FitzHugh-Nagumo神经元网络的同步振荡与联想记忆

本文研究由FitzHugh—Nagumo神经元所组成的脉动神经元网络的同步与联想记忆恢复。基于神经元微观生理结构，本文给出具有空间随机分布延时的神经元间耦合，而这种随机分布延时描述了脉动信号从突触前神经元到突触后神经元在轴突上传播所需要的时间。记忆由空时发放的神经元集群表达，在噪声涨落的作用下，系统取得了对不完整输入的记忆恢复。     

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.     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

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。     

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。     

周期系统中概周期解的数值计算

非线性系统的长期规则运动除了平衡点和周期解以外,概周期解,或有时表现为拟周期解也是一种长期规则运动。无论是在理论上,还是在实际计算中,确定概周期解这种规则运动远比前二者困难。本文用简单胞映射方法结合一定的技巧,数值计算周期非线性系统的概周期解。     

一类CML模型的同步解和时空周期结构

本文由扩散反应方程提出一类ＣＭＬ模型，并讨论了它的同步解和时空周期结构。     

基于脉冲控制下的复杂时滞动力网络的鲁棒同步

近年来复杂网络已经引起了科学和工程技术等各个领域的广泛关注,尤其是复杂网络中的非线性动力学行为,以及网络的拓扑结构如何影响它的动力学行为的研究,已成为当前一项极其重要的战略课题.本文主要讨论基于脉冲控制下复杂时滞动力网络的同步动力学,应用时滞动力系统的脉冲控制理论,给出了复杂时滞动力网络的一些简单而又一般的鲁棒同步化准则,这些准则能够提供一个新的和有效的控制方法来同步一个任意给定的时滞动力网络到一个期望的同步态,进一步地将所获得的结果应用到由混沌FHN神经元振子为动力节点所构成的一个具有近邻耦合结构的复杂动力网络,数值模拟表明了所获理论结果的正确性和控制方法的有效性.     

一类三维慢变振荡器周期解的奇摄动解

本文给出一类三维慢变振荡器周期解的求解方法,实例计算结果与文献[1]一致,利用本文的方法可研究文献[1]不能研究的问题。     

时滞动力系统的稳定性与分岔:从理论走向应用

本文综述了近年来时滞动力系统稳定性与分岔方面的研究进展, 重点阐述了作者及其团队在稳定性分析、Hopf分岔计算、利用时滞改善系统稳定性等方面的一些理论和方法研究结果, 介绍了时滞对颤振主动控制系统、不稳定系统镇定、网络系统的影响等方面的研究. 基于研究体会, 对进一步的研究提出了若干展望.     

Existence and Attractivity of Almost Periodic Solution for Recurrent Neural Networks with Unbounded Delays and Variable Coefficients

This paper presents several sufficient conditions for the existence and attractivity of almost periodic solution for a new class of recurrent neural networks with unbounded delays and variable coefficients. Different from the normal approach, that's to say, without resorting to any Lyapunov function, these results are obtained by utilizing generalized Halanay inequality technique and combining the theory of exponential dichotomy with fixed point method. Some existing results are found to be special case of this paper. In addition, the exponential stability of the almost periodic solution, which is not studied in the earlier references, is also considered for the system. An example is given to illustrate the feasibility of our results.This work was jointly supported by the National Natural Science Foundation of China under Grant 60373067, the Hong Kong Special Administrative Region, China with Project No. 7001146, the Natural Science Foundation of Jiangsu Province, China under Grant BK2003053, Qing-Lan Engineering Project of Jiangsu Province, China.     

Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation

This paper deals with the most important characteristics of a generalized Van der Pol–Duffing oscillator in resonance with a periodic excitation. We use an asymptotic perturbation method based on Fourier expansion and time rescaling and demonstrate through a second order perturbation analysis the existence of one or two limit cycles. Moreover, we identify a sufficient condition to obtain a doubly periodic motion, when a second low frequency appears, in addition to the forcing frequency. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.     

一种新的求解非线性系统周期解方法

本文利用切比雪夫多项式的若干良好性质,对非自治强非线性动力系统进行分析。将状态矢量在主周期上先展开谐波级数的形式,再将各谐波展开为切比雪夫多项式的形式,从而将求周期解的问题转变为非线性代数方程组的求解问题,得出一种可以方便、迅速地获得近似周期解的解析方法。此方法不依赖于小参数假设,可以用于分析强非线性问题和高维问题,而且对参数激励系统同样有效。以Duffing系统周期解的计算为例,通过与标准谐波平衡方法和四阶Runge-Kutta数值积分结果比较,说明此方法的有效性。     

Mathematical Phenomenology of Neural Stimulation by Periodic Fields

Neuron synchronization has been hypothesized as the basic mechanism leading neurological phenomena like low electroencephalographic rhythm dimension or high coherence. Cognitive processes, such as associative memory, can also be explained in terms of neuron synchronization. Inspired by the analysis of an experiment on cortex periodic photostimulation, in resonance conditions, a simple network of integrate & fire (i & f) neurons, has been used to simulate cognitive perturbations by oscillatory and pulsate stimulation of the central nervous system (CNS). In view of realistic simulations of transcranial magnetic stimulation (TMS) phenomena, a discrete extension of the FitzHug–Nagumo nervous fiber model, endowed with regenerative nodes, has been developed too.     

Estimation of Attraction Domain and Exponential Convergence Rate of Continuous Feedback Associative Memory

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Delay-induced bifurcations in collocated position control of an elastic arm

The interference of the elasticity of a single robotic arm and the unavoidable time delay of its position control is analysed from nonlinear vibrations viewpoint. The simplified mechanical model of two blocks and a connecting spring considers the first vibration mode of the arm, while the collocated proportional-derivative (PD) control uses the state of the first block only and actuates also there. It is assumed that the relevant nonlinearity is the saturation of the delayed control force. The linear stability analysis proves that stabilizable and non-stabilizable parameter regions follow each other periodically even for large spring stiffnesses and for tiny time delays. Hopf bifurcation calculation is carried out after an infinite-dimensional centre manifold reduction, and closed-form algebraic expressions are given for the amplitudes of the emerging oscillations. These results support the experimental tuning of the control gains since the parameters of the arising and often unexpected self-excited vibrations can serve as a guide for this practical procedure.

     

Twist Character of the Fourth Order Resonant Periodic Solution

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov–Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844–867.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

非自治系统用瞬态响应求周期解稳定度的方法

根据Floquet理论关于线性周期系数系统解的性质及稳定性条件 ,定义了衡量非线性非自治系统周期解受扰后的衰减速率指标—稳定度。从动力系统流的概念出发 ,给出了利用非线性非自治系统稳态周期解受扰后的瞬态响应信息计算周期解稳定度的方法。以不平衡滑动轴承 弹性转子系统为例 ,说明了该方法的有效性。将稳定度等于零作为临界判据 ,该方法不仅解决了工频周期解失稳边界的确定问题 ,而且解决了渐进稳定域的估计和抗冲击扰动裕度的计算问题     

Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.