神经网络动力系统吸引子对称性的变化

用余维２的双Ｈｏｐｆ分叉的规范形方程设计了期望存储振荡型记忆模式的模拟四阶关系神经网络，所设计的网络向量场具有中心对称性。研究表明该网络发生二阶双Ｈｏｐｆ分叉后可以出现不变二环面。观察二环面上的销相运动，发现了系统出现对称破裂的规律，邓把双Ｈｏｐｆ分叉的两个频率比表示成既约分数的形式，当该分数的分子和分母均为奇数时，网络的吸引子保持对称性，而当分子主分母中任一个为偶数时，就会发生对称破裂。     

规范形网络中的混沌吸引子

讨论多余维Hopf分叉三阶规范形的普适开折形成的网络更进一步的复杂动力学行为．通过对余维二Hopf分叉的规范形网络多级分叉的分析，发现在参数空间的某个区域会出现二环面，将S形非线性加入规范形网络，在出现二环面的区域内可以出现混沌．本文给出了该混沌吸引子的相图及其二阶Poincare映射的图景．由这些图可以看到该混沌吸引子具有非常奇妙的形态：某些二阶Poincare映射像一只逼真的蝴蝶．     

规范形网络中的混沌吸引子

讨论多余维Hopf分叉三阶规范形的普适开折形成的网络更进一步的复杂动力学行为．通过对余维二Hopf分叉的规范形网络多级分叉的分析，发现在参数空间的某个区域会出现二环面，将S形非线性加入规范形网络，在出现二环面的区域内可以出现混沌．本文给出了该混沌吸引子的相图及其二阶Poincare映射的图景．由这些图可以看到该混沌吸引子具有非常奇妙的形态：某些二阶Poincare映射像一只逼真的蝴蝶．     

双零加一对纯虚根特征值系统的最简规范形

针对双零加一对纯虚根特征值系统高维分岔问题,用矩阵表示法研究其系统的最简规范形。在传统规范形基础上,证明了只有线性算子值域补空间上的近恒同变换才能化简高阶规范形。引入这些近恒同变换,代入传统规范形中,利用待定系数法逐次化简规范形,得出最简规范形系数和传统规范形系数的关系。通过对k(k≥3)阶规范形系数方程的分析发现:化简规范形的个数完全由恒同变换个数决定,对确定的最简规范形式,其系数可由原方程系数唯一确定。得到了该类系统最简规范形的形式,并编制了不经中心流形降维,直接计算该类系统最简规范形系数的Mathematica程序。     

随机平均规范形方法

计算随机规范形系数是应用随机规范形方法的关键．提出一种应用随机平均计算随机规范形系数的方法．为了说明方法的有效性,对白噪声激励的Duffing系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致．当非线性项系数较大时,规范形方法所得解与精确解相差不大．     

非半单分叉的Normal Form计算

在文［１］基础上，提出一种仅知道派生线性系统零实部特征值时求解非线性系统非半单分叉ＮｏｒｍａｌＦｏｒｍ的方法．通过适当的分类，将要求解的线性代数方程组分为若干相互独立的方程组．将所求系数向量按字典序列排列后，各独立方程组的系数矩阵是上三角矩阵．在非共振情形，各系数向量可按反字典序列递推求出．在共振情形，根据文中的二个定理，巧妙地由一简单的常数矩阵的最大秩子矩阵，定位其系数矩阵的满秩子矩阵，解决了这类方程组的降维简化．通过消元法，把简化后的方程化成类似于半单分叉ＮｏｒｍａｌＦｏｒｍ求解过程中方程的形式，其解法也类似．该方法非常易于在计算机代数软件平台上程序化．     

Van der Pol-Duffing方程的非线性动力学分叉特性研究

应用平均法研究Van der Pol-Duffing方程的幅频响应特性，并通过奇异性理论分析其静态分叉现象。进一步的动态分叉研究对系统参数空间进行了划分，发现在不同的参数区域内，系统相空间具有完全不同的拓扑特性，并应用胞映射方法分析了特定参数区域内的多吸引子共存现象。     

人工神经网络非线性动力学及应用

综述神经网络系统非线性动力学行为及其与网络性能的联系和在科学和工程中的应用；讨论了神经网络的权值动力学系统和状态动力学系统的动态过程，神经网络的稳定性和鲁棒稳定性，高阶关联网络的性能、张量描述、吸引性和复杂性，神经网络设计和综合的分叉理论；说明了神经网络动力学分析的意义和重要性．     

一类强非线性振动系统的分叉

对于参数激励和强迫激励共同作用的一类强非线性系统，本文先用改进的Ｌ－Ｐ方法求出了变换参数，使该系统的解能展为小参数的幂级数．然后利用多尺度法求出了该系统的分叉响应方程．研究了这类强非线性系统的余维１分叉问题，画出了转迁集和分叉图     

多孔介质中热对流的分叉机理研究

本文利用解析分析方法研究了数值模拟发现的多孔介质层中出现的对流分叉机理，指出控制方程中的Ｒａｙｌｅｉｇｈ数，是决定流动的特征参数。当Ｒａｙｌｅｉｇｈ数小于临界数值时，多孔介质内流动处于静止传热状态，并且这种状态是稳定的。如果Ｒａｙｌｅｉｇｈ数大于临界数值，非线性方程出现分叉解，文中指出，存在多个使平凡解失稳而分叉的临界Ｒａｙｌｅｉｇｈ数，当Ｒａｙｌｅｉｇｈ数由小到大经历这些临界数值时，其由平凡解发展起来的分叉解的流态，依次由单回流区转变为双回流区及三回流区。理论分析给出了分叉解和分叉解的振幅方程，阐明了分叉的机理，其结论和数值结果定性一致．     

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.     

神经网络时滞系统非共振双Hopf分岔及其广义同步

本文建立了具有自连接和抑制-兴奋型他连接的两个同性神经元模型。其中自连接是由于兴奋型的突触产生,而他连接则分别对应于两神经元兴奋、抑制型的突触。发现如果有兴奋型自连接就会有双Hopf分岔,而没有时滞自连接时双Hopf分岔就会消失,因此自连接引起了双Hopf分岔。作为一个例子,通过变动连接中的时滞和他连接中的比重,1／√2双Hopf分岔得到了详细研究。通过中心流形约化,分岔点邻域内各种不同的动力学行为得到了分类,并以解析形式表出。神经元活动的分岔路径得以表明。从得到的解析近似解可以发现,本文所研究的具有兴奋一抑制型他连接的两相同神经元的节律不能完全同步而只能广义同步。时滞也可以使其节律消失,两神经元变为非活动的。这些结果在控制神经网络关联记忆和设计人工神经网络方面有着潜在的应用。     

Dynamics of a multiplex neural network with delayed couplings

Multiplex networks have drawn much attention since they have been observed in many systems,e.g.,brain,transport,and social relationships.In this paper,the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied.The stability and bifurcation of the network equilibrium are discussed,and interesting neural activities of the network are explored.Based on the neuron circuit,transfer function circuit,and time delay circuit,a circuit platform of the network is constructed.It is shown that delayed couplings play crucial roles in the network dynamics,e.g.,the enhancement and suppression of the stability,the patterns of the synchronization between networks,and the generation of complicated attractors and multi-stability coexistence.     

Dynamical analysis of two coupled parametrically excited van der Pol oscillators

The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to different types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.     

3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.     

阵发型神经元网络的联想记忆与分割

以Hindmarsh-Rose(HR)神经元组成网络,研究这些网络的记忆(或模式)存储与时间分割问题．给定一个输入模式,它是几种模式的叠加,网络能够以一部分神经元同步发放的形式一个接一个地分割出每一种模式．如果输入的模式是缺损的,系统能够恢复为原型,即网络具有联想记忆功能．模拟也得到据作者所知至今还未报道的一些现象．     

Hopf and Bogdanov�CTakens bifurcations in a coupled FitzHugh�CNagumo neural system with delay

In this paper, Hopf bifurcation and Bogdanov?CTakens bifurcation with codimension 2 in a coupled FitzHugh?CNagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov?CTakens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov?CTakens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of Bogdanov?CTakens bifurcation indicates that the neuron signals can transit between resting and spiking.     

脉动型神经元网络的联想记忆与分割

以广泛讨论的Hodgk in-Hux ley神经元节点组成脉动神经元网络,从神经系统空时模式编码理论研究网络的记忆(或模式)存储与时间分割问题。给定一个输入模式,它是几种模式的叠加,网络能够以一部分神经元同步发放的形式一个接一个地在时间域分割出每一种模式。如果输入的模式是缺损的,系统能够把它们恢复成完好的原型,即神经网络的联想记忆功能。