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.     

近哈密顿系统的Hopf分岔

简化了Wiggins提出的关于近哈密顿系统的Hopf分岔条件,并结合硬弹簧Duffing系统,研究了该类系统的Hopf分岔行为,并用数值积分的方法验证了结果的正确性。     

Quasi-periodic and chaotic behaviour of a two-degree-of-freedom impact system in a strong resonance case

A two-degree-of-freedom system contacting a single stop is considered. Quasi-periodic and chaotic behavior of the system in a strong resonance case is investigated by theoretical analysis and numerical simulation This work was supported by the National Natural Science Foundation of China(19672052).     

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.     

机电耦联系统余维3动态分岔研究

以r_{sl}, r_{f}以及x_{c}为分岔参数,对具有串补电容的单 机无穷大电力系统的失稳振荡问题,运用动态分岔理论进行了研究. 对系统同时出现有3对 纯虚根特征值的一类多参数高余维分岔情况,运用中心流行方法降维后得到约化方程,对此 强非线性约化方程的求解难点,运用多参数稳定性理论、谐波平衡法、归一化技术和Normal Form方法,得到了系统的解析解. 由分析得知,系统会出现3种Hopf分岔情况、二维环面 情况,以及三维环面分岔解,甚至会出现四维环面,或者更高维的环面分岔. 详细讨论 了系统各种分岔解的稳定性条件和稳定区域,并作了详细的数值分析加以验证.     

单调激活函数惯性项神经耦合系统的混沌共存

混沌及其稳态共存是神经网络系统中一个重要研究热点问题．本文基于惯性项神经元模型,利用非线性单调激活函数构造了一个惯性项神经耦合系统,采用理论分析和数值模拟相结合的方法,研究了系统平衡点以及静态分岔的类型,分析了系统两种不同模式的混沌及其稳态共存．具体来说,我们通过选取不同的初始值,利用相应的相位图和时间历程图,展现了系统混沌对初值的敏感依赖性．进一步,采用耦合强度作为动力学的分岔参数,研究了混沌产生的倍周期分岔机制,得到了单调激活函数耦合下的惯性项神经元系统混沌共存现象．     

Detection of Bifurcation Structures by Higher-Order Averaging for Duffing's Equation

Applying the higher-order averaging method, we study the periodically forced, standard Duffing oscillator. A package of the computer algebra system, Mathematica, recently developed by the author and a coworker, is improved and used to implement the tedious but necessary computations for application of higher-order averaging. We detect many types of subharmonic, superharmonic and ultra-subharmonic motions and their bifurcations. A theoretical exposition for a previous numerical observation of a superstructure of bifurcation sets is partly given. A numerical example is also presented and the theoretical predictions are compared with the corresponding simulation results.     

碰撞振动系统分岔与混沌的研究进展

针对工程实际中普遍存在的碰撞振动系统这种典型的非光滑动力系统, 其研究具有重要的理论意义和工程实用价值. 碰撞振动系统动力学的分析与研究方法主要有理论分析、数值模拟以及应用与实验研究. 为了研究碰撞振动系统的周期运动稳定性、分岔及混沌, 采用的手段有建立Poincar\'{e}映射、中心流形和范式方法, 映射的分岔与混沌理论是碰撞振动系统研究的理论基础. 首先简述了碰撞振动系统的分析与研究方法, 光滑非线性系统动力学的分析方法部分可以推广到碰撞振动系统, 碰撞振动的不连续性导致一些方法的适用性和有效性问题. 进一步综述了碰撞振动系统周期运动稳定性、分岔、混沌及奇异性的理论研究和工程应用现状. 最后着重结合相关离散型映射系统的动力学发展, 对碰撞振动系统的分岔与混沌研究及存在的主要问题进行了讨论, 并展望了其发展趋势.      

Non-linear dynamic behaviour of multi-folding microstructure systems based on origami skill

This paper presents the theoretical basis for both static and dynamic numerical approaches to the elastic stability and elasto-plastic stability of a folding multi-layered truss. Both analyses are based on bifurcation theory and include geometrical non-linearity. The dynamic analysis includes an allowance for contact between nodes. Comparisons are made between elastic behaviour and elasto-plastic behaviour obtained from both numerical dynamic methods in which bifurcations are demonstrated as a theoretically elastic analysis. Our analysis reveals that several folding behaviours in the pantographic truss are identified during the elastic and elasto-plastic instabilities, where the folding behaviour of the truss is shown to be a function of the initial geometry and velocity of the dynamic loading. The authors suggest that understanding this behaviour will be very useful for developing light-weight structures subject to dynamic loading based on static bifurcation analysis and dynamic analysis.     

应用突变理论分析Euler压杆的稳定性

建立了Euler压杆问题的能量表达式。对此模型，应用突变理论进行稳定分析，得出了系统全部的分叉集与突变流形。突变流形为一族层层嵌套、互不交叉的抛物线。分析的结论要比有限差分法、有限单元法、大范围非线形分析等数值计算的结果简洁、明了，有助于压杆后屈曲问题的研究。     

Small-signal amplification of period-doubling bifurcations in smooth iterated maps

Various authors have shown that, near the onset of a period-doubling bifurcation, small perturbations in the control parameter may result in much larger disturbances in the response of the dynamical system. Such amplification of small signals can be measured by a gain defined as the magnitude of the disturbance in the response divided by the perturbation amplitude. In this paper, the perturbed response is studied using normal forms based on the most general assumptions of iterated maps. Such an analysis provides a theoretical footing for previous experimental and numerical observations, such as the failure of linear analysis and the saturation of the gain. Qualitative as well as quantitative features of the gain are exhibited using selected models of cardiac dynamics.     

Group delay induced instabilities and Hopf bifurcations, of a controlled double pendulum

Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability.     

封闭方腔内空气和水自然对流每一次分岔数值预报

采用二阶全展开ETG分裂步有限元方法,通过对流动拓扑的详细分析,在排除网格密度影响的基础上,结合二分法给出封闭方腔内空气和水两种典型流体自然对流发生第一次分岔时的临界Rayleigh数。计算结果表明,该方法可用于进行不同Prandtl数条件下方腔内自然对流流动第一次分岔的数值预报,可作为后续各阶分岔及转捩数值预报研究的基础。在相应的条件下,封闭方腔内空气比水更容易发生分岔,且空气的流动结构相对于水表现出一定的倾斜性。     

时滞Rucklidge系统Hopf分岔分析与电路实现

以一类新的单时滞Rucklidge系统为分析对象,通过计算时滞系统的平衡点,分析该系统在各平衡点的稳定性和Hopf分岔的存在性,得到其发生Hopf分岔的条件。Matlab多组数值仿真验证了理论分析的正确性。基于此设计了一种可切换时滞与非时滞的混沌电路,并运用Multisim14.0进行仿真,实验结果表明,该电路可行且有效。     

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.     

载流圆板在磁场中的随机稳定性分析

本文根据大挠度板壳力学基础理论和电磁弹性力学理论,建立了载流圆板的非线性磁弹性随机振动力学模型,采用伽辽金变分法将其变换成非线性常微分动力学方程．通过拟不可积哈密顿系统的平均理论将该方程等价为一个一维伊藤随机微分方程．通过计算该方程的最大Lyapunov 指数判断该系统的局部随机稳定性,并进一步采用基于随机扩散过程的奇异边界理论判断该系统的全局稳定性．最后通过讨论该系统的稳态概率密度函数图的形状变化讨论了该动力系统的随机Hopf分岔的变化规律,并采用数值模拟对理论分析进行了验证．     

非线性系统周期强迫不平衡响应的稳定性分析

多自由度强非线性系统是工程实际中经常遇见的一类模型,利用非线性动力学分析中的打靶法求解该类系统的周期解,并对Flopuet主导特征值判断周期解的失稳方式,利用该方法对旋转机械中的一个具体模型;双盘县臂柔性转子-非同心型挤压油膜阻尼器（SFD）系统周期强迫不平衡响应的稳定性和分岔行为进行了分析,分析表明,在该系统中存在着第二Hopf分岔、倍周期分岔、鞍-结分岔三种分岔形式。     

一类非线性磁流变系统局部分岔特性研究

讨论了一类基于磁流变阻尼器非线性系统的局部分岔与控制问题，建立了该系统的动力 学模型，运用中心流形定理和范式理论，得到该系统双零特征值问题的规范形及其普适开折， 进而探讨了此系统的分岔行为和稳定性；给出了分岔曲线、转迁集；并给出了此类非线性系 统的数值仿真结果.     

Criticality of Hopf bifurcation in state-dependent delay model of turning processes

In this paper the non-linear dynamics of a state-dependent delay model of the turning process is analyzed. The size of the regenerative delay is determined not only by the rotation of the workpiece, but also by the vibrations of the tool. A numerical continuation technique is developed that can be used to follow the periodic orbits of a system with implicitly defined state-dependent delays. The numerical analysis of the model reveals that the criticality of the Hopf bifurcation depends on the feed rate. This is in contrast to simpler constant delay models where the criticality does not change. For small feed rates, subcritical Hopf bifurcations are found, similar to the constant delay models. In this case, periodic orbits coexist with the stable stationary cutting state and so there is the potential for large amplitude chatter and bistability. For large feed rates, the Hopf bifurcation becomes supercritical for a range of spindle speeds. In this case, stable periodic orbits instead coexist with the unstable stationary cutting state, removing the possibility of large amplitude chatter. Thus, the state-dependent delay in the model has a kind of stabilizing effect, since the supercritical case is more favorable from a practical viewpoint than the subcritical one.     

Beading instability in soft cylindrical gels with capillary energy: Weakly non-linear analysis and numerical simulations

Soft cylindrical gels can develop a long-wavelength peristaltic pattern driven by a competition between surface tension and bulk elastic energy. In contrast to the Rayleigh–Plateau instability for viscous fluids, the macroscopic shape in soft solids evolves toward a stable beading, which strongly differs from the buckling arising in compressed elastic cylinders.This work proposes a novel theoretical and numerical approach for studying the onset and the non-linear development of the elasto-capillary beading in soft cylinders, made of neo-Hookean hyperelastic material with capillary energy at the free surface, subjected to axial stretch. Both a theoretical study, deriving the linear and the weakly non-linear stability analyses for the problem, and numerical simulations, investigating the fully non-linear evolution of the beaded morphology, are performed. The theoretical results prove that an axial elongation can not only favour the onset of beading, but also determine the nature of the elastic bifurcation. The fully non-linear phase diagrams of the beading are also derived from finite element numerical simulations, showing two peculiar morphological transitions when varying either the axial stretch or the material properties of the gel. Since the bifurcation is found to be subcritical for very slender cylinders, an imperfection sensitivity analysis is finally performed. In this case, it is shown that a surface sinusoidal imperfection can resonate with the corresponding marginally stable solution, thus selecting the emerging beading wavelength.In conclusion, the results of this study provide novel guidelines for controlling the beaded morphology in different experimental conditions, with important applications in micro-fabrication techniques, such as electrospun fibres.