余维数为2的时滞FHN和BAM神经网络的分岔研究

发布时间：2018-08-25 12:27

【摘要】：近年来,如Hopfied、Bidirectional associative memory(BAM)、Cohen-Grossberg、Fitzhugh-Nagumo(FHN)以及细胞神经网络等多种神经网络被很好的应用于解决一些模式识别、优化组合、信号处理、自动控制和图像处理复杂的问题.而这些应用主要是基于神经网络丰富的动力学行为.所以,神经网络动力学逐渐发展成为当今生命科学研究领域的一个重要前沿课题,受到了国际上很多学者的关注.尤其是FHN和BAM神经网络系统稳定性以及分岔现象受到了格外的关注.由于在生物神经系统的信号传输中,时滞出现的不可避免性和神经元突触连接强度的可变性,会使神经网络系统发生更加复杂的动力学行为.因此,对余维数为2的时滞神经网络的分岔研究,有助于改进神经网络系统和拓展其相关的应用领域.本文对时滞耦合FHN神经网络的Hopf-pitchfork分岔和中立型时滞BAM神经网络的Bogdanov-Takens(B-T)分岔进行了深入的研究,现将主要工作与创新点概括如下:一、对于时滞耦合FHN神经系统.首先,通过分析该系统在平衡点线性化系统的特征方程,给出了系统发生余维数为2的Hopf-pitchfork分岔的充分条件.其次,以时滞和耦合强度为分岔参数,运用中心流形定理和规范型方法,计算系统在中心流形上Hopf-pitchfork分岔的规范型用来分析系统分岔的性质.最后,给出了Hopf-pitchfork分岔的分岔图并进行了详细分析,且为了验证理论分析的正确性进行了数值模拟.通过分析我们发现,随着两个分岔参数的变化,在Hopfpitchfork分岔临界点附近系统可能会存在一个稳定极限环、一对稳定的平衡点或是一个稳定极限环与一对稳定的平衡点共存在的多稳态现象.这意味着在神经系统中的神经元不只会处于静息状态或放电状态,还会处于静息态和周期放电共存的状态.此外,还发现随着两个分岔参数的改变,神经元不仅可以实现从静息态到周期放电的转变,还可以实现从周期放电到静息态的转变.二、对于中立型时滞BAM神经网络模型.首先,根据系统在平衡点线性化特征方程根的分布,得到了系统发生余维数为2的B-T分岔和余维数为3的Triplezero分岔的临界条件.其次,选择神经元之间的两个连接权重为分岔参数,利用与上面类似的方法分别进行计算系统在中心流形上B-T分岔的二阶和三阶规范型用来分析系统的分岔现象.最后,对B-T分岔的二阶和三阶规范型的分岔图进行了详细分析,且为了验证理论分析的正确性和有效性进行了数值模拟.通过分析,我们发现,随着分岔参数的改变,在B-T分岔的临界点附近可能会存在一对稳定点共存、稳定的周期解或同宿轨道等有趣的现象.
[Abstract]:In recent years, many kinds of neural networks, such as Hopfied,Bidirectional associative memory (BAM) Cohen-Grossberg Fitzhugh-Nagumo (FHN) and cellular neural networks, have been used to solve the complex problems of pattern recognition, optimal combination, signal processing, automatic control and image processing. These applications are mainly based on the rich dynamic behavior of neural networks. Therefore, neural network dynamics has gradually developed into an important frontier topic in the field of life science, which has been paid attention to by many scholars in the world. Especially, the stability and bifurcation of FHN and BAM neural networks are paid more and more attention. Due to the inevitability of time delay and the variability of synaptic connection strength in the biological nervous system, the neural network system will have more complex dynamic behavior. Therefore, the research on the bifurcation of delayed neural networks with codimension 2 is helpful to improve the neural network system and expand its related application fields. In this paper, the Hopf-pitchfork bifurcation of the delayed coupled FHN neural network and the Bogdanov-Takens (B-T) bifurcation of the neutral delayed BAM neural network are studied. The main work and innovations are summarized as follows: first, for the delayed coupled FHN neural system. Firstly, by analyzing the characteristic equations of the linearized system at the equilibrium point, the sufficient conditions for the Hopf-pitchfork bifurcation of the system with codimension 2 are given. Secondly, with time delay and coupling strength as bifurcation parameters, the normal form of Hopf-pitchfork bifurcation on the center manifold is calculated by using the center manifold theorem and the normal form method to analyze the properties of the system bifurcation. Finally, the bifurcation diagram of Hopf-pitchfork bifurcation is given and analyzed in detail, and the numerical simulation is carried out to verify the correctness of the theoretical analysis. Through analysis, we find that with the change of two bifurcation parameters, there may be a stable limit cycle near the critical point of Hopfpitchfork bifurcation. A pair of stable equilibrium points or a stable limit cycle and a pair of stable equilibrium points co-exist multi-stable phenomenon. This means that neurons in the nervous system will not only be in a state of rest or discharge, but also in a state of coexistence of resting and periodic discharge. In addition, it is found that with the change of two bifurcation parameters, the neuron can not only realize the transition from the static state to the periodic discharge, but also realize the transition from the periodic discharge to the rest state. Second, for neutral delay BAM neural network model. Firstly, according to the root distribution of the eigenequation of linearization of the system at the equilibrium point, the critical conditions for the B-T bifurcation with codimension 2 and the Triplezero bifurcation with codimension 3 for the system are obtained. Secondly, two connecting weights between neurons are chosen as bifurcation parameters, and the second and third order normal forms of B-T bifurcation on the central manifold are used to analyze the bifurcation phenomena of the system respectively. Finally, the bifurcation diagrams of the second and third order normal forms of B-T bifurcation are analyzed in detail, and the numerical simulation is carried out to verify the correctness and validity of the theoretical analysis. Through analysis, we find that with the change of bifurcation parameters, there may be some interesting phenomena such as coexistence of stable points, stable periodic solutions or homoclinic orbits near the critical point of B-T bifurcation.
【学位授予单位】：云南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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