带自反馈环状时滞细胞神经网络的分支与稳定性分析
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摘要
本硕士学位论文利用中心流定理和规范型理论,对带自反馈的环状时滞细胞神经网络的稳定性和Hopf分支进行了分析,其中中心流定理起降低维数的作用,而规范型理论则是将所研究的问题尽可能在等价意义下从形式上予以简化,全文共分三章.
第一章简单介绍该问题研究的背景和发展的现状,以及它的广泛应用和研究工作的意义.
第二章讨论单向环状带自反馈的时滞细胞神经网络模型:的稳定性和Hopf分支,得到了一些相关的结果,并就这些结果做了数值模拟实验进行验证.
第三章讨论双向环状带自反馈的时滞细胞神经网络模型:的稳定性和Hopf分支,得到了一些相关的结果,并就这些结果做了数值模拟实验进行验证.
在第二章和第三章中,每章由三部分组成,第一部分为引言和模型部分;第二部分主要分析平凡解的线性稳定性,给出几个判断网络稳定性和不稳定的充分条件,讨论分支的存在性,并以自反馈时滞作为分支参数,给出分支值的计算公式;第三部分利用中心流定理和规范型理论来确定分支方向、分支周期解的稳定性等性质,给出例子以及数值模拟实验验证本文主要结果.
This thesis analyzes the Hopf bifurcation and stability of annular delayed neural networks with self-connection, by applying the center manifold theorem and the normal form theory, in which the center manifold theorem is applied to reducing dimensions, while the normal form theory for simplifying the form of our models.
In Chapter 1, we briefly introduce the background and the recent development of the issue we researched , in which its applying and the purpose of the investigation are denoted, too.
In Chapter 2, we discuss the stability and bifurcation of an annular delayed neural network with self-connection:We obtaine some relevant results, and then make the numerical simulation experimentsto verify them.
In Chapter 3, we investigate the stability and bifurcation for a class of threeneuronnetwork with two delay and self-feedback:We obtaine some relevant results, and then make the numerical simulation experimentsto verify them.
In Chapters 2 and 3, each chapter consists of three parts, the first part is for the introduction and model ; the second part mainly analyzes the linear stability of trivial solution, gives some sufficient conditionsstability and instability ,discusses the existence of a branch and gives the formula for calculating the value of branch through regarding the self-feedback delay as the parameter of branch ; the third part uses the center manifold theorem and the normal form theory to determine the branch direction and stability of periodic solutions nature, and gives examples of numerical simulation experiments to verify the main results of this paper.
第一章简单介绍该问题研究的背景和发展的现状,以及它的广泛应用和研究工作的意义.
第二章讨论单向环状带自反馈的时滞细胞神经网络模型:的稳定性和Hopf分支,得到了一些相关的结果,并就这些结果做了数值模拟实验进行验证.
第三章讨论双向环状带自反馈的时滞细胞神经网络模型:的稳定性和Hopf分支,得到了一些相关的结果,并就这些结果做了数值模拟实验进行验证.
在第二章和第三章中,每章由三部分组成,第一部分为引言和模型部分;第二部分主要分析平凡解的线性稳定性,给出几个判断网络稳定性和不稳定的充分条件,讨论分支的存在性,并以自反馈时滞作为分支参数,给出分支值的计算公式;第三部分利用中心流定理和规范型理论来确定分支方向、分支周期解的稳定性等性质,给出例子以及数值模拟实验验证本文主要结果.
This thesis analyzes the Hopf bifurcation and stability of annular delayed neural networks with self-connection, by applying the center manifold theorem and the normal form theory, in which the center manifold theorem is applied to reducing dimensions, while the normal form theory for simplifying the form of our models.
In Chapter 1, we briefly introduce the background and the recent development of the issue we researched , in which its applying and the purpose of the investigation are denoted, too.
In Chapter 2, we discuss the stability and bifurcation of an annular delayed neural network with self-connection:We obtaine some relevant results, and then make the numerical simulation experimentsto verify them.
In Chapter 3, we investigate the stability and bifurcation for a class of threeneuronnetwork with two delay and self-feedback:We obtaine some relevant results, and then make the numerical simulation experimentsto verify them.
In Chapters 2 and 3, each chapter consists of three parts, the first part is for the introduction and model ; the second part mainly analyzes the linear stability of trivial solution, gives some sufficient conditionsstability and instability ,discusses the existence of a branch and gives the formula for calculating the value of branch through regarding the self-feedback delay as the parameter of branch ; the third part uses the center manifold theorem and the normal form theory to determine the branch direction and stability of periodic solutions nature, and gives examples of numerical simulation experiments to verify the main results of this paper.
引文
[1] Ruan, s. & J. Wei. Periodic solutions of planar systems with two delays[J].Proceedings of the Royal Society of Edinburgh A, 1999, (129):1017-1032.
[2] Chen, Y. & Y. Huang, J. Wu. Desynchronization of large scale delayed neural networks[J].Proc. Amer. Math. Soc, 2000, 128(8): 2365-2371.
[3] Chen, Y. Existence and unstable sets of oscillating periodic orbits for delayed excitatory networks of two neurons[J].Diff. Eq. Dynam. Syst., 2001, (9):169-185.
[4] Chen, Y. & J. Wu, T. Krisztin. Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system[J].J. Diff. Equns., 2000, (163):130-173.
[5] Chen, Y. & J. Wu. The asymptotic shapes of periodic solutions of a singular delay differential system[J]. J. Diff. Equns., 2001, (169):614-632.
[6] Hu, J. S.& J. Wang. Global stability of a class of disrete-time recurrent neural networks[J].IEEE Trans. Circuits Syst. I, 2002,(49):1104-1116.
[7] Huang, L. & J. Wu. Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation[J].SIAM J. Math.Anal, 2003,34(4):836-860.
[8] Liao, X. & K. Wong, Z. Wu. Bifurcation analysis on a two-neuron system with distributed delay[J].Physica D, 2001, (149):123-141.
[9] Lin, L. & M. M. Gupta. Globally asymptotical stability of discrete time analogy neural networks[J].IEEE Trans. neural Networks, 1996, (7):1024-1031.
[10] Van den Driessche, P. & J. Wu, X. Zou. Stabilization role of inhibitory self-connections in a delayed neural network[J].Physica D, 2001, (150):84-90.
[11] Erbe, L. H. & W. Krawcewicz, K. Geba, J. Wu. S~l-degree and global Hopf bifurcation theory of functional differential equations[J].J. Diff. Equns., 1992, (98):227-298.
[12] Wu, J. Theory and Applications of Partial Functional differential Equations[M].New York: Springer, 1996, (119).
[13] Hopfield, J. Neurons with graded response have collective computational properties like those of two-state neurons[J].Proc. Nat. Acad. Sci. USA. 1984, (82):3088-3092.
[14] Cohen, M. & S. Grossberg. Absolute stability and global pattern formation and parallel memory storage by competitive neural networks[J]. IEEE Trans, Syst, Man Cybern. 1983, 13(5):815-826.
[15] Chen, Y. & J. Wu. Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network[J].Physica D, 1999, (134):185-199.
[16] Chen, Y. & J. Wu. Existence and attraction of a phase-locked oscillation in a delayed network of two neurons[J].Diff. Integr. Eqs., 2001, (14):1181-1236.
[17] Faria, T. On a planar system modeling a neuron nwtwork with memory[J]. J.Diff. Eqs., 2000 (168):129-149.
[18] Wei, J. & S. Ruan. Stability and bifurcation in a neural network model with two delays[J].Physica D, 1999, (130):255-272.
[19] Wei, J. & M. Velarde. Bifurcation analysis and existence of periodic solutions in a simple neural network with delay[J]. Chaos, 2004, (14):940-953.
[20] Li, X. & J. Wei. Stability and bifurcation analysis on a delayed neural network model[J].Int Journal of Bifurcation and Chaos, 2005, 15(9):2883-2893.
[21] Balde, P. & A. Atiya. How delays affect neural dynamics and learning[J].IEEE Tran. NN. 1994, (5):612-621.
[22] Campbell, S. A. Stability and bifurcation of a simple neural network with multiple time delays[J].Fields Institute Common.Series, 1999, (21):65-79.
[23] Alexander, J. C. Bifurcation of zeros of parameterized functions[J].J. Funct.Anal., 1978, (29):37-53.
[24] Alexander, J. C. & G. Auchmuty. Global bifurcations of phase-locked oscillators[J].Arch. Rational Mech. Anal., 1986, 93:253-270.
[25] Golubitsky, M. & I. Stewart , D. G. Schaeffer. Singularities and Groups in Bifurcation Theory[M].Springer-Verlag, New York, 1988.
[26] Albertini, F. & D. D'Alessandro. Further conditions on the stability of continuous time systems with saturation[J].IEEE Trans. Circuits Syst. I, 2000, (47):723-729.
[27] Cao, J. D. & J. L. Hang, J. Lam. Exponential stability of high order bidirectional associatine memory neural networks with time delays[J].Physica D, 2004, 199(3-4):425-436.
[28] Cao, J. & Q. Tao. Estimation on domain of attraction and convergence rate of Hopfield continuous feedback neural networks[J].J. Comput. Syst. Sci., 2001, (62):528-534.
[29] Pakdaman, K. & C. P. Malta, C. Grotta-R, agazzo, O. Arino . Vibert Transient oscillations in continuous-time excitatory ring neural networks with delay[J].Physical Review E, 1997, (55):3234-3247.
[30] Belair, J. & S. A. Campbell, P. van den Driessche. Frustration, stability, and delay-induced oscillations in a neural network model[J].SIAM J. Appl. Math., 1996, (56):245-255.
[31] Driessche, P. V. & X. Zou. Global attractivety in delayed Hopfield neural network models[J].SIAM J. Appl. Math., 1998, (58):1878-1890.
[32] Gopalsamy, K. & I. Leung. Delay induced periodicity in a neural network of excitation and inhibition[J].Physica D, 1996, (89):395-426.
[33] Huang, L. & J. Wu, Dynamics of inhibitory artificial neural networks with threshold nonlinearity[J].fields Ins. Commun., 2001, (29):235-243.
[34] Marcus, C. M. & R. M. Westervelt. Stability of analog neural nwtworks with delay[J].Phys. Rev. A, 1989, (39):347-359.
[35] Wu, J. Introduction to Neural Dynamics and Signal Transmission Delay[M]. Walter de Gruyter, Berlin, 2001.
[36] Hassard, B. D. & N. D. Kazarinoff, Y. H. Wan. Theory and Applications of Hopf Bifurcation[M].Cambridge University Press, Cambridge, 1981.
[2] Chen, Y. & Y. Huang, J. Wu. Desynchronization of large scale delayed neural networks[J].Proc. Amer. Math. Soc, 2000, 128(8): 2365-2371.
[3] Chen, Y. Existence and unstable sets of oscillating periodic orbits for delayed excitatory networks of two neurons[J].Diff. Eq. Dynam. Syst., 2001, (9):169-185.
[4] Chen, Y. & J. Wu, T. Krisztin. Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system[J].J. Diff. Equns., 2000, (163):130-173.
[5] Chen, Y. & J. Wu. The asymptotic shapes of periodic solutions of a singular delay differential system[J]. J. Diff. Equns., 2001, (169):614-632.
[6] Hu, J. S.& J. Wang. Global stability of a class of disrete-time recurrent neural networks[J].IEEE Trans. Circuits Syst. I, 2002,(49):1104-1116.
[7] Huang, L. & J. Wu. Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation[J].SIAM J. Math.Anal, 2003,34(4):836-860.
[8] Liao, X. & K. Wong, Z. Wu. Bifurcation analysis on a two-neuron system with distributed delay[J].Physica D, 2001, (149):123-141.
[9] Lin, L. & M. M. Gupta. Globally asymptotical stability of discrete time analogy neural networks[J].IEEE Trans. neural Networks, 1996, (7):1024-1031.
[10] Van den Driessche, P. & J. Wu, X. Zou. Stabilization role of inhibitory self-connections in a delayed neural network[J].Physica D, 2001, (150):84-90.
[11] Erbe, L. H. & W. Krawcewicz, K. Geba, J. Wu. S~l-degree and global Hopf bifurcation theory of functional differential equations[J].J. Diff. Equns., 1992, (98):227-298.
[12] Wu, J. Theory and Applications of Partial Functional differential Equations[M].New York: Springer, 1996, (119).
[13] Hopfield, J. Neurons with graded response have collective computational properties like those of two-state neurons[J].Proc. Nat. Acad. Sci. USA. 1984, (82):3088-3092.
[14] Cohen, M. & S. Grossberg. Absolute stability and global pattern formation and parallel memory storage by competitive neural networks[J]. IEEE Trans, Syst, Man Cybern. 1983, 13(5):815-826.
[15] Chen, Y. & J. Wu. Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network[J].Physica D, 1999, (134):185-199.
[16] Chen, Y. & J. Wu. Existence and attraction of a phase-locked oscillation in a delayed network of two neurons[J].Diff. Integr. Eqs., 2001, (14):1181-1236.
[17] Faria, T. On a planar system modeling a neuron nwtwork with memory[J]. J.Diff. Eqs., 2000 (168):129-149.
[18] Wei, J. & S. Ruan. Stability and bifurcation in a neural network model with two delays[J].Physica D, 1999, (130):255-272.
[19] Wei, J. & M. Velarde. Bifurcation analysis and existence of periodic solutions in a simple neural network with delay[J]. Chaos, 2004, (14):940-953.
[20] Li, X. & J. Wei. Stability and bifurcation analysis on a delayed neural network model[J].Int Journal of Bifurcation and Chaos, 2005, 15(9):2883-2893.
[21] Balde, P. & A. Atiya. How delays affect neural dynamics and learning[J].IEEE Tran. NN. 1994, (5):612-621.
[22] Campbell, S. A. Stability and bifurcation of a simple neural network with multiple time delays[J].Fields Institute Common.Series, 1999, (21):65-79.
[23] Alexander, J. C. Bifurcation of zeros of parameterized functions[J].J. Funct.Anal., 1978, (29):37-53.
[24] Alexander, J. C. & G. Auchmuty. Global bifurcations of phase-locked oscillators[J].Arch. Rational Mech. Anal., 1986, 93:253-270.
[25] Golubitsky, M. & I. Stewart , D. G. Schaeffer. Singularities and Groups in Bifurcation Theory[M].Springer-Verlag, New York, 1988.
[26] Albertini, F. & D. D'Alessandro. Further conditions on the stability of continuous time systems with saturation[J].IEEE Trans. Circuits Syst. I, 2000, (47):723-729.
[27] Cao, J. D. & J. L. Hang, J. Lam. Exponential stability of high order bidirectional associatine memory neural networks with time delays[J].Physica D, 2004, 199(3-4):425-436.
[28] Cao, J. & Q. Tao. Estimation on domain of attraction and convergence rate of Hopfield continuous feedback neural networks[J].J. Comput. Syst. Sci., 2001, (62):528-534.
[29] Pakdaman, K. & C. P. Malta, C. Grotta-R, agazzo, O. Arino . Vibert Transient oscillations in continuous-time excitatory ring neural networks with delay[J].Physical Review E, 1997, (55):3234-3247.
[30] Belair, J. & S. A. Campbell, P. van den Driessche. Frustration, stability, and delay-induced oscillations in a neural network model[J].SIAM J. Appl. Math., 1996, (56):245-255.
[31] Driessche, P. V. & X. Zou. Global attractivety in delayed Hopfield neural network models[J].SIAM J. Appl. Math., 1998, (58):1878-1890.
[32] Gopalsamy, K. & I. Leung. Delay induced periodicity in a neural network of excitation and inhibition[J].Physica D, 1996, (89):395-426.
[33] Huang, L. & J. Wu, Dynamics of inhibitory artificial neural networks with threshold nonlinearity[J].fields Ins. Commun., 2001, (29):235-243.
[34] Marcus, C. M. & R. M. Westervelt. Stability of analog neural nwtworks with delay[J].Phys. Rev. A, 1989, (39):347-359.
[35] Wu, J. Introduction to Neural Dynamics and Signal Transmission Delay[M]. Walter de Gruyter, Berlin, 2001.
[36] Hassard, B. D. & N. D. Kazarinoff, Y. H. Wan. Theory and Applications of Hopf Bifurcation[M].Cambridge University Press, Cambridge, 1981.
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