时滞动力系统的某些动力学行为研究
|
摘要
随着科学技术的快速发展,在物理、力学、控制理论、生物学、医学和经济学等自然科学及边缘学科的研究领域提出了大量由时滞动力方程(也称时滞动力系统)所描述的具体数学模型,因而对时滞动力系统进行研究在理论和实际应用方面都有重要意义。
时滞动力系统属于非线性动力学的分支。经典非线性动力学是以扰动、渐近分析的方法研究弱非线性弱耦合的系统。而现代非线性动力学与经典非线性动力学不同,研究的是系统的定性与定量的变化规律,其使用的方法是精确方法,所研究的系统具有强非线性性,其研究对象主要包括稳定性、周期解、吸引子以及分叉、混沌、孤立子等新的现象,其主要任务是探索非线性科学的复杂性。
本文对著名的Duffing型, Liénard型方程以及区间细胞神经网络、反应扩散神经网络、联想记忆神经元模型进行了研究和推广,考虑时滞对系统稳定性的影响,运用多种方法研究了几种时滞泛函微分方程所表示的模型的全局动力行为,包括:周期解的存在性,鲁棒稳定性,分叉等,并利用Matlab求解泛函微分方程和对神经网络进行了仿真。本文共由6章组成,主要内容如下:
首先是第1章绪论,概述了时滞动力方程的研究背景和发展状况,并简要介绍了本文的主要工作。
第2章,利用重合度理论(Mawhin延拓定理)、不等式理论以及各种变换技巧研究了推广的时滞Duffing型、Liénard型方程,考虑其周期解的存在性问题,得到了某些充分条件,所得结果推广和改进了相关文献的结果。
第3章,一方面,在有限区间情况下,采用拓扑度理论和建立适当的Liapunov-函数方法,讨论区间细胞神经网络的渐近鲁棒稳定性问题。另一方面,在无穷区间情况下,在对无穷区间上的Lebesgue-Stieltjes积分的连续性、可微性进行研究的基础上,应用新的分析技巧和构建适当的带有Lebesgue-Stieltjes积分的Liapunov泛函,讨论区间细胞神经网络的全局指数鲁棒稳定性。两种情况均得到了简单有效的判别准则并给出了实例。所得结论有助于系统结构稳定性分析。
第4章,通过截断函数和截断方程,给出了S-分布时滞的反应扩散神经细胞网络模型(RDCNNs)具有全局指数稳定性的平衡点唯一存在的充分性条件,去掉有关文献中要求信号函数f j和g j, j = 1, 2,,n,有界性、单调性和可微性的苛刻条件。给出实用有效的M ?矩阵判断S-分布时滞RDCNNs稳定性的代数方法。所的结果通过实例得到了验证。
第5章,研究了具有非单调动力系统的神经元联想记忆模型,其中的输出函数不是sigmoid函数而是非单调的。通过分析非单调系统的分叉相图,获得基于特征方程和Hassard技巧的渐近稳定原则。利用正规型理论和中心流形定理分析了Hopf分叉的稳定性和方向。最后,利用数值模拟对时滞变化引起的相图的变化进行了仿真。所得到的网络具有一定的应用价值。
最后,对本文所研究的内容和主要结果进行了总结,并对研究工作的前景做了展望。
With the increasing development of science and technology, many mathematical models which are described by delay dynamical equations(delay dynamical systems)are proposed in the field of natural science and edging field including physics, mechanics, control theory, biology, medicine and economics. Therefore, the study of delay dynamical systems is important in theory and practical application.
Delay dynamical systems belong to nonlinear dynamics. Classic nonlinear dynamics focus on weakly nonlinear and weakly coupled systems by the methods of disturbance and asymptotic analysis. While modern nonlinear dynamics is different from classic one, it studies the rules of the qualitative and quantitative change of systems. The research methods are precision. The systems are strongly nonlinear. The research objectives are stability, periodic solution, attractor, bifurcation, chaos, and solitons. The aim is to discovery the complexity of nonlinear dynamics.
In this paper, the author studies further on some mathematics models.
Considering the effecting of time delay to the stability of the systems and applying many methods, the author investigates global dynamical behaviors of several kinds delay functional differential equations. The author studys the existence of periodic solutions, robust stability, bifurcation, and introduce the solution of functional differential equations and neural network simulation with Matlab.
This paper is composed of six chapters and main results are described as follows: First, the author introduces the historical background and the recent development of delay dynamic systems and neural networks. Moreover, the main results of this paper are also briefly introduced.
Second, by applying coincidence degree theory,the existence of periodic solution of Duffing-type and generalized Liénard-type equation with delay is investigated, and some sufficient criteria are established,which improve and generalize some known results.
Third, one hand, applying topological degree theory and constructing suitable Liapunov functional, the author investigates asymptotic robust stability of interval cellular neural networks with S-type distributed delays on finite intervals. The other hand, on the base of studying of continuity and differentiability of Lebesgue-Stieltjes integration on infinitely interval, the author investigates global exponential robust stability of interval cellular networks with S-type distributed time delay by applying the new analysis techniques and constructing suitable Liapunov functional with Lebesgue -Stieltjes integration. In two cases, the convenient criteria and examples are presented. The conclusions are useful to analyze the stability of systems.
Fourth, via the intercept functions and intercept equation, the author presents some sufficient conditions for the existence of a unique globally exponentially stable equilibrium of reaction-diffusion cellular neural networks (RDCNNs) with S-type distributed time delay, in which the boundedness, monotonicity and differentiability of signal functions f j and g j,j = 1,2,,n defined on R are not required. The results are verifiable and the method of M-matrix is more practical.
Fifth, the associative memory of neurons model with nonmonotone dynamics is considered, where the output function is not sigmoid but nonmonotonic. After analyze the bifurcating phase plot of the nonmonotonic system, the author obtains the asymptotically stability criteria based on the characteristic equation and technique in Hassard. The stability and direction of the Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Finally, numerical simulations have been used to demonstrate the change of phase plot with respect to the time delay. The networks studied above have value in application to some extend.
Finally, the studying contents and the main results of this paper are briefly summarized, furthermore, some research prospects are also proposed.
时滞动力系统属于非线性动力学的分支。经典非线性动力学是以扰动、渐近分析的方法研究弱非线性弱耦合的系统。而现代非线性动力学与经典非线性动力学不同,研究的是系统的定性与定量的变化规律,其使用的方法是精确方法,所研究的系统具有强非线性性,其研究对象主要包括稳定性、周期解、吸引子以及分叉、混沌、孤立子等新的现象,其主要任务是探索非线性科学的复杂性。
本文对著名的Duffing型, Liénard型方程以及区间细胞神经网络、反应扩散神经网络、联想记忆神经元模型进行了研究和推广,考虑时滞对系统稳定性的影响,运用多种方法研究了几种时滞泛函微分方程所表示的模型的全局动力行为,包括:周期解的存在性,鲁棒稳定性,分叉等,并利用Matlab求解泛函微分方程和对神经网络进行了仿真。本文共由6章组成,主要内容如下:
首先是第1章绪论,概述了时滞动力方程的研究背景和发展状况,并简要介绍了本文的主要工作。
第2章,利用重合度理论(Mawhin延拓定理)、不等式理论以及各种变换技巧研究了推广的时滞Duffing型、Liénard型方程,考虑其周期解的存在性问题,得到了某些充分条件,所得结果推广和改进了相关文献的结果。
第3章,一方面,在有限区间情况下,采用拓扑度理论和建立适当的Liapunov-函数方法,讨论区间细胞神经网络的渐近鲁棒稳定性问题。另一方面,在无穷区间情况下,在对无穷区间上的Lebesgue-Stieltjes积分的连续性、可微性进行研究的基础上,应用新的分析技巧和构建适当的带有Lebesgue-Stieltjes积分的Liapunov泛函,讨论区间细胞神经网络的全局指数鲁棒稳定性。两种情况均得到了简单有效的判别准则并给出了实例。所得结论有助于系统结构稳定性分析。
第4章,通过截断函数和截断方程,给出了S-分布时滞的反应扩散神经细胞网络模型(RDCNNs)具有全局指数稳定性的平衡点唯一存在的充分性条件,去掉有关文献中要求信号函数f j和g j, j = 1, 2,,n,有界性、单调性和可微性的苛刻条件。给出实用有效的M ?矩阵判断S-分布时滞RDCNNs稳定性的代数方法。所的结果通过实例得到了验证。
第5章,研究了具有非单调动力系统的神经元联想记忆模型,其中的输出函数不是sigmoid函数而是非单调的。通过分析非单调系统的分叉相图,获得基于特征方程和Hassard技巧的渐近稳定原则。利用正规型理论和中心流形定理分析了Hopf分叉的稳定性和方向。最后,利用数值模拟对时滞变化引起的相图的变化进行了仿真。所得到的网络具有一定的应用价值。
最后,对本文所研究的内容和主要结果进行了总结,并对研究工作的前景做了展望。
With the increasing development of science and technology, many mathematical models which are described by delay dynamical equations(delay dynamical systems)are proposed in the field of natural science and edging field including physics, mechanics, control theory, biology, medicine and economics. Therefore, the study of delay dynamical systems is important in theory and practical application.
Delay dynamical systems belong to nonlinear dynamics. Classic nonlinear dynamics focus on weakly nonlinear and weakly coupled systems by the methods of disturbance and asymptotic analysis. While modern nonlinear dynamics is different from classic one, it studies the rules of the qualitative and quantitative change of systems. The research methods are precision. The systems are strongly nonlinear. The research objectives are stability, periodic solution, attractor, bifurcation, chaos, and solitons. The aim is to discovery the complexity of nonlinear dynamics.
In this paper, the author studies further on some mathematics models.
Considering the effecting of time delay to the stability of the systems and applying many methods, the author investigates global dynamical behaviors of several kinds delay functional differential equations. The author studys the existence of periodic solutions, robust stability, bifurcation, and introduce the solution of functional differential equations and neural network simulation with Matlab.
This paper is composed of six chapters and main results are described as follows: First, the author introduces the historical background and the recent development of delay dynamic systems and neural networks. Moreover, the main results of this paper are also briefly introduced.
Second, by applying coincidence degree theory,the existence of periodic solution of Duffing-type and generalized Liénard-type equation with delay is investigated, and some sufficient criteria are established,which improve and generalize some known results.
Third, one hand, applying topological degree theory and constructing suitable Liapunov functional, the author investigates asymptotic robust stability of interval cellular neural networks with S-type distributed delays on finite intervals. The other hand, on the base of studying of continuity and differentiability of Lebesgue-Stieltjes integration on infinitely interval, the author investigates global exponential robust stability of interval cellular networks with S-type distributed time delay by applying the new analysis techniques and constructing suitable Liapunov functional with Lebesgue -Stieltjes integration. In two cases, the convenient criteria and examples are presented. The conclusions are useful to analyze the stability of systems.
Fourth, via the intercept functions and intercept equation, the author presents some sufficient conditions for the existence of a unique globally exponentially stable equilibrium of reaction-diffusion cellular neural networks (RDCNNs) with S-type distributed time delay, in which the boundedness, monotonicity and differentiability of signal functions f j and g j,j = 1,2,,n defined on R are not required. The results are verifiable and the method of M-matrix is more practical.
Fifth, the associative memory of neurons model with nonmonotone dynamics is considered, where the output function is not sigmoid but nonmonotonic. After analyze the bifurcating phase plot of the nonmonotonic system, the author obtains the asymptotically stability criteria based on the characteristic equation and technique in Hassard. The stability and direction of the Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Finally, numerical simulations have been used to demonstrate the change of phase plot with respect to the time delay. The networks studied above have value in application to some extend.
Finally, the studying contents and the main results of this paper are briefly summarized, furthermore, some research prospects are also proposed.
引文
[1] J.K. Hale, S.V. Lunel. Introduction to functional differential equations. New York: Springer -Verlag Inc., 1993
[2] Z. X. Zheng.Theory of functional differential equations.Hefei: Anhui education press, 1994
[3]秦元勋,刘永清,王联,郑祖庥.带有时滞的动力系统的运动稳定性(第2版).北京:科学出版社,1989
[4]廖晓昕.稳定性的理论,方法和应用.武汉:华中理工大学出版社,1999
[5]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法.济南:山东科技出版社,2005
[6] Gaines R E,Mawhin J. Coincidence Degree and Nonlinear Differential Equations. 568. Berlin: Spring-Verlag, 1977
[7] M.W.Hirsch, S.Smale著,黄傑，刘世伟合译.《Differential Equations, Dynamical Systems, and Linear Algebra》.北京:高等教育出版社,1986
[8]高普云.非线性动力学-分叉、混沌与孤立子.长沙:国防科技大学出版社.2005
[9]阎平凡,张长水.人工神经网络与模拟进化计算.北京:清华大学出版社,2000
[10]罗四维.大规模人工神经网络理论基础.北京:清华大学出版社;北方交通大学出版社2004
[11] Hopfield J. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Science, USA (Biophysics), 1982, 79:2554 -2558
[12] Hopfield J. Neurons with graded response have collective computational properties likes those of two-state neurons. Proceedings of the National Academy of Science, USA (Biophysics), 1984, 81: 3088-3092
[13] Marcus C M, Westervelt R M. Stability of analog neural network with delay. Physics Review A, 1989, 39: 347-359
[14] Gopalsamy K, He X. Stability in asymmetric Hopfield nets with transmission delays. Physica D, 1994, 76: 344-358
[15] M.Forti, Tesi.w. Conditions for global stability of neural networks with application to linearand quadratic programming problems. IEEE, Transactions on Circuits and Systems, 1995, 42: 354-366
[16]赵洪涌,王广兰.具有变时滞Hopfield神经网络的概周期解存在性与全局吸引性.数学物理学报,2004,24(6):723-729
[17]陈万义.一类Hopfield型时滞神经网络模型的全局渐近稳定性.生物数学学报,2004,19 (2): 175-179
[18]陈安平,黄立宏.Hopfield神经网络的概周期解存在性和全局吸引性,数学物理学报,2001,21 (4): 505-511
[19] Liao X, Chen G, Sanchez E. LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans Circuit system, 2002, 49 (7): 1033-1039
[20] Arik S. Stability analysis of delayed neural networks. IEEE Trans Circuit system, 2000, 47(7): 1092-1098
[21] Zhou J, Liu Z, Chen G. Dynamics of periodic delayed neural networks. Neural Networks, 2004, 17: 87-101
[22] Tank D W, Hopfield J. Neural computation by concentrating information in time. Proceedings of the National Academy of Science, USA (Biophysics), 1987, 84: 1896-1991
[23]张强,马润年.具有连续分布时滞的双向联想记忆神经网络的全局稳定性.中国科学(E辑),2003,33(6):481-487
[24] Li T, Fei S M. Exponential state estimation for recurrent neural networks with distributed delays. Neurocomputing, 2007, 71(1-3): 428-438
[25] Yang H F, Chu T G. LMI conditions for stability of neural networks with distributed delays. Chaos, Solitons and Fractals, 2007, 34(2): 557-563
[26] Huang Z T, Luo X S, Yang Q G. Global asymptotic stability analysis of bidirectional associative memory neural networks with distributed delays and impulse. Chaos, Solitons and Fractals, 2007, 34(3): 878-885
[27] Wang L S, Xu D Y. Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays. International Journal of System Science, 2002, 33(11): 869-877
[28] Wang M, Wang L S. Global asymptotic robust stability of static neural network models with S-type distributed delays. Mathematical and Computer Modeling, 2006, 44(1-2): 218-222
[29]邓谨,王林山,徐道义.具有S型分布时滞的Hopfield神经网络模型的渐近行为.四川大学学报,2004,41(1):19-23
[30]王珉,王林山.S-分布时滞静态神经网络模型的全局渐近稳定性.山东大学学报(理学版), 2005,40:5-8
[31] Liao Xiaoxin, Fu Yuli, Cao Jian, Zhao Xinquan. Stability of Hopfield neural networks with reaction-diffusion terms, Acta Electronica Sinica, 2002, 28(1): 78-81
[32]廖晓昕,杨叔子等.具有反应扩散的广义神经网络的稳定性.中国科学(E辑),2002, 32(1):87-94
[33]王林山,徐道义.变时滞反应扩散Hopfield神经网络全局指数稳定性.中国科学(E辑), 2003,33(6):488-495
[34] Linshan wang. Comments on“Robust stability for interval neural networks with time delay by X.F Liao”. IEEE Trans on Neural Networks, 2002, 13(1): 250-252
[35] Linshan wang, Daoyi Xu. Stability for Hopfield neural networks with delay, Journal of Vibration and Control, 2002, 8(1): 13-18
[36] D.M. Zhou. Global asymptotic stability of bi-directional associative memory neural networks with delays. Journal of Biomathematics, 2000, 15: 88–92
[37] L.S. Wang, D.Y. Xu. Global asymptotic stability of bi-directional associative memory neural networks with S-type distributed delays. International Journal of Systems Science 2002, 33: 869–877
[38] Linshan wang, Daoyi Xu. Asymptotic behavior of a class of reaction-diffusion equations with delays, Journal of Mathematical Analysis and applications, 2003, 281(2): 439-453
[39] Linshan wang, Daoyi Xu. Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays. Science in China (SeriesF), 2003, 46 (6): 466-474
[40] Shengfan Zhou, Linshan wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete and Continuous Dynamical Systems, 2003, 9 (2): 394-412
[41] Linshan wang, Yuying Gao. On global robust stability for interval Hopfield neural networks with time delay. Ann. Of Diff. Eqs., 2003, 19 (2): 1-6
[42]王林山.时滞递归神经网络.北京:科学出版社,2007
[43] L.O.Chua and L.Yang, Cellular neural networks theory, IEEE, Transactions on Circuits andSystems,1988, 35: 1257-1272.
[44] Yang T, Yang L. The global stability of fuzzy cellular neural network. IEEE Trans Circ Syst. I 1996, 43(10): 880-883
[45] Liu Y, Tang W. Exponential stability of fuzzy cellular neural networks with constant and time-varying delays. Phys Lett A, 2004, 323(3-4): 224-233
[46] Barbounis T G, Theocharis J B. A locally recurrent fuzzy neural network with application to the wind speed prediction using spatial correlation.Neurocomputing,2007,70(7-9):1525-1542
[47] Huang T. Exponential stability of delayed fuzzy cellular neural networks with diffusion. Chaos, Solitons and Fractals, 2007, 31(3): 658-664
[48] J. Cao, Q.Song, Stability in Cohen-Grossberg type BAM neural networks with time-varying delays. Nonlinearity 2006, 19: 1601-1617
[49] Zhao H, Ding N. Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays. Applied Mathematics and Computation, 2006, 183(1): 464-470
[50] Wang Z, Lauria S, Fang J, Liu X. Exponential stability of uncertain stochastic neural networks with mixed time-delays. Chaos, Solitons and Fractals, 2007, 32(1): 62-72
[51] Yonghui Sun, Jinde Cao. Pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Analysis, 2007, 8:1171-1185
[52] Wang Z D, Fang J A, Liu X H. Global stability of stochastic high-order neural networks with discrete and distributed delays. Chaos,Solitons and Fractals, 2008, 36(2): 388-396
[53] Demuth H, Beale M. Neural network toolbox user’s guide---neural network toolbox for use with MATLAB. The MathWorks, Inc., 1992
[54] A Dhooge, W Goverts, Kuznetsov Yu. A. Matcont. A MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 2003, 29(2): 141-164
[55]张立明.人工神经网络的模型及其应用.上海:复旦大学出版社,1993
[56]王伟.人工神经网络原理―入门及应用.北京:北京航天航空大学出版社,1995
[57]丛爽.面向MATLAB工具箱的神经网络理论与应用.合肥:中国科学技术大学出版社,1998
[58]周开利,康耀红.神经网络模型及其MATLAB仿真程序设计.北京:清华大学出版社,2005
[59]丁同仁.在共振点的非线性振动[J].中国科学(A辑),1982, 1: 1-13
[60] Ianacci R,Nkashama M N.Lecture Notes in Math[M].New York:Springer-Verlag,1984
[61] Ward J.Asymptotic conditions for periodic solutions of ordinary differential equations. Proc Amer Math Soc,1981,81:415-420
[62] Mawhin J.Periodic Solutions of some vector retarded functional differential equations .J Math Anal Appl,1974,45:588-603
[63]郑祖麻.泛函微分方程理论.安徽:安徽教育出版社,1994
[64]周纪卿,朱因远.非线性振动.西安:西安交通大学出版社,1998
[65]刘式适,刘式达.物理学中的非线性方程.北京:北京大学出版社, 2000
[66]黄先开,向子贵.具有时滞的Duffing型方程x + g( x(t?τ))=p(t)的2π周期解.科学通报,1994,39(3):201-203
[67]张正球,庾建设.一类时滞Duffing型方程周期解.高校应用数学学报,1998, 13(4): 398-392
[68]王根强,燕居让.多变时滞非线性自治微分方程的周期解.数学物理学报,2000,20(4): 431-435
[69]刘锡平,葛渭高.一类具复杂偏差变元的Duffing型方程的周期解.高校应用数学学报,2003,18(1): 51-56
[70]李永坤.具有偏差变元的Liénard型方程的周期解.数学研究与评论,1998,18:565-570
[71]鲁世平,葛渭高.一类二阶具偏差变元的微分方程周期解.数学学报,2002,45(4): 811- 818
[72] Lu Shiping, Ge Weigao. Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument. Nonlinear Analysis, 2004, 56: 501-514
[73] Lequn Peng. Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments. Mathematical and Compter Modelling,2007, 45(3): 378-386
[74]孙德献,陈凤德,陈晓星.具有时滞和反馈控制的Logistic增长模型的正周期解.生物数学学报, 2005, 20(4): 447- 455
[75] Lu Shi-ping,Ge-Weigao. Periodic solutions of second order differential equations with deviating arguments. J. Math. Anal. Appl.,2004, 249: 231- 243
[76]孟华,刘炳文.具有偏差变元的Liénard型方程周期解.数学研究与评论,2006, 26(3):547- 552
[77]刘炳文,黄立宏.一类具时滞的Liénard型方程的周期解.数学物理学报, 2006, 26A(3): 329-336
[78]范猛王克.一类具有HollingⅡ型功能性反应的捕食者—食饵系统全局周期解的存在性.数学物理学报(A辑),2001, 21(4): 492- 497 [ 79]王在洪.跨共振点Duffing方程周期解的多解性.数学年刊(A辑), 1997, 1(4): 421-432
[80] Zhong-Ping Jiang. Advanced feedback control of the chaotic Duffing equation. IEEE Transactions on Circuits and Systems. 2002, 49(2): 244-249
[81] Temple H. Fay. Duffing’s equation and nonlinear resonance. INT. J. MATH. EDUC. SCI. TECHNOL., 2003, 34(6):871-880
[82] Zhaosheng Feng. Monotonous property of non-oscillations of the damped Duffing,s equation. Chaos, Solitons & Fractals, 2006, 28: 463-471
[83] Qiyuan Zhou, Bingwen Liu. New result on almost periodic solutions for a class of nonlinear Duffing equtions with a deviating arqument. Applied Mathematics Letters, 2009, 22:6-11
[84] Qinsheng Bi. Chaos crisis in coupled Duffing's systems with initial phase difference. Physics Letters A, 2007, 369(5-6): 418-431
[85] Zhujun Jing, Jicai Huang, Jin Deng. Complex dynamics in three-well duffing system with two external forcings. Chaos, Solitons & Fractals, 2007, 33(3): 795-812
[86] Sihem A. Lazzouni, Samuel Bowong, F.M. Moukam Kakmeni, Brahim Cherki, Noureddine Ghouali. Chaos control using small-amplitude damping signals of the extended Duffing equation. Communications in Nonlinear Science and Numerical Simulation. 2007, 12(5): 804-813
[87] Long-Jye Sheu, Hsien-Keng Chen, Juhn-Horng Chen, Lap-Mou Tam. Chaotic dynamics of the fractionally damped Duffing equation. Chaos, Solitons & Fractals, 2007, 32(4): 1459-1468
[88] Bashir Ahmad, Badra S. Alghamdi. Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions. Computer Physics Communications, 2008, 179(6): 409-416
[89] Xin-Ge Liu, Mei-Lan Tang, Ralph R. Martin. Periodic solutions for a kind of Liénard equation. Journal of Computational and Applied Mathematics, 2008, 219(1): 263-275
[90] Vasile Marinca, Nicolae Heri?anu. Periodic solutions of Duffing equation with strong non- linearity. Chaos, Solitons & Fractals, 2008, 37(1): 144-149
[91] Fuxing Zhang, Ya Li. Existence and uniqueness of periodic solutions for a kind of duffing type p-Laplacian equation. Nonlinear Analysis: Real World Applications, 2008, 9(3): 985-989
[92] Y.Z.Chen,X.Y.Lin.A successive integration technique for the solution of the Duffing oscillator equation with damping and excitation. Nonlinear Analysis, 2009, 70: 3603-3608
[93] E.Tafo Wembe, R.Yamapi. Chaos synchronization of resistively coupled Duffing systems: Numerical and experimental investigations. Communications in Nonlinear Science and Numerical Simulation, 2009, 14: 1439-1453
[94] Fabao Gao, Shiping Lu, Wei Zhang. Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument. Nonlinear Analysis, 2009, 70: 3567-3574
[95] Liu Z H, Yuan R. Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response. Journal of Mathematics Analysis and Applications, 2004, 296: 521-537
[96] Yan X P,Chu Y D. Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system. Journal of Computational and Applied Mathematics, 2006, 196: 198-210
[97] Meng X Z, Han D, Song Y L. Stability and bifurcation in a non-Kolmogorov type prey-predator system with time delay. Mathematical and Computer Modelling, 2005, 41: 1445-1455
[98] Li Y L,Xiao D M. Bifurcations of a predator–prey system of Holling and Leslie types. Chaos, Solitons and Fractals, 2007, 34(2): 606-620
[99] Liao X, Wong K, Wu Z. Bifurcation analysis on a two-neuron system with distributed delays. Physica D, 2001, 149: 123-141
[100] Zhao H Y, Wang L. Hopf bifurcation in Cohen-Grossberg neural network with distributed delays. Nonlinear Analysis: Real World Applications, 2007, 8(1): 73-89
[101] Liao, X. F and Ran, J. H. Hopf bifurcation in love dynamical models with nonlinear couples and time delays, Chaos, solitons and Fractals., 2007, 31: 853-865
[102] Belair, J and Campbell, S. A., Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math., 1994, 54: 1402-1424
[103] Cooke K, Grossman Z. Discrete delay, distributed delay and stability switches. J Math AnalAppl, 1982, 86: 592-627
[104] R Seydel. Practical Bifurcation and Stability Analysis. Springer-Verlag, New York, Inc, 1994
[105] Hassard B D, Kazarinoff N D, Wan Y H. Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981
[106] Morita, M. Associative Memory with nonmonotone dynamics, Neural Networks., 1993, 6 : 115-126
[107] Liao XF, Li C, Zhou S. Hopf bifurcation and chaos, in macroeconomic models with policy lag. Chaos, solitons and Fractals., 2005, 25: 91-108
[108] Liao XF, Chen G. Hopf bifurcation and chaos analysis of Chen’s system with distributed delays. Chaos, solitons and Fractals., 2005, 25: 197-220
[109] Wei J, Ruan S. Stability and bifurcation in a neural network model with two delays. Physica D 1999, 130: 255-272
[110] Yoshizawa, S., Morita, M. and Amari, S Capacity of associative memory using a nonmono–tonic neuron model, Neural networks., 1993, 6 :167-176
[111]徐伟,都琳,许勇.非线性随机动力系统研究的若干进展.工程数学学报,2006:23(6):951-960
[112]龚光鲁.随机微分方程及其应用概要.北京:清华大学出版社,2008
[2] Z. X. Zheng.Theory of functional differential equations.Hefei: Anhui education press, 1994
[3]秦元勋,刘永清,王联,郑祖庥.带有时滞的动力系统的运动稳定性(第2版).北京:科学出版社,1989
[4]廖晓昕.稳定性的理论,方法和应用.武汉:华中理工大学出版社,1999
[5]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法.济南:山东科技出版社,2005
[6] Gaines R E,Mawhin J. Coincidence Degree and Nonlinear Differential Equations. 568. Berlin: Spring-Verlag, 1977
[7] M.W.Hirsch, S.Smale著,黄傑，刘世伟合译.《Differential Equations, Dynamical Systems, and Linear Algebra》.北京:高等教育出版社,1986
[8]高普云.非线性动力学-分叉、混沌与孤立子.长沙:国防科技大学出版社.2005
[9]阎平凡,张长水.人工神经网络与模拟进化计算.北京:清华大学出版社,2000
[10]罗四维.大规模人工神经网络理论基础.北京:清华大学出版社;北方交通大学出版社2004
[11] Hopfield J. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Science, USA (Biophysics), 1982, 79:2554 -2558
[12] Hopfield J. Neurons with graded response have collective computational properties likes those of two-state neurons. Proceedings of the National Academy of Science, USA (Biophysics), 1984, 81: 3088-3092
[13] Marcus C M, Westervelt R M. Stability of analog neural network with delay. Physics Review A, 1989, 39: 347-359
[14] Gopalsamy K, He X. Stability in asymmetric Hopfield nets with transmission delays. Physica D, 1994, 76: 344-358
[15] M.Forti, Tesi.w. Conditions for global stability of neural networks with application to linearand quadratic programming problems. IEEE, Transactions on Circuits and Systems, 1995, 42: 354-366
[16]赵洪涌,王广兰.具有变时滞Hopfield神经网络的概周期解存在性与全局吸引性.数学物理学报,2004,24(6):723-729
[17]陈万义.一类Hopfield型时滞神经网络模型的全局渐近稳定性.生物数学学报,2004,19 (2): 175-179
[18]陈安平,黄立宏.Hopfield神经网络的概周期解存在性和全局吸引性,数学物理学报,2001,21 (4): 505-511
[19] Liao X, Chen G, Sanchez E. LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans Circuit system, 2002, 49 (7): 1033-1039
[20] Arik S. Stability analysis of delayed neural networks. IEEE Trans Circuit system, 2000, 47(7): 1092-1098
[21] Zhou J, Liu Z, Chen G. Dynamics of periodic delayed neural networks. Neural Networks, 2004, 17: 87-101
[22] Tank D W, Hopfield J. Neural computation by concentrating information in time. Proceedings of the National Academy of Science, USA (Biophysics), 1987, 84: 1896-1991
[23]张强,马润年.具有连续分布时滞的双向联想记忆神经网络的全局稳定性.中国科学(E辑),2003,33(6):481-487
[24] Li T, Fei S M. Exponential state estimation for recurrent neural networks with distributed delays. Neurocomputing, 2007, 71(1-3): 428-438
[25] Yang H F, Chu T G. LMI conditions for stability of neural networks with distributed delays. Chaos, Solitons and Fractals, 2007, 34(2): 557-563
[26] Huang Z T, Luo X S, Yang Q G. Global asymptotic stability analysis of bidirectional associative memory neural networks with distributed delays and impulse. Chaos, Solitons and Fractals, 2007, 34(3): 878-885
[27] Wang L S, Xu D Y. Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays. International Journal of System Science, 2002, 33(11): 869-877
[28] Wang M, Wang L S. Global asymptotic robust stability of static neural network models with S-type distributed delays. Mathematical and Computer Modeling, 2006, 44(1-2): 218-222
[29]邓谨,王林山,徐道义.具有S型分布时滞的Hopfield神经网络模型的渐近行为.四川大学学报,2004,41(1):19-23
[30]王珉,王林山.S-分布时滞静态神经网络模型的全局渐近稳定性.山东大学学报(理学版), 2005,40:5-8
[31] Liao Xiaoxin, Fu Yuli, Cao Jian, Zhao Xinquan. Stability of Hopfield neural networks with reaction-diffusion terms, Acta Electronica Sinica, 2002, 28(1): 78-81
[32]廖晓昕,杨叔子等.具有反应扩散的广义神经网络的稳定性.中国科学(E辑),2002, 32(1):87-94
[33]王林山,徐道义.变时滞反应扩散Hopfield神经网络全局指数稳定性.中国科学(E辑), 2003,33(6):488-495
[34] Linshan wang. Comments on“Robust stability for interval neural networks with time delay by X.F Liao”. IEEE Trans on Neural Networks, 2002, 13(1): 250-252
[35] Linshan wang, Daoyi Xu. Stability for Hopfield neural networks with delay, Journal of Vibration and Control, 2002, 8(1): 13-18
[36] D.M. Zhou. Global asymptotic stability of bi-directional associative memory neural networks with delays. Journal of Biomathematics, 2000, 15: 88–92
[37] L.S. Wang, D.Y. Xu. Global asymptotic stability of bi-directional associative memory neural networks with S-type distributed delays. International Journal of Systems Science 2002, 33: 869–877
[38] Linshan wang, Daoyi Xu. Asymptotic behavior of a class of reaction-diffusion equations with delays, Journal of Mathematical Analysis and applications, 2003, 281(2): 439-453
[39] Linshan wang, Daoyi Xu. Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays. Science in China (SeriesF), 2003, 46 (6): 466-474
[40] Shengfan Zhou, Linshan wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete and Continuous Dynamical Systems, 2003, 9 (2): 394-412
[41] Linshan wang, Yuying Gao. On global robust stability for interval Hopfield neural networks with time delay. Ann. Of Diff. Eqs., 2003, 19 (2): 1-6
[42]王林山.时滞递归神经网络.北京:科学出版社,2007
[43] L.O.Chua and L.Yang, Cellular neural networks theory, IEEE, Transactions on Circuits andSystems,1988, 35: 1257-1272.
[44] Yang T, Yang L. The global stability of fuzzy cellular neural network. IEEE Trans Circ Syst. I 1996, 43(10): 880-883
[45] Liu Y, Tang W. Exponential stability of fuzzy cellular neural networks with constant and time-varying delays. Phys Lett A, 2004, 323(3-4): 224-233
[46] Barbounis T G, Theocharis J B. A locally recurrent fuzzy neural network with application to the wind speed prediction using spatial correlation.Neurocomputing,2007,70(7-9):1525-1542
[47] Huang T. Exponential stability of delayed fuzzy cellular neural networks with diffusion. Chaos, Solitons and Fractals, 2007, 31(3): 658-664
[48] J. Cao, Q.Song, Stability in Cohen-Grossberg type BAM neural networks with time-varying delays. Nonlinearity 2006, 19: 1601-1617
[49] Zhao H, Ding N. Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays. Applied Mathematics and Computation, 2006, 183(1): 464-470
[50] Wang Z, Lauria S, Fang J, Liu X. Exponential stability of uncertain stochastic neural networks with mixed time-delays. Chaos, Solitons and Fractals, 2007, 32(1): 62-72
[51] Yonghui Sun, Jinde Cao. Pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Analysis, 2007, 8:1171-1185
[52] Wang Z D, Fang J A, Liu X H. Global stability of stochastic high-order neural networks with discrete and distributed delays. Chaos,Solitons and Fractals, 2008, 36(2): 388-396
[53] Demuth H, Beale M. Neural network toolbox user’s guide---neural network toolbox for use with MATLAB. The MathWorks, Inc., 1992
[54] A Dhooge, W Goverts, Kuznetsov Yu. A. Matcont. A MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 2003, 29(2): 141-164
[55]张立明.人工神经网络的模型及其应用.上海:复旦大学出版社,1993
[56]王伟.人工神经网络原理―入门及应用.北京:北京航天航空大学出版社,1995
[57]丛爽.面向MATLAB工具箱的神经网络理论与应用.合肥:中国科学技术大学出版社,1998
[58]周开利,康耀红.神经网络模型及其MATLAB仿真程序设计.北京:清华大学出版社,2005
[59]丁同仁.在共振点的非线性振动[J].中国科学(A辑),1982, 1: 1-13
[60] Ianacci R,Nkashama M N.Lecture Notes in Math[M].New York:Springer-Verlag,1984
[61] Ward J.Asymptotic conditions for periodic solutions of ordinary differential equations. Proc Amer Math Soc,1981,81:415-420
[62] Mawhin J.Periodic Solutions of some vector retarded functional differential equations .J Math Anal Appl,1974,45:588-603
[63]郑祖麻.泛函微分方程理论.安徽:安徽教育出版社,1994
[64]周纪卿,朱因远.非线性振动.西安:西安交通大学出版社,1998
[65]刘式适,刘式达.物理学中的非线性方程.北京:北京大学出版社, 2000
[66]黄先开,向子贵.具有时滞的Duffing型方程x + g( x(t?τ))=p(t)的2π周期解.科学通报,1994,39(3):201-203
[67]张正球,庾建设.一类时滞Duffing型方程周期解.高校应用数学学报,1998, 13(4): 398-392
[68]王根强,燕居让.多变时滞非线性自治微分方程的周期解.数学物理学报,2000,20(4): 431-435
[69]刘锡平,葛渭高.一类具复杂偏差变元的Duffing型方程的周期解.高校应用数学学报,2003,18(1): 51-56
[70]李永坤.具有偏差变元的Liénard型方程的周期解.数学研究与评论,1998,18:565-570
[71]鲁世平,葛渭高.一类二阶具偏差变元的微分方程周期解.数学学报,2002,45(4): 811- 818
[72] Lu Shiping, Ge Weigao. Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument. Nonlinear Analysis, 2004, 56: 501-514
[73] Lequn Peng. Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments. Mathematical and Compter Modelling,2007, 45(3): 378-386
[74]孙德献,陈凤德,陈晓星.具有时滞和反馈控制的Logistic增长模型的正周期解.生物数学学报, 2005, 20(4): 447- 455
[75] Lu Shi-ping,Ge-Weigao. Periodic solutions of second order differential equations with deviating arguments. J. Math. Anal. Appl.,2004, 249: 231- 243
[76]孟华,刘炳文.具有偏差变元的Liénard型方程周期解.数学研究与评论,2006, 26(3):547- 552
[77]刘炳文,黄立宏.一类具时滞的Liénard型方程的周期解.数学物理学报, 2006, 26A(3): 329-336
[78]范猛王克.一类具有HollingⅡ型功能性反应的捕食者—食饵系统全局周期解的存在性.数学物理学报(A辑),2001, 21(4): 492- 497 [ 79]王在洪.跨共振点Duffing方程周期解的多解性.数学年刊(A辑), 1997, 1(4): 421-432
[80] Zhong-Ping Jiang. Advanced feedback control of the chaotic Duffing equation. IEEE Transactions on Circuits and Systems. 2002, 49(2): 244-249
[81] Temple H. Fay. Duffing’s equation and nonlinear resonance. INT. J. MATH. EDUC. SCI. TECHNOL., 2003, 34(6):871-880
[82] Zhaosheng Feng. Monotonous property of non-oscillations of the damped Duffing,s equation. Chaos, Solitons & Fractals, 2006, 28: 463-471
[83] Qiyuan Zhou, Bingwen Liu. New result on almost periodic solutions for a class of nonlinear Duffing equtions with a deviating arqument. Applied Mathematics Letters, 2009, 22:6-11
[84] Qinsheng Bi. Chaos crisis in coupled Duffing's systems with initial phase difference. Physics Letters A, 2007, 369(5-6): 418-431
[85] Zhujun Jing, Jicai Huang, Jin Deng. Complex dynamics in three-well duffing system with two external forcings. Chaos, Solitons & Fractals, 2007, 33(3): 795-812
[86] Sihem A. Lazzouni, Samuel Bowong, F.M. Moukam Kakmeni, Brahim Cherki, Noureddine Ghouali. Chaos control using small-amplitude damping signals of the extended Duffing equation. Communications in Nonlinear Science and Numerical Simulation. 2007, 12(5): 804-813
[87] Long-Jye Sheu, Hsien-Keng Chen, Juhn-Horng Chen, Lap-Mou Tam. Chaotic dynamics of the fractionally damped Duffing equation. Chaos, Solitons & Fractals, 2007, 32(4): 1459-1468
[88] Bashir Ahmad, Badra S. Alghamdi. Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions. Computer Physics Communications, 2008, 179(6): 409-416
[89] Xin-Ge Liu, Mei-Lan Tang, Ralph R. Martin. Periodic solutions for a kind of Liénard equation. Journal of Computational and Applied Mathematics, 2008, 219(1): 263-275
[90] Vasile Marinca, Nicolae Heri?anu. Periodic solutions of Duffing equation with strong non- linearity. Chaos, Solitons & Fractals, 2008, 37(1): 144-149
[91] Fuxing Zhang, Ya Li. Existence and uniqueness of periodic solutions for a kind of duffing type p-Laplacian equation. Nonlinear Analysis: Real World Applications, 2008, 9(3): 985-989
[92] Y.Z.Chen,X.Y.Lin.A successive integration technique for the solution of the Duffing oscillator equation with damping and excitation. Nonlinear Analysis, 2009, 70: 3603-3608
[93] E.Tafo Wembe, R.Yamapi. Chaos synchronization of resistively coupled Duffing systems: Numerical and experimental investigations. Communications in Nonlinear Science and Numerical Simulation, 2009, 14: 1439-1453
[94] Fabao Gao, Shiping Lu, Wei Zhang. Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument. Nonlinear Analysis, 2009, 70: 3567-3574
[95] Liu Z H, Yuan R. Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response. Journal of Mathematics Analysis and Applications, 2004, 296: 521-537
[96] Yan X P,Chu Y D. Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system. Journal of Computational and Applied Mathematics, 2006, 196: 198-210
[97] Meng X Z, Han D, Song Y L. Stability and bifurcation in a non-Kolmogorov type prey-predator system with time delay. Mathematical and Computer Modelling, 2005, 41: 1445-1455
[98] Li Y L,Xiao D M. Bifurcations of a predator–prey system of Holling and Leslie types. Chaos, Solitons and Fractals, 2007, 34(2): 606-620
[99] Liao X, Wong K, Wu Z. Bifurcation analysis on a two-neuron system with distributed delays. Physica D, 2001, 149: 123-141
[100] Zhao H Y, Wang L. Hopf bifurcation in Cohen-Grossberg neural network with distributed delays. Nonlinear Analysis: Real World Applications, 2007, 8(1): 73-89
[101] Liao, X. F and Ran, J. H. Hopf bifurcation in love dynamical models with nonlinear couples and time delays, Chaos, solitons and Fractals., 2007, 31: 853-865
[102] Belair, J and Campbell, S. A., Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math., 1994, 54: 1402-1424
[103] Cooke K, Grossman Z. Discrete delay, distributed delay and stability switches. J Math AnalAppl, 1982, 86: 592-627
[104] R Seydel. Practical Bifurcation and Stability Analysis. Springer-Verlag, New York, Inc, 1994
[105] Hassard B D, Kazarinoff N D, Wan Y H. Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981
[106] Morita, M. Associative Memory with nonmonotone dynamics, Neural Networks., 1993, 6 : 115-126
[107] Liao XF, Li C, Zhou S. Hopf bifurcation and chaos, in macroeconomic models with policy lag. Chaos, solitons and Fractals., 2005, 25: 91-108
[108] Liao XF, Chen G. Hopf bifurcation and chaos analysis of Chen’s system with distributed delays. Chaos, solitons and Fractals., 2005, 25: 197-220
[109] Wei J, Ruan S. Stability and bifurcation in a neural network model with two delays. Physica D 1999, 130: 255-272
[110] Yoshizawa, S., Morita, M. and Amari, S Capacity of associative memory using a nonmono–tonic neuron model, Neural networks., 1993, 6 :167-176
[111]徐伟,都琳,许勇.非线性随机动力系统研究的若干进展.工程数学学报,2006:23(6):951-960
[112]龚光鲁.随机微分方程及其应用概要.北京:清华大学出版社,2008
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |