离散神经网络模型平衡点的稳定性与分支
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摘要
本文主要研究动力系统中几类神经网络模型平衡点的稳定性及分支情况。全文共分五章。
第一章简单介绍了神经网络的学科发展背景及离散神经网络模型的研究现状,并给出了本文将要用到的一些符号和定义。
第二章,第三章分别研究了二元、四元离散神经网络模型的平衡点稳定性及局部Neimark-Sacker-分支情况,并利用Matlab数学软件,对文中讨论的模型进行数值模拟,证明了本文结论的有效性和可行性。
第四章我们讨论了一类三元具时滞离散神经网络模型,利用数学分析技巧,对线性化系统的特征根进行分析,获得了平衡点的局部稳定性及分支点,通过选取适当的参数,讨论了该模型平衡点的稳定性,Pitchfork-分支,Flip-分支和Neimark-Sacker-分支,并用中心流形理论得出了决定分支方向和稳定性的计算公式。
第五章总结了本文所做的工作,得到的主要结果及对离散神经网络模型的进一步研究方向。
The stability of the equilibrium point and bifurcation for a kind of discrete neural network model is concerned in this thesis. The thesis contains five chapters.
As the introduction, in Chapter 1, the develop background and research status of discrete neural networks model are briefly addressed, and some notations and definitions are given in this thesis.
The stability of the equilibrium point and Neimark-Sacker bifurcation for discrete neural networks of two、four neurous are studied in Chapter 2、3, respectively. The calculating formulae of direction and stability of the bifurcation are obtained by using the normal form theory and the center manifold theorem. By using the mathematical software of Matlab, results of thesis are displayed graphically which show the validity and feasibility of the theory.
In Chapter 4, we discussed a kind of three neurous delay discrete neural networks model. By using the skill of mathematical analysis, we analysis the characteristic roots of corresponding linearization system and obtain local stability of the equilibrium point and bifurcation point. Choosing an appropriate bifurcation parameter, the stability of the equilibrium point, Pitchfork/Flip/Neimark-Sacker bifurcation are discussed. We obtained the calculating formula of direction by using the center manifold theorem.
In Chapter 5, we summarized the main results of this thesis and further research direction of the discrete neural networks model.
第一章简单介绍了神经网络的学科发展背景及离散神经网络模型的研究现状,并给出了本文将要用到的一些符号和定义。
第二章,第三章分别研究了二元、四元离散神经网络模型的平衡点稳定性及局部Neimark-Sacker-分支情况,并利用Matlab数学软件,对文中讨论的模型进行数值模拟,证明了本文结论的有效性和可行性。
第四章我们讨论了一类三元具时滞离散神经网络模型,利用数学分析技巧,对线性化系统的特征根进行分析,获得了平衡点的局部稳定性及分支点,通过选取适当的参数,讨论了该模型平衡点的稳定性,Pitchfork-分支,Flip-分支和Neimark-Sacker-分支,并用中心流形理论得出了决定分支方向和稳定性的计算公式。
第五章总结了本文所做的工作,得到的主要结果及对离散神经网络模型的进一步研究方向。
The stability of the equilibrium point and bifurcation for a kind of discrete neural network model is concerned in this thesis. The thesis contains five chapters.
As the introduction, in Chapter 1, the develop background and research status of discrete neural networks model are briefly addressed, and some notations and definitions are given in this thesis.
The stability of the equilibrium point and Neimark-Sacker bifurcation for discrete neural networks of two、four neurous are studied in Chapter 2、3, respectively. The calculating formulae of direction and stability of the bifurcation are obtained by using the normal form theory and the center manifold theorem. By using the mathematical software of Matlab, results of thesis are displayed graphically which show the validity and feasibility of the theory.
In Chapter 4, we discussed a kind of three neurous delay discrete neural networks model. By using the skill of mathematical analysis, we analysis the characteristic roots of corresponding linearization system and obtain local stability of the equilibrium point and bifurcation point. Choosing an appropriate bifurcation parameter, the stability of the equilibrium point, Pitchfork/Flip/Neimark-Sacker bifurcation are discussed. We obtained the calculating formula of direction by using the center manifold theorem.
In Chapter 5, we summarized the main results of this thesis and further research direction of the discrete neural networks model.
引文
[1]W. S. Mcculloch, W. Pitts. A logical calculus of the ideas imuninet in nervous activity [J]. Bulletin of Mathematical Biophysics,1943, (5):115-133.
[2]M. Xiao, J. Cao. Bifurcation analysis on a discrete-time tabu learning model. Comput [J]. Math. Appl.,2008, (220):725-738.
[3]W. He, J. Cao. Stability and bifurcation of a class of discrete-time neural networks [J]. Math. Comput. Model,2007 (31):2111-2122.
[4]L. Sharyer, S. A. Campbell. Stability bifurcation and multistability in a system of two coupled neurons with multiple time delays [J]. SIAM J. Appl. Math.,2000 (61):673-700.
[5]Y. A. Kuznetsov. Elements of Applied Bifurcation Theory [M]. New York Springer,1998:21-46.
[6]K. Gopalsamy, I. Leung. Delay induced periodicity in a neural network of excitation and inhibition [J]. Phys. D,1996,(89):395-426.
[7]L. Olien, J. Belair. Bifurcations, stability and monotonicity properties of a delayed neural network model [J]. Phys. D,1997 (102):349-363.
[8]J. Wei, S. Ruan. Stability and bifurcation in a neural network with two delays [J]. Phys. D,1999 (130):255-272.
[9]Y. Lin, R. Lemmert, P. Volkmann. Bifurcation of periodic solution in a three-unit neural network with delay [J]. Acta. Math. Appl. Sin.,2001 (17):375-382.
[10]S. Guo, L. Huang. Stability and bifurcation in a discrete system of two neurons with delays [J]. Nonlinear Anal. Real World Appl.,2008,9(4):1323-1335.
[11]S. Guo, Y. Chen. Stability and bifurcation of a discrete-time three-neuron system with delays [J]. Int. J. Appl. Math. Eng. Sci.,2007 (1):103-115.
[12]B. D. Hassard, N. D. Kazarinoff, Y. H. Wan. Theory and Applications of Hopf Bifurcation [M]. Cambridge:Cambridge University Press,1981:221-235.
[13]T. Faria. On a planar system modeling a neuron network with memory [J]. J. Differential Equations,2000 (168):129-149.
[14]Y. Chen, J. Wu. The asymptotic shapes of periodic solutions of a singular delay differential system [J]. Differential Equations,2001 (169):614-632.
[15]F. Ren, J. Cao. LMI-based criteria for stability of high-order neural networks with time-varying delay[J]. Nonlinear Anal. Real World Appl.,2006 (7):967-979.
[16]J. Cao, Q. Song. Stability in Cohen-Grossberg type bidirectional associative memory neural networks with time-varying delays[J]. Nonlinearity,2006 (19):1601-1617.
[17]C. Li, G. Chen, X. Liao. Hopf bifurcation and chaos in tabu learning neuron models [J]. Int. Bifur. Chaos,2005 (15):2633-2642.
[18]C. Zhang, B. Zheng. Hopf bifurcation in numerical approximation of a n-dimension neural network model with multi-delays[J]. Chaos Solitons&Fractals,2005 (25):129-146.
[19]A. Stuart, A. Humphries. Dynamical Systems and Numerical Analysis [M]. Cambridge:Cambridge University Press,1996:30-56.
[20]Z. H. Yuan, J. S. Yu, L. H. Huang. Asymptotic behavior of delay difference systems [J]. Comput. Math. Appl.,2001 (42):283-290.
[21]M. B. D'Amico, J. L. Moiola. Hopf bifurcation for maps:a frequency-domain approach [J]. IEEE Trans. Circuits and Systems,2002(49):281-288.
[22]A. N. Michel, J. A. Farrel, W. Porod. Qualitative analysis of neural networks [J]. IEEE Trans. Circuits and Systems,1989 (36):229-243.
[23]J. Cao, J. Liang. Boundedness and stability for Cohen-Grossberg neural networks with time-varying delays [J]. J. Math. Anal. Appl., 2004(296):665-685.
[24]郑祖庥.泛函微分方程理论[M].合肥：安徽教育出版社,1994：2-56.
[25]廖晓昕.细胞神经网络的数学理论(I)[J].中国科学(A辑),1994,24(9)：90-910.
[26]廖晓听.细胞神经网络的数学理论(11)[J].中国科学(A辑),1994,24(10)：1037-1046.
[27]阮炯, 顾凡及,蔡志杰.神经动力学模型方法和应用[M].武汉：科学出版社,2002：68-73.
[28]袁曾任.人工神经元网络及其应用[M].北京：清华大学出版社,2000：106-109.
[29]焦李成.神经网络计算[M].西安：西安电子科技大学出版社,1993：99-102.
[30]张筑生.微分动力系统原理[M].武汉：科学出版社,1999：12-67.
[31]唐云.对称分岔理论基础[M].武汉：科学出版社,2000：1-63.
[32]Hale. J., Lunel Sv. Introduction to Functional Differential Equations [M]. New York:Spring-verlag,1993:223-245.
[33]S. Guo, L. Huang. Hopf bifurcating periodic orbits in a ring of neurons with delays [J]. Phys.D,2003(183):19-44.
[34]S. Ruan, J. Wei. On the zeros of transcendental functions to stability of delay differential equations with two delays [J]. Math.Anal.,2003 (10):863-874.
[35]Y. Yuan, S. A. Campbell. Stability and synchronization of a ring of identical cells with delayed coupling [J]. J. Dynam.Differential Equations,2004 (16):709-744.
[36]J. Wei, Y. Yuan. Synchronized Hopf bifurcation analysis in a neural network model with delays [J]. J. Math. Anal. Appl.,2005 (312):205-229.
[37]J. Wu. Symmetric functional differential equation sand neural networks with memory [J]. Trans. Amer. Math. Soc.,1998 (350):4799-483.
[38]X. Yan. Hopf bifurcation and stability for a delayed tri-neuron neural network model [J]. J. Comput. Appl. Math.,2006 (196):579-595.
[39]T. Faria. On a planar system modeling a neuron network with memory [J]. Differential Equations,2000 (168):129-149.
[40]M. W. Hirsch. Convergent activation dynamics in continuous-time networks neural network [J].1989 (2):331-334.
[41]S. Guo, X. Tang, L. Huang. Stability and bifurcation in a discrete system of two neurons with delays [J]. Nonlinear Anal. Real World,2007 (10):1016-1023.
[42]X. Liao. Mathematical Theory and Application of Stability [M]. Wuhan:Hua
zhong Normal University Press,2001:10-29.
[43]J. Cao. On stability of delayed cellular neural networks [J]. Physics Letters A, 1999(261):303-308.
[44]Z. Zhou, J. Wu. Stable periodic orbits in nonlinear discrete-time neural networks with delayed feedback [J]. Comput. Math. Appl.,2003 (45):935-942.
[45]Y. Song, M. Han, J. Wei. Stability and Hopf bifurcation on a simplied BAM neural network with delays [J]. Physica D,2005 (200):185-204.
[46]L. Wang, X. Zou. Hopf bifurcation in bidirectional associative memory neural networks with delays [J]. Appl. Math.,2004 (167):73-90.
[47]J. K. Hale. Theory of Functional Differential Equations [M]. New York:Springer, 1977:245-267.
[48]J. Wei, M. Li. Global existence of periodic solutions in a tri-neuron network model with delays [J]. Physica D,2004 (198):106-119.
[49]J. Wei, S. Ruan. Stability and bifurcation in a neural network modelwith two delays [J]. Physica D,1999 (130):255-272.
[50]D. Xu, H. Zhao. Invariant and attracting sets of Hopied neural networks with delay [J]. Int. J. System Sci.,2001 (32):863-866.
[51]C. Li, X. Liao, J. Yu. Tabu learning method for multiuser detection in CDMA systems [J]. Neuro. computing,2002 (49):411-415.
[52]K. Gopalsamy, X. He. Stability in a symmetric Hopied neural networks with transmission delay [J]. Phys.D.,1994 (76):344-358.
[53]李绍荣,廖晓峰.连续时延神经网络的Hopf分叉现象研究[J].电子科技大学学报,2002(31)：163-167.
[54]时宝,张德存,盖明久.微分方程理论及其应用[M],北京：科学出版社,1999：44-67.
[55]廖晓听,昌莉.离散Hopfield神经网络的稳定性研究[J].自动化学报,1999(25)：721-727.
[56]高莹,熊志欢.判定离散Hopfield神经网络稳定性的新方法[J].武汉大学学报(理学版),2002(48)：18-22.
[2]M. Xiao, J. Cao. Bifurcation analysis on a discrete-time tabu learning model. Comput [J]. Math. Appl.,2008, (220):725-738.
[3]W. He, J. Cao. Stability and bifurcation of a class of discrete-time neural networks [J]. Math. Comput. Model,2007 (31):2111-2122.
[4]L. Sharyer, S. A. Campbell. Stability bifurcation and multistability in a system of two coupled neurons with multiple time delays [J]. SIAM J. Appl. Math.,2000 (61):673-700.
[5]Y. A. Kuznetsov. Elements of Applied Bifurcation Theory [M]. New York Springer,1998:21-46.
[6]K. Gopalsamy, I. Leung. Delay induced periodicity in a neural network of excitation and inhibition [J]. Phys. D,1996,(89):395-426.
[7]L. Olien, J. Belair. Bifurcations, stability and monotonicity properties of a delayed neural network model [J]. Phys. D,1997 (102):349-363.
[8]J. Wei, S. Ruan. Stability and bifurcation in a neural network with two delays [J]. Phys. D,1999 (130):255-272.
[9]Y. Lin, R. Lemmert, P. Volkmann. Bifurcation of periodic solution in a three-unit neural network with delay [J]. Acta. Math. Appl. Sin.,2001 (17):375-382.
[10]S. Guo, L. Huang. Stability and bifurcation in a discrete system of two neurons with delays [J]. Nonlinear Anal. Real World Appl.,2008,9(4):1323-1335.
[11]S. Guo, Y. Chen. Stability and bifurcation of a discrete-time three-neuron system with delays [J]. Int. J. Appl. Math. Eng. Sci.,2007 (1):103-115.
[12]B. D. Hassard, N. D. Kazarinoff, Y. H. Wan. Theory and Applications of Hopf Bifurcation [M]. Cambridge:Cambridge University Press,1981:221-235.
[13]T. Faria. On a planar system modeling a neuron network with memory [J]. J. Differential Equations,2000 (168):129-149.
[14]Y. Chen, J. Wu. The asymptotic shapes of periodic solutions of a singular delay differential system [J]. Differential Equations,2001 (169):614-632.
[15]F. Ren, J. Cao. LMI-based criteria for stability of high-order neural networks with time-varying delay[J]. Nonlinear Anal. Real World Appl.,2006 (7):967-979.
[16]J. Cao, Q. Song. Stability in Cohen-Grossberg type bidirectional associative memory neural networks with time-varying delays[J]. Nonlinearity,2006 (19):1601-1617.
[17]C. Li, G. Chen, X. Liao. Hopf bifurcation and chaos in tabu learning neuron models [J]. Int. Bifur. Chaos,2005 (15):2633-2642.
[18]C. Zhang, B. Zheng. Hopf bifurcation in numerical approximation of a n-dimension neural network model with multi-delays[J]. Chaos Solitons&Fractals,2005 (25):129-146.
[19]A. Stuart, A. Humphries. Dynamical Systems and Numerical Analysis [M]. Cambridge:Cambridge University Press,1996:30-56.
[20]Z. H. Yuan, J. S. Yu, L. H. Huang. Asymptotic behavior of delay difference systems [J]. Comput. Math. Appl.,2001 (42):283-290.
[21]M. B. D'Amico, J. L. Moiola. Hopf bifurcation for maps:a frequency-domain approach [J]. IEEE Trans. Circuits and Systems,2002(49):281-288.
[22]A. N. Michel, J. A. Farrel, W. Porod. Qualitative analysis of neural networks [J]. IEEE Trans. Circuits and Systems,1989 (36):229-243.
[23]J. Cao, J. Liang. Boundedness and stability for Cohen-Grossberg neural networks with time-varying delays [J]. J. Math. Anal. Appl., 2004(296):665-685.
[24]郑祖庥.泛函微分方程理论[M].合肥：安徽教育出版社,1994：2-56.
[25]廖晓昕.细胞神经网络的数学理论(I)[J].中国科学(A辑),1994,24(9)：90-910.
[26]廖晓听.细胞神经网络的数学理论(11)[J].中国科学(A辑),1994,24(10)：1037-1046.
[27]阮炯, 顾凡及,蔡志杰.神经动力学模型方法和应用[M].武汉：科学出版社,2002：68-73.
[28]袁曾任.人工神经元网络及其应用[M].北京：清华大学出版社,2000：106-109.
[29]焦李成.神经网络计算[M].西安：西安电子科技大学出版社,1993：99-102.
[30]张筑生.微分动力系统原理[M].武汉：科学出版社,1999：12-67.
[31]唐云.对称分岔理论基础[M].武汉：科学出版社,2000：1-63.
[32]Hale. J., Lunel Sv. Introduction to Functional Differential Equations [M]. New York:Spring-verlag,1993:223-245.
[33]S. Guo, L. Huang. Hopf bifurcating periodic orbits in a ring of neurons with delays [J]. Phys.D,2003(183):19-44.
[34]S. Ruan, J. Wei. On the zeros of transcendental functions to stability of delay differential equations with two delays [J]. Math.Anal.,2003 (10):863-874.
[35]Y. Yuan, S. A. Campbell. Stability and synchronization of a ring of identical cells with delayed coupling [J]. J. Dynam.Differential Equations,2004 (16):709-744.
[36]J. Wei, Y. Yuan. Synchronized Hopf bifurcation analysis in a neural network model with delays [J]. J. Math. Anal. Appl.,2005 (312):205-229.
[37]J. Wu. Symmetric functional differential equation sand neural networks with memory [J]. Trans. Amer. Math. Soc.,1998 (350):4799-483.
[38]X. Yan. Hopf bifurcation and stability for a delayed tri-neuron neural network model [J]. J. Comput. Appl. Math.,2006 (196):579-595.
[39]T. Faria. On a planar system modeling a neuron network with memory [J]. Differential Equations,2000 (168):129-149.
[40]M. W. Hirsch. Convergent activation dynamics in continuous-time networks neural network [J].1989 (2):331-334.
[41]S. Guo, X. Tang, L. Huang. Stability and bifurcation in a discrete system of two neurons with delays [J]. Nonlinear Anal. Real World,2007 (10):1016-1023.
[42]X. Liao. Mathematical Theory and Application of Stability [M]. Wuhan:Hua
zhong Normal University Press,2001:10-29.
[43]J. Cao. On stability of delayed cellular neural networks [J]. Physics Letters A, 1999(261):303-308.
[44]Z. Zhou, J. Wu. Stable periodic orbits in nonlinear discrete-time neural networks with delayed feedback [J]. Comput. Math. Appl.,2003 (45):935-942.
[45]Y. Song, M. Han, J. Wei. Stability and Hopf bifurcation on a simplied BAM neural network with delays [J]. Physica D,2005 (200):185-204.
[46]L. Wang, X. Zou. Hopf bifurcation in bidirectional associative memory neural networks with delays [J]. Appl. Math.,2004 (167):73-90.
[47]J. K. Hale. Theory of Functional Differential Equations [M]. New York:Springer, 1977:245-267.
[48]J. Wei, M. Li. Global existence of periodic solutions in a tri-neuron network model with delays [J]. Physica D,2004 (198):106-119.
[49]J. Wei, S. Ruan. Stability and bifurcation in a neural network modelwith two delays [J]. Physica D,1999 (130):255-272.
[50]D. Xu, H. Zhao. Invariant and attracting sets of Hopied neural networks with delay [J]. Int. J. System Sci.,2001 (32):863-866.
[51]C. Li, X. Liao, J. Yu. Tabu learning method for multiuser detection in CDMA systems [J]. Neuro. computing,2002 (49):411-415.
[52]K. Gopalsamy, X. He. Stability in a symmetric Hopied neural networks with transmission delay [J]. Phys.D.,1994 (76):344-358.
[53]李绍荣,廖晓峰.连续时延神经网络的Hopf分叉现象研究[J].电子科技大学学报,2002(31)：163-167.
[54]时宝,张德存,盖明久.微分方程理论及其应用[M],北京：科学出版社,1999：44-67.
[55]廖晓听,昌莉.离散Hopfield神经网络的稳定性研究[J].自动化学报,1999(25)：721-727.
[56]高莹,熊志欢.判定离散Hopfield神经网络稳定性的新方法[J].武汉大学学报(理学版),2002(48)：18-22.