分布式时延神经网络系统Hopf分岔的频域分析
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摘要
神经网络是一个非常复杂的非线性动力学系统，在神经网络动力学性质中有可能出现稳定、不稳定、振荡和混沌行为。人体控制呼吸和心跳等有规律的运动功能是靠周期性的神经脉冲，因此，研究神经网络中的周期解具有现实意义。包含持续振荡的神经网络可应用于模式识别与联想记忆。
在生物和人工神经网络模型中，有时必须考虑到其内在的时延。目前，时延神经网络模型的动力学现象是一个热门的研究课题。而分布式时延模型比离散时延模型应用更加广泛。本文讨论分布式时延的双神经元模型，以平均时延作为分岔参数，研究连续时延的神经网络模型的Hopf分岔现象，也就是当分岔参数通过某一临界值时，一族周期解从平衡点处产生。
需要指出的是，研究Hopf分岔的工作通常是在时延微分方程的状态空间中讨论的，通常称为“时域”方法。最近，在一些文献中提出了一种研究微分方程的新方法，应用反馈系统的理论与方法，即在状态空间中作拉普拉斯变换后在复数域中进行分析，称之为“频域”方法。频域方法首先是由陈关荣教授等提出的。频域方法相比传统的时域方法具有一定优势，利用图示方法避开了复杂的数学计算和分析。时延神经网络模型用时域方法研究Hopf分岔非常复杂，特别是强核分布式时延模型用时域方法研究尤其困难，本文用频域方法很好地解决了这个问题。
本文用频域方法确定分岔点的存在性，以平均时延作为分岔参数，研究Hopf分岔现象，当分岔参数超过某一临界值时发生分岔，并利用图示Hopf分岔定理分析分岔方向与周期解的稳定性，并给出了一些数值模拟例子和频域图以验证所得结论的正确性。
在第一章，我们首先简述非线性动力学系统的一些基本的数学概念和结果，介绍关于传统的Hopf分岔理论时域方法和频域方法，并说明频域方法与时域方法相比所具有的优势。
在第二章，分别给出弱核分布式时延神经网络模型和强核分布式时延神经网络模型，通过引入状态反馈控制，取得带非线性反馈的一个线性系统，并以平均时延作为分岔参数，用频域方法分析相关的特征方程，讨论Hopf分岔点的存在性，给出了计算Hopf分岔点的代数方程。
在第三章，利用奈奎斯特准则和Hopf分岔图示定理分析上述两个模型的Hopf分岔方向与分岔周期解的稳定性。引入曲率系数，根据曲率系数确定分岔方向。并作出频域图，即在复平面上作特征值轨线和一条射线L1，根据频域图判断周期解的稳定性。
在第四章，给出数值模拟例子，并作出频域图，以说明我们所得结果的正确性与有效性。
It is well known that neural networks are complex and large-scale nonlinear dynamical systems. The dynamical characteristics of neural networks include stable, unstable, oscillatory, and chaotic behavior. The periodic nature of neural impulses is of fundamental significance in the control of regular dynamical functions such as breathing and heart beating. Neural networks involving persistent oscillations such as limit cycle may be applied to pattern recognition and associative memory.
In modeling biological or artificial neural networks, it is sometimes necessary to take into account the inherent time delays. The dynamics of neural networks with time delays have been discussed by many researchers. It is well known that a neural network model with distributed delay is more general than one with discrete delay. In this paper, a more general two-neuron model with distributed delays is investigated.
We notice that, the works about Hopf bifurcation usually use the state-space formulation for delayed differential equations, referred to as the "time domain" approach. Yet there is another interesting formulation for the differential equations in the literature. This alternative representation applies the familiar theory and methodology of feedback engineering systems: an approach described in the "frequency domain", the complex domain after the standard Laplace transforms have been taken in the time domain state-space system. The frequency domain approach was initiated by Allwright [1], Mees and Chua [2], Moiola and Chen [3, 4]. This new methodology has the advantage over classical time domain methods. A typical one is its pictorial characteristic that utilizes advanced computer graphical capabilities and so bypasses quite a lot of sophistical mathematical analysis. It is very difficult to analyze the Hopf bifurcation on a neural network with time delays by applying the time domain approach, especially, in the case of a two-neuron model with distributed delays and the strong kernel. In this paper, these models are analyzed by means of the frequency domain approach.
In this paper, by means of the frequency domain approach proposed in [5], the
existence of Hopf bifurcation parameter is determined. If the mean delay used as a bifurcation parameter, it is found that Hopf bifurcation occurs when the bifurcation parameter exceeds a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are analyzed by means of the Graphical Hopf Bifurcation Theorem. Some numerical simulation results and the frequency-domain graphs are given, verifying the theoretical analysis results.
In Chapter one, we firstly give a briefing of some fundamental mathematical concepts and results of nonlinear dynamical systems, then both the time domain and the frequency domain approach to the classical Hopf bifurcation theorem will be introduced. Finally the advantages of the frequency-domain approach over the classical time-domain approach are addressed.
In Chapter two, a general two-neuron model with distributed delays for a weak kernel / strong kernel is investigated. By introducing a "state-feedback control", one obtains a linear system with a nonlinear feedback. The mean delay is used as a bifurcation parameter. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter for the model is proven. The algebraic equations for computing the Hopf bifurcation are obtained.
In Chapter three, the direction and stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. The curvature coefficient will be introduced for determining the direction of the Hopf bifurcation. By drawing the frequency graph of the eigenvalue-locus and the half-line L1, one can determine the stability of the bifurcating periodic solution.
In Chapter four, some numerical simulation results and the frequency graph are presented to justify the theoretical analysis results.
在生物和人工神经网络模型中，有时必须考虑到其内在的时延。目前，时延神经网络模型的动力学现象是一个热门的研究课题。而分布式时延模型比离散时延模型应用更加广泛。本文讨论分布式时延的双神经元模型，以平均时延作为分岔参数，研究连续时延的神经网络模型的Hopf分岔现象，也就是当分岔参数通过某一临界值时，一族周期解从平衡点处产生。
需要指出的是，研究Hopf分岔的工作通常是在时延微分方程的状态空间中讨论的，通常称为“时域”方法。最近，在一些文献中提出了一种研究微分方程的新方法，应用反馈系统的理论与方法，即在状态空间中作拉普拉斯变换后在复数域中进行分析，称之为“频域”方法。频域方法首先是由陈关荣教授等提出的。频域方法相比传统的时域方法具有一定优势，利用图示方法避开了复杂的数学计算和分析。时延神经网络模型用时域方法研究Hopf分岔非常复杂，特别是强核分布式时延模型用时域方法研究尤其困难，本文用频域方法很好地解决了这个问题。
本文用频域方法确定分岔点的存在性，以平均时延作为分岔参数，研究Hopf分岔现象，当分岔参数超过某一临界值时发生分岔，并利用图示Hopf分岔定理分析分岔方向与周期解的稳定性，并给出了一些数值模拟例子和频域图以验证所得结论的正确性。
在第一章，我们首先简述非线性动力学系统的一些基本的数学概念和结果，介绍关于传统的Hopf分岔理论时域方法和频域方法，并说明频域方法与时域方法相比所具有的优势。
在第二章，分别给出弱核分布式时延神经网络模型和强核分布式时延神经网络模型，通过引入状态反馈控制，取得带非线性反馈的一个线性系统，并以平均时延作为分岔参数，用频域方法分析相关的特征方程，讨论Hopf分岔点的存在性，给出了计算Hopf分岔点的代数方程。
在第三章，利用奈奎斯特准则和Hopf分岔图示定理分析上述两个模型的Hopf分岔方向与分岔周期解的稳定性。引入曲率系数，根据曲率系数确定分岔方向。并作出频域图，即在复平面上作特征值轨线和一条射线L1，根据频域图判断周期解的稳定性。
在第四章，给出数值模拟例子，并作出频域图，以说明我们所得结果的正确性与有效性。
It is well known that neural networks are complex and large-scale nonlinear dynamical systems. The dynamical characteristics of neural networks include stable, unstable, oscillatory, and chaotic behavior. The periodic nature of neural impulses is of fundamental significance in the control of regular dynamical functions such as breathing and heart beating. Neural networks involving persistent oscillations such as limit cycle may be applied to pattern recognition and associative memory.
In modeling biological or artificial neural networks, it is sometimes necessary to take into account the inherent time delays. The dynamics of neural networks with time delays have been discussed by many researchers. It is well known that a neural network model with distributed delay is more general than one with discrete delay. In this paper, a more general two-neuron model with distributed delays is investigated.
We notice that, the works about Hopf bifurcation usually use the state-space formulation for delayed differential equations, referred to as the "time domain" approach. Yet there is another interesting formulation for the differential equations in the literature. This alternative representation applies the familiar theory and methodology of feedback engineering systems: an approach described in the "frequency domain", the complex domain after the standard Laplace transforms have been taken in the time domain state-space system. The frequency domain approach was initiated by Allwright [1], Mees and Chua [2], Moiola and Chen [3, 4]. This new methodology has the advantage over classical time domain methods. A typical one is its pictorial characteristic that utilizes advanced computer graphical capabilities and so bypasses quite a lot of sophistical mathematical analysis. It is very difficult to analyze the Hopf bifurcation on a neural network with time delays by applying the time domain approach, especially, in the case of a two-neuron model with distributed delays and the strong kernel. In this paper, these models are analyzed by means of the frequency domain approach.
In this paper, by means of the frequency domain approach proposed in [5], the
existence of Hopf bifurcation parameter is determined. If the mean delay used as a bifurcation parameter, it is found that Hopf bifurcation occurs when the bifurcation parameter exceeds a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are analyzed by means of the Graphical Hopf Bifurcation Theorem. Some numerical simulation results and the frequency-domain graphs are given, verifying the theoretical analysis results.
In Chapter one, we firstly give a briefing of some fundamental mathematical concepts and results of nonlinear dynamical systems, then both the time domain and the frequency domain approach to the classical Hopf bifurcation theorem will be introduced. Finally the advantages of the frequency-domain approach over the classical time-domain approach are addressed.
In Chapter two, a general two-neuron model with distributed delays for a weak kernel / strong kernel is investigated. By introducing a "state-feedback control", one obtains a linear system with a nonlinear feedback. The mean delay is used as a bifurcation parameter. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter for the model is proven. The algebraic equations for computing the Hopf bifurcation are obtained.
In Chapter three, the direction and stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. The curvature coefficient will be introduced for determining the direction of the Hopf bifurcation. By drawing the frequency graph of the eigenvalue-locus and the half-line L1, one can determine the stability of the bifurcating periodic solution.
In Chapter four, some numerical simulation results and the frequency graph are presented to justify the theoretical analysis results.
引文
[1] D. J. Allwright. Harmonic balance and the Hopf bifurcation theorem. Math. Proc. of Cambridge Phil. Soc, 1977, Vol.82: 453~467
[2] A. I. Mees, L. O. Chua. The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems. IEEE Trans. Circ. Sys, 1979, Vol.26: 235~254
[3] J. L. Moiola, G. Chen. Computation of limit cycles via higher-order harmonic balance approximation. IEEE Trans. Auto. Contr, 1993, Vol.38: 782~790
[4] J. L. Moiola, G. Chen. Frequency domain approach to computational analysis of bifurcations and limit cycles: A tutorial. Int. J. of Bifur. Chaos, 1993, Vol.3: 843~867
[5] J. L. Moiola, G. Chen. Hopf Bifurcation Analysis: A Frequency Domain Approach. Singapore: World Scientific, 1996. 21~41
[6] 叶彦谦. 常微分方程讲义. 北京: 高等教育出版社, 1979. 280~297
[7] J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1997. 117~156
[8] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan. Theory and Applications of Hopf Bifurcation. London: Cambridge University Press, 1981. 1~117
[9] K. Gopalsamy, I. Leung. Convergence under dynamical thresholds with delays. IEEE Trans. on NN, 1997, Vol.8, No.2: 341~348
[10] 李绍荣, 廖晓峰. 连续时延神经网络的Hopf分岔现象研究 . 电子科技大学学报, 2002, Vol.31, No.2: 163~167
[11] X. F. Liao, Z. F. Wu, J. B. Yu. Stability switches and bifurcation analysis of a neural network with continuous delay. IEEE Trans. SMC A, 1999, Vol.29, No.6: 692~696
[12] X. F. Liao, K. W. Wong, C. S. Leung. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function. Chaos, Solitons and Fractals, 2001, Vol.12: 1535~1547
[13] K. Gopalsamy, I. Leung. Delay-induced Periodicity in a neural network of excitation and inhibition. Physica D, 1996, Vol.89: 395~426
[14] K. Gopalsamy, I. Leung, P. Liu. Global Hopf-bifurcation in a neural netlet. Appl. Math. Comput, 1998, Vol.94: 171~192
[15] K. L. Babcock, R. M. Westervelt. Dynamics of simple electronic neural networks with added inertia. Physica D, 1986, Vol.23: 464~469
[16] K. L. Babcock, R. M. Westervelt. Dynamics of simple electronic neural networks. Physica D, 1987, Vol.28: 305~316
[17] H. R. Wiillson, J. D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J, 1972, Vol.12: 1~24
[18] U. An der Heiden. Delays in physiological systems. J. Math. Biol, 1979, Vol.8: 345~364
[19] L. Olien, J. Belair. Bifurcation, stability and monotonicity properties of a delayed neural model. Physica D, 1997, Vol.102: 349~363
[20] J. Wei, S. Ruan. Stability and bifurcation in a neural network model with two delays. Physica D, 1999, Vol.130: 255~272
[21] X. F. Liao, K. W. Wong, Z. F. Wu. Bifurcation analysis on a two-neuron system with distributed delays. Physica D, 2001, Vol.149: 123~141
[2] A. I. Mees, L. O. Chua. The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems. IEEE Trans. Circ. Sys, 1979, Vol.26: 235~254
[3] J. L. Moiola, G. Chen. Computation of limit cycles via higher-order harmonic balance approximation. IEEE Trans. Auto. Contr, 1993, Vol.38: 782~790
[4] J. L. Moiola, G. Chen. Frequency domain approach to computational analysis of bifurcations and limit cycles: A tutorial. Int. J. of Bifur. Chaos, 1993, Vol.3: 843~867
[5] J. L. Moiola, G. Chen. Hopf Bifurcation Analysis: A Frequency Domain Approach. Singapore: World Scientific, 1996. 21~41
[6] 叶彦谦. 常微分方程讲义. 北京: 高等教育出版社, 1979. 280~297
[7] J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1997. 117~156
[8] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan. Theory and Applications of Hopf Bifurcation. London: Cambridge University Press, 1981. 1~117
[9] K. Gopalsamy, I. Leung. Convergence under dynamical thresholds with delays. IEEE Trans. on NN, 1997, Vol.8, No.2: 341~348
[10] 李绍荣, 廖晓峰. 连续时延神经网络的Hopf分岔现象研究 . 电子科技大学学报, 2002, Vol.31, No.2: 163~167
[11] X. F. Liao, Z. F. Wu, J. B. Yu. Stability switches and bifurcation analysis of a neural network with continuous delay. IEEE Trans. SMC A, 1999, Vol.29, No.6: 692~696
[12] X. F. Liao, K. W. Wong, C. S. Leung. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function. Chaos, Solitons and Fractals, 2001, Vol.12: 1535~1547
[13] K. Gopalsamy, I. Leung. Delay-induced Periodicity in a neural network of excitation and inhibition. Physica D, 1996, Vol.89: 395~426
[14] K. Gopalsamy, I. Leung, P. Liu. Global Hopf-bifurcation in a neural netlet. Appl. Math. Comput, 1998, Vol.94: 171~192
[15] K. L. Babcock, R. M. Westervelt. Dynamics of simple electronic neural networks with added inertia. Physica D, 1986, Vol.23: 464~469
[16] K. L. Babcock, R. M. Westervelt. Dynamics of simple electronic neural networks. Physica D, 1987, Vol.28: 305~316
[17] H. R. Wiillson, J. D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J, 1972, Vol.12: 1~24
[18] U. An der Heiden. Delays in physiological systems. J. Math. Biol, 1979, Vol.8: 345~364
[19] L. Olien, J. Belair. Bifurcation, stability and monotonicity properties of a delayed neural model. Physica D, 1997, Vol.102: 349~363
[20] J. Wei, S. Ruan. Stability and bifurcation in a neural network model with two delays. Physica D, 1999, Vol.130: 255~272
[21] X. F. Liao, K. W. Wong, Z. F. Wu. Bifurcation analysis on a two-neuron system with distributed delays. Physica D, 2001, Vol.149: 123~141
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