进行了计算分析：随着系统参数的变化，该系统将出现稳态周期解、准周期分岔、倍周期分岔。文中也对系统平衡点的稳定性进行了分析，讨论了其Ｈｏｐｆ分岔行为     

Bifurcation of a shaft with hysteretic-type internal friction force of material

IntroductionItwasfoundlongtimeagothattheinternalfrictionofmaterialcancauseinstabilityofrotatingshaft.Soitisalwaysoneoftheimportantsubjectsinrotordynamics[1].Earlyinvestigationswerefocusedonthedynamicalstabilityproblemofrotorinfluencedbythelinearinternalfrictionofmaterial,aimingtoobtainthecriterionofstability[2～4 ].Asthedevelopmentofnonlineardynamics,moreandmoreattentionswerepaidtothestudyoftheself_excitedmotionofrotatingshaft,thatisthebifurcation .Thestabilityregionsandbifurcationsofbothanau…     

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.     

Stability and bifurcation in an electromechanical system with time delays

The effect of time delays occurring in a proportional-integral-derivative feedback controller on the linear stability of a simple electromechanical system is investigated by analyzing the characteristic transcendental equation. It is found that the trivial fixed point of the system can lose its stability through Hopf bifurcations when the time delay crosses certain critical values. Codimension two bifurcations, which result from non-resonant and resonant Hopf–Hopf bifurcation interactions, are also found to exist in the system.     

Bifurcation and stability analysis of a neural network model with distributed delays

In this paper, the dynamics of a generalized two-neuron model with self-connections and distributed delays are investigated, together with the stability of the equilibrium. In particular, the conditions under which the Hopf bifurcation occurs at the equilibrium are obtained for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Explicit algorithms for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [20]. Some numerical simulations are given to illustrate the effectiveness of the results found. The obtained results are new and they complement previously known results.This work was supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053.     

四维超混沌系统Hopf分岔分析与反控制

对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。     

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.     

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.