Dynamics of an inertial two-neuron system with time delay

In this paper, we considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By analyzing the distribution of the eigenvalues of the corresponding transcendental characteristic equation of its linearized equation, local stability criteria are derived for various model parameters and time delay. By choosing time delay as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. Furthermore, the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Also, resonant codimension-two bifurcation is found to occur in this model. Some numerical examples are finally given for justifying the theoretical results. Chaotic behavior of this inertial two-neuron system with time delay is found also through numerical simulation, in which some phase plots, waveform plots, power spectra and Lyapunov exponent are computed and presented.     

Complexity analysis of a Cournot–Bertrand duopoly game with different expectations

In this paper, we consider a Cournot–Bertrand mixed duopoly model with different expectations, where the market has linear demand and the firms have fixed marginal cost functions. Two firms choose output and price as decision variables, respectively, under the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market is occupied by the one that applies a lower price. The two players are considered to have bounded rational and static expectations. The existence and local stability condition of Nash equilibrium is investigated. We find the stability region of Cournot–Bertrand system is bigger than that of Cournot or Bertrand system under the same conditions. Furthermore, there are two different kinds of bifurcations when the parameters pass through the different boundary curves of the stability region, which is different from the Cournot or Bertrand model. Numerical simulation method is used to display the dynamic behaviors of the dynamical system, such as periodic cycles, bifurcation diagrams and strange attractors of the systems. The economic explanations of the complex dynamic behaviors are also given.     

Positive position feedback (PPF) controller for suppression of nonlinear system vibration

In this paper, a study for positive position feedback controller is presented that is used to suppress the vibration amplitude of a nonlinear dynamic model at primary resonance and the presence of 1:1 internal resonance. We obtained an approximate solution by applying the multiple scales method. Then we conducted bifurcation analyses for open and closed loop systems. The stability of the system is investigated by applying the frequency-response equations. The effects of the different controller parameters on the behavior of the main system have been studied. Optimum working conditions of the system were extracted to be used in the design of such systems. Finally, numerical simulations are performed to demonstrate and validate the control law. We found that all predictions from analytical solutions are in good agreement with the numerical simulation. A comparison with the available published work is included at the end of the work.     

基于POD-Galerkin降维方法的热毛细对流分岔分析

作为流动与传热相互耦合的非线性过程, 热毛细对流有着复杂的转捩过程, 探究流场和温度场随参数变化而发生的分岔现象, 是热毛细对流研究的一个重要课题. 基于本征正交分解的POD-Galerkin降维方法可以通过提取特征模态, 构建低维模型, 实现流场的快速计算. 数值分岔方法可以通过求解含参数动力系统的分岔方程, 直接计算稳定解和分岔点. 探究了将直接数值模拟方法、POD-Galerkin降维方法、数值分岔方法的优势结合, 以提高热毛细对流转捩过程分析效率的可行性. 利用直接数值模拟得到的流场和温度场数据, 构建了不同体积比下, 二维有限长液层热毛细对流的POD-Galerkin低维模型, 在低维模型上采用数值积分及数值分岔方法计算了分岔点, 得到了低维方程的分岔图. 在一定参数范围内, 在低维模型上模拟热毛细对流, 对雷诺数和体积比进行参数外推, 通过与直接数值模拟的结果对比, 验证了低维模型的准确性与鲁棒性. 说明了低维方程可以定性反映原高维系统的流动特性, 而定量方面, 由低维模型和直接数值模拟计算得到的周期解频率的相对误差大约为5%. 验证了利用POD-Galerkin降维方法研究热毛细对流的可行性.      

界面张力梯度驱动对流向湍流转捩的研究

作为空间自然对流热质输运的基本形式,界面张力梯度驱动对流是流动和传热强耦合的复杂非线性过程,也是一个多控制参数耦合作用的过程,表现出丰富的流动时空特征.界面张力梯度驱动对流是微重力流体物理的重要研究内容和学科前沿,同时在空间燃料输运过程和空间能源或热管利用等空间流体管理问题中均有重要应用.本文综述了界面张力梯度驱动对流...     

弹性转子实验台的响应分析

建立了考虑轴承和隔振垫弹性的非对称支承转子实验台系统的非线性动力学碰摩模型, 应用数值分析的方法对其进行研究. 以转速为分叉参数,结合Poincar\'{e}截面和自相关函数图等, 分析隔振垫刚度对系统分叉与混沌动力学行为的影响. 分析结果表明, 隔振垫刚度对系统动力学行为有较大影响, 系统通向混沌的道路主要是阵发性分叉和倍周期分叉. 实验分析所得到的系统运动性质与数值模拟结果一致.     

The research on Cournot–Bertrand duopoly model with heterogeneous goods and its complex characteristics

This paper details the research of the Cournot–Bertrand duopoly model with the application of nonlinear dynamics theory. We analyze the stability of the fixed points by numerical simulation; from the result we found that there exists only one Nash equilibrium point. To recognize the chaotic behavior of the system, we give the bifurcation diagram and Lyapunov exponent spectrum along with the corresponding chaotic attractor. Our study finds that either the change of output modification speed or the change of price modification speed will cause the market to the chaotic state which is disadvantageous for both of the firms. The introduction of chaos control strategies can bring the market back to orderly competition. We exert control on the system with the application of the state feedback method and the parameter variation control method. The conclusion has great significance in theory innovation and practice.     

The application and complexity analysis about a high-dimension discrete dynamical system based on heterogeneous triopoly game with multi-product

In production practice, firms usually produce multi-products rather than single products to obtain cost-saving advantages, cater for the diversity of consumer tastes and provide a barrier to entry. Based on nonlinear and economics theories, this paper establishes a discrete triopoly dynamical model which considers multi-product firms with heterogeneous expectations: naïve, adaptive and bounded rationality expectations. The discrete model is described by a 6-dimensional dynamical system. Thesis explores the path of the complexity of evolution and its intrinsic regularity, studies the influence of parameter change on the sensitivity level. The stable conditions of Cournot Nash equilibrium point are analyzed. The route to complex dynamics is investigated using 2-D and creative 3-D bifurcation diagrams by numerical experiment. The results show: the adaptive parameters can modify the stability of the market, but cannot lead to chaos independently; the bounded rationality parameters can arouse chaos for the whole market; larger differentiated degree of multi-product can suppress instability which is caused by adaptive and bounded rationality adjustment parameters. These results have significant theoretical and practical value to multi-product triopoly game with heterogeneous expectations in related markets.     

Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays

In this paper, we consider the effect of distributed delays in a three-neuron unidirectional ring. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. Taking the average delay as a bifurcation parameter, we find two critical values at which the system undergoes Hopf bifurcations. The orbital asymptotic stability of the Hopf bifurcating periodic solutions is investigated by using the method of multiple scales. The global Hopf bifurcation is also studied. Finally, the theoretical results are illustrated by some numerical simulations.     

一类忆阻神经元的电活动多模振荡及Hamilton 能量反馈控制

根据法拉第电磁感应定律,在离子穿越细胞膜或者在外界电磁辐射下,细胞内外的电生理环境会产生电磁感应效应,继而会影响神经元的电活动行为. 基于此,本文考虑电磁感应影响下的 Hindmarsh-Rose (HR) 神经元模型,研究了其混合模式振荡放电特征,并设计一个 Hamilton 能量反馈控制器,将其控制到不同的周期簇放电状态. 首先,通过理论分析发现磁通 HR 神经元系统的 Hopf 分岔使其平衡点的稳定性发生了改变,并产生极限环,进而研究了 Hopf 分岔点附近膜电压的放电特征. 基于双参数数值仿真发现该系统具有丰富的分岔结构,在不同的参数平面上存在倍周期分岔、伴有混沌的加周期分岔、无混沌的加周期分岔以及共存的混合模式振荡. 最后,为了有效控制膜电压的混合模式振荡,利用亥姆霍兹理论计算出磁通 HR 神经元系统的 Hamilton 能量函数并设计 Hamilton 能量反馈控制器,通过数值仿真分析了膜电压在不同反馈增益下的簇放电状态,发现该控制器能够有效地控制膜电压到不同的周期簇放电模式. 本文的研究结果为探究电磁感应下神经元的分岔结构及其能量控制领域提供了有用的理论支撑.      

Stability and bifurcation for limit cycle oscillations of an airfoil with external store

In this paper, stability and local bifurcation behaviors for the nonlinear aeroelastic model of an airfoil with external store are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of bifurcation response equations are considered. They are characterized as (1) one pair of purely imaginary eigenvalues and two pairs of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary eigenvalues in nonresonant case and one pair of conjugate complex roots with negative real parts; (3) three pairs of purely imaginary eigenvalues in nonresonant case. With the aid of Maple software and normal form theory, the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves are obtained, which can lead to static bifurcation and Hopf bifurcation. Under certain conditions, 2-D tori motion may occur. The complex dynamical motions are considered in this paper. Finally, the numerical solutions achieved by the fourth-order Runge–Kutta method agree with the analytic results.     

Numerical simulation of particle migration in asymmetric bifurcation channel

In this work we provide numerical validation of the particle migration during flow of concentrated suspension in asymmetric T-junction bifurcation channel observed in a recent experiment [1]. The mathematical models developed to explain particle migration phenomenon basically fall into two categories, namely, suspension balance model and diffusive flux model. These models have been successfully applied to explain migration behavior in several two-dimensional flows. However, many processes often involve flow in complex 3D geometries. In this work we have carried out numerical simulation of concentrated suspension flow in 3D bifurcation geometry using the diffusive flux model. The simulation method was validated with available experimental and theoretical results for channel flow. After validation of the method we have applied the simulation technique to study the flow of concentrated suspensions through an asymmetric T-junction bifurcation composed of rectangular channels. It is observed that in the span-wise direction inhomogeneous concentration distribution that develops upstream persists throughout the inlet and downstream channels. Due to the migration of particles near the bifurcation section there is almost equal partitioning of flow in the two downstream branches. The detailed comparison of numerical simulation results is made with the experimental data.     

Dynamical properties and simulation of a new Lorenz-like chaotic system

This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rössler system, Chen system, and includes Lü systems as its special case. By using the center manifold theorem, the stability character of its non-hyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.     

Optimal harvesting and complex dynamics in a delayed eco-epidemiological model with weak Allee effects

In this article, an eco-epidemiological system with weak Allee effect and harvesting in prey population is discussed by a system of delay differential equations. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system is analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section and maximum Lyapunov exponent. Our simulation results suggest that for increasing the delay parameter, the system undergoes chaotic oscillation via period doubling. We also observe a quasi-periodicity route to chaos and complex dynamics with respect to Allee parameter; such behavior can be subdued by the strength of the Allee effect and harvesting effort through period-halving bifurcation. To find out the optimal harvesting policy for the time delay model, we consider the profit earned by harvesting of both the prey populations. The effect of Allee and gestation delay on optimal harvesting policy is also discussed.     

MULTI-MODE OSCILLATIONS AND HAMILTON ENERGY FEEDBACK CONTROL OF A CLASS OF MEMRISTOR NEURON$^{\bf 1)}$

根据法拉第电磁感应定律,在离子穿越细胞膜或者在外界电磁辐射下,细胞内外的电生理环境会产生电磁感应效应,继而会影响神经元的电活动行为. 基于此,本文考虑电磁感应影响下的 Hindmarsh-Rose (HR) 神经元模型,研究了其混合模式振荡放电特征,并设计一个 Hamilton 能量反馈控制器,将其控制到不同的周期簇放电状态. 首先,通过理论分析发现磁通 HR 神经元系统的 Hopf 分岔使其平衡点的稳定性发生了改变,并产生极限环,进而研究了 Hopf 分岔点附近膜电压的放电特征. 基于双参数数值仿真发现该系统具有丰富的分岔结构,在不同的参数平面上存在倍周期分岔、伴有混沌的加周期分岔、无混沌的加周期分岔以及共存的混合模式振荡. 最后,为了有效控制膜电压的混合模式振荡,利用亥姆霍兹理论计算出磁通 HR 神经元系统的 Hamilton 能量函数并设计 Hamilton 能量反馈控制器,通过数值仿真分析了膜电压在不同反馈增益下的簇放电状态,发现该控制器能够有效地控制膜电压到不同的周期簇放电模式. 本文的研究结果为探究电磁感应下神经元的分岔结构及其能量控制领域提供了有用的理论支撑.     

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。     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.     

评《分岔问题及其计算方法》

本文评述了武际可教授和黄克服教授的著作《分岔问题及其计算方法》。该书从生活中常见的分岔问题出发,概述了各类分岔问题及其研究进展;该书的突出特点是以等价类定义分岔,由此对动力系统的稳定性、局部和全局分岔、静分岔与霍普夫分岔等问题进行了深入的论述,并详细介绍了弧长方法为代表的数值计算方法在求解分岔问题方面的有效性。该书是一本非常值得相关科研工作者学习的参考书。

     

Multiple stability switches and Hopf bifurcation in a damped harmonic oscillator with delayed feedback

This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package.