Bifurcation analyses and hardware experiments for bursting dynamics in non-autonomous memristive FitzHugh-Nagumo circuit

In this paper, a non-autonomous memristive Fitz Hugh-Nagumo(FHN) circuit is constructed using a second-order memristive diode bridge with LC network. For convenience of circuit implementation, an AC voltage source is adopted to substitute the original AC current stimulus. Stimulated by the slowly varying AC voltage source, the number, locations and stabilities of the equilibrium points slowly evolve with the time, which are thus indicated as the AC equilibrium points. Different sequences of fold and/or Hopf bifurcations are encountered in a full period of time series evolutions, leading to various kinds of chaotic or periodic bursting activities. To figure out the related bifurcation mechanisms, the fold and Hopf bifurcation sets are mathematically formulated to locate the critical bifurcation points. On this basis, the transitions between the resting and repetitive spiking states are clearly illustrated by the time series of the AC equilibrium points and state variables, from which Hopf/sub Hopf, Hopf/Hopf, and Hopf/fold bursting oscillations are identified in the specified parameter regions. Finally, based on a fabricated hardware circuit, the experimental measurements are executed. The results verify that the presented memristive FHN circuit indeed exhibits complex bursting activities, which enriches the family of memristor-based FHN circuits with bursting dynamics.     

以滞量为参数的广义Lienard方程的数值逼近

利用欧拉方法研究了对以滞量为参数的具有Hopf分支的广义Lienard方程的数值逼近问题。首先,利用欧拉方法将得到的时滞差分方程表示为映射,然后以时滞r为分支参数,利用离散动力系统的分支理论,在广义Lienard方程具有Hopf分支的条件下,给出了差分方程Hopf分支存在的条件,及连续系统与其数值逼近间的关系,证明了当该系统在r=r0产生Hopf分支时,其数值逼近也在相应的参数rh处具有Hopf分支,并且rh=r0+o(h),最后给出了一个数值仿真的例子,仿真结果表明Euler离散后的系统依旧保持了原系统的动力学性质,从而验证了理论结果的正确性.     

Bursting of Morris-Lecar neuronal model with current-feedback control

The Morris-Lecar (ML) neuronal model with current-feedback control is considered as a typical fast-slow dynamical system to study the combined influences of the reversal potential V _Ca of Ca²⁺ and the feedback current I on the generation and transition of different bursting oscillations. Two-parameter bifurcation analysis of the fast subsystem is performed in the parameter (I, V _Ca)-plane at first. Three important codimension-2 bifurcation points and some codimension-1 bifurcation curves are obtained which enable one to determine the parameter regions for different types of bursting. Next, we further divide the control parameter (V ₀, V _Ca)-plane into five different bursting regions, namely, the “fold/fold” bursting region R1, the “fold/Hopf” bursting region R2, the “fold/homoclinic” bursting region R3, the “subHopf/homoclinic” bursting region R4 and the “subHopf/subHopf” bursting region R5, as well as a silence region R6. Codimension-1 and -2 bifurcations are responsible for explanation of transition mechanisms between different types of bursting. The results are instructive for further understanding the dynamical behavior and mechanisms of complex firing activities and information processing in biological nervous systems. Supported by the National Natural Science Foundation of China (Grant Nos. 10872014 and 10702002)     

具有限时滞Van der pol方程Hopf分支的数值逼近

研究了欧拉方法对以滞量为参数的具有Hopf分支的Van der pol方程的数值逼近问题。首先,利用欧拉方法将得到的时滞差分方程表示为映射,然后以滞量为分支参数,利用离散动力系统的分支理论,在Van der pol方程具有Hopf分支的条件下,给出了差分方程Hopf分支存在的条件及连续系统与其数值逼近间的关系,证明了当该系统在r=r0产生Hopf分支时,其数值逼近也在相应的参数rh处具有Hopf分支,并且rh=r0＋o（h）.     

具时滞物价瑞利方程的Neimark-Sacker分支

研究了以滞量为参数的具时滞物价瑞利方程的数值Hopf分支问题。首先利用欧拉方法将得到的时滞差分方程表示为映射,然后利用离散动力系统的分支理论,在瑞利方程具有Hopf分支的条件下,讨论了差分方程Hopf分支存在的条件及连续系统与其数值逼近间的关系,最后证明了当连续系统产生Hopf分支时,其Euler离散将产生Neimark-Sacker分支,进而得到结论：Euler离散使得方程的Hopf分支性质得以保持。     

时滞BAM神经网络的数值逼近

研究了一类重要的多时滞BAM神经网络模型的Hopf分支的数值逼近问题.将时滞差分方程表示为映射,然后利用离散动力系统的分支理论,给出了差分方程的Hopf分支存在的条件,得到了连续模型的Hopf分支与其数值逼近的关系,证明了当步长充分小时,数值Hopf分支值逼近于原方程的Hopf分支值.     

Potassium-induced bifurcations and chaos of firing patterns observed from biological experiment on a neural pacemaker

Changes of neural firing patterns and transitions between firing patterns induced by the introduction of external stimulation or adjustment of biological parameter have been demonstrated to play key roles in information coding.In this paper,bifurcation processes of bursting patterns were observed from an experimental neural pacemaker,through the adjustment of potassium parameter including ion concentration and calcium-dependent channel conductance.The adjustment of calcium-dependent potassium channel conductance was achieved by changing the extracellular tetraethylammonium concentration.The deterministic dynamics of chaotic bursting patterns induced by period-doubling bifurcation and intermittency,and lying between two periodic bursting patterns in a period-adding bifurcation process was investigated with a nonlinear prediction method.The bifurcations included period-doubling and period-adding bifurcations of bursting patterns.The experimental bifurcations and chaos closely matched those previously simulated in the theoretical neuronal model by adjusting potassium parameter,which demonstrated the simulation results of the theoretical model.The experimental results indicate that the potassium concentration and conductance of calcium-dependent potassium channel can induce bifurcations of the neural firing patterns.The potential role of these bifurcation structures in neural information coding mechanism is discussed.     

双时滞能源价格模型分支性的数值逼近

研究了一类双时滞能源价格模型的Hopf分支的数值逼近问题。利用欧拉方法和离散动力系统的分支理论,证明了当模型在r1=r10处有H0pf分支时,其数值逼近在相应的rl=rlh处也产Hopf分支。并且数值Hopf分支值与原连续系统的H0pf分支值之间满足rlh=rl0＋O（h）。     

Bifurcation analysis for Hindmarsh-Rose neuronal model with time-delayed feedback control and application to chaos control

This paper is concerned with bifurcations and chaos control of the Hindmarsh-Rose(HR)neuronal model with the time-delayed feedback control.By stability and bifurcation analysis,we find that the excitable neuron can emit spikes via the subcritical Hopf bifurcation,and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current.For the purpose of control of chaos,we adopt the time-delayed feedback control,and convert chaos control to the Hopf bifurcation of the delayed feedback system.Then the analytical conditions under which the Hopf bifurcation occurs are given with an explicit formula.Based on this,we show the Hopf bifurcation curves in the two-parameter plane.Finally,some numerical simulations are carried out to support the theoretical results.It is shown that by appropriate choice of feedback gain and time delay,the chaotic orbit can be controlled to be stable.The adopted method in this paper is general and can be applied to other neuronal models.It may help us better understand the bifurcation mechanisms of neural behaviors.     

分布式时延范台坡方程的Hopf分岔

分析了分布式时延的范台坡方程,将平均时延作为分岔参数,证明了模型经历了Hopf分岔过程,用图示Hopf理论获得了判定分岔周期解的稳定性和分岔方向的准则。并应用数字仿真的例子证明了理论分析的正确性。     

一个类Lorenz系统的动力学分析

针对新提出的三维自治连续时间类Lorenz系统进行研究,求得该系统的平衡点,并分析平衡点的稳定性。对平衡点进行了Hopf分岔分析,得出Hopf分岔的参数条件。通过对系统的第一Lyapunov系数的分析,推导出系统发生余维二退化Hopf分岔的参数条件。借助计算机对类Lorenz系统进行数值仿真,得到该系统发生退化Hopf分岔的分岔图,与理论推导结果相符合,从而验证了理论推导的正确性。     

连续时延神经网络的Hopf分岔现象研究

讨论了带连续时延神经网络的Hopf分岔现象。对于强核和弱核的情况,利用平均时延作为分岔参数,证明了模型经历了Hopf分岔过程。在带弱核的神经网络模型中,得到了分岔周期解稳定性准则。给出了一些数值例子,通过计算机仿真验证了所得结论的正确性。     

一类碰撞振动系统的倍周期分岔研究

为了研究倍化分岔与Hopf分岔之间的联系,研究了一类碰撞振动系统因周期运动失稳而产生倍化分岔的问题。首先给出了该系统周期1-1运动的Poincaré映射建立过程,然后根据其映射的线性化矩阵的特征值穿越单位圆情况分析其映射不动点发生倍化分岔的可能性,最后通过数值计算加以验证。研究表明:系统存在典型倍周期分岔,另外单参数变化产生非共振条件下的Hopf分岔时,当参数进一步变化而越过共振点附近的某个共振区时,系统会产生非典型的倍周期分岔,其倍化分岔序列的分支数取决于强（弱）共振的阶数。     

大型水轮发电机组轴系扭振的分岔和稳定域分析

应用拉格朗日－麦克斯韦方程建立大型水轮发电机组数学模型，研究该系统平衡位置的稳定性和局部分岔，进一步证明轴承扭振不会发生Hopf分岔，利用构造李雅普诺夫函数对稳定域进行估计，并对三峡大型水轮发电机进行数值仿真。     

An analytical method for studying double Hopf bifurcations induced by two delays in nonlinear differential systems

An analytical method is introduced to investigate double Hopf bifurcations induced by two delays qualitatively and quantitatively. As an illustrative example, the clear procedure is demonstrated to study delay-induced weak resonant double Hopf bifurcation in a nonlinear system with multiple delays. When two delays are close to double Hopf bifurcation point, all solutions derived from the bifurcation are classified qualitatively and expressed explicitly. Numerical simulations are a good agreement with our theoretical analysis, and also already work in references. The results show that our work in this paper proposes a simple and valid method for investigating delay-induced double Hopf bifurcations. The important feature of our work is that the explicit expression of periodic solutions is easy to be obtained by solving algebraic equations.     

内共振条件下大范围直线运动梁的分岔分析

利用解析方法研究了一类3∶1内共振条件下大范围直线运动梁的稳定性与分岔行为.利用稳定性分析和特征值分析等方法,得到了梁系统的静态分岔、Hopf分岔、2维胎面以及3维胎面等分岔解及其稳定性情况,并给出了相应的临界分岔曲线.     

内共振条件下大范围直线运动梁的分岔分析

利用解析方法研究了一类3∶1内共振条件下大范围直线运动梁的稳定性与分岔行为.利用稳定性分析和特征值分析等方法,得到了梁系统的静态分岔、Hopf分岔、2维胎面以及3维胎面等分岔解及其稳定性情况,并给出了相应的临界分岔曲线.     

简单记忆混沌系统的动力学分析与电路实现

为了研究记忆混沌系统的动力学特性,提出了一个含广义记忆元件的简单三维自治混沌系统。分析了混沌记忆系统的记忆效应和混沌特性,进行了电路设计和相应的实验验证;并在平衡点及稳定性分析的基础上,讨论了记忆混沌系统随参数变化的动力学行为。结果表明,与基于忆阻器的混沌电路不同,新提出的系统有一个确定的平衡点,其动力学行为仅依赖于系统控制参数,有着分岔、混沌和阵发混沌等复杂非线性现象。     

一类广义Camassa-Holm方程的相图分析

为了研究一类广义Camassa-Holm方程的行波解问题,通过行波变换及平面非线性动力系统的分岔理论,借助Maple符号计算软件程序,研究了一类广义Camassa-Holm方程的分岔特性,给出了该系统在不同分支参数空间中的相图构型.     

一类改进的含分布时滞金融系统的动力学分析

针对一类改进的含分布时滞金融系统进行了动力学分析.通过增维将系统转化为等价的四维连续自治系统,并对该系统的平衡点进行分析.利用劳斯-霍维茨定理得出系统在平衡点达到稳定的参数条件.同时,利用Lyapunov指数和Hopf分岔定理等,对该系统进行了进一步分析,并得出Hopf分岔的参数条件.借助计算机对该四维系统进行数值模拟,得到相应的相图、时序图、Hopf分岔图以及Poincare截面图,验证了理论推导的准确性.