利用复合矩阵和对称群研究动力系统的稳定性
摘要
本学位论文主要利用复合矩阵和对称群研究了动力系统的稳定性,尤其是Hopf分支问题.复合矩阵和对称群是代数中两个重要概念,这两个概念在动力系统的稳定性研究中起重要作用.复合矩阵不仅是研究矩阵稳定性的很好的工具,也是建立从低维动力系统到高维动力系统联系的桥梁,在解决连续动力系统的全局稳定性、周期轨道的存在性等问题中独具魅力.对称群是研究对称性动力系统稳定性,尤其是Hopf分支的基石.
分支问题是动力系统中最重要的研究课题之一,本论文通过复合矩阵的谱性质给出了利用复合矩阵有效地判断矩阵Schur稳定的方法,进而可以判断离散动力系统的平衡解的渐近稳定性及Hopf分支的存在性.
时滞微分方程用于描述既依赖于当前状态也依赖于过去状态的发展系统.毫无疑问,时滞所引起的无限维空间性质的研究是一项很困难的任务,所以对时滞微分方程分支问题的研究既要用到微分方程理论,又要用到代数、泛函、拓扑等相关知识.虽然基于复合矩阵和对称群对研究时滞微分方程中的Hopf分支问题已有一些一般性的理论,如n维Bendixson准则、对称性泛函微分方程Hopf分支定理等.然而将这些一般性的结果应用到具体的动力系统模型中,却可能遇到其中以下困难:(I)计算系统的线性化矩阵的二阶可加复合矩阵和复合方程,目前只能计算3维、4维动力系统的复合方程; (II)特征方程通常是超越的而且是依赖于参数的,很难分析它的零点分布; (III)讨论线性化系统所产生的连续半群的无穷小生成元的广义特征空间; (IV)系统的对称性分析.本论文解决了上述部分问题,内容如下:
(1)给出了任意n阶矩阵的二阶可加复合矩阵的简单、直接算法.用此算法可以计算任意n维动力系统的复合方程.作为此算法的应用,再结合Muldowney等人的n维Bendixson准则和J. Wu的全局Hopf分支定理,讨论了一类具时滞的BAM神经网络模型的Hopf分支,得出该系统不存在以某一特殊常数为周期的非常数周期解,讨论了其Hopf分支周期解的全局存在性.
(2)给出了分块循环矩阵的特征值与特征向量,解决了所有以二面体群Dn为对称群的动力系统的广义特征空间的计算问题.作为此算法的应用,结合J. Wu的对称性时滞微分方程Hopf分支定理,论述了一类具时滞的n个BVP振子模型和一类具时滞的带耦合振子的神经元模型的稳定性和Hopf分支.讨论了系统零解的一般Hopf分支,给出了出现等变Hopf分支的充分条件.
(3)利用对称群及其表示论,研究了具体动力系统模型的对称性.通过刻画对称群的迷向子群的不动点子空间,描述了上述两类模型的Hopf分支的时空存在模式.
(4)借助空间分解,讨论了对称模型相应的超越方程的零点分布,从而给出系统平衡解渐近稳定的条件.
The focus of this dissertation is to study issues related to stability of dynamicsystems on the base of compound matrices and symmetric groups, especially the bi-furcation problem. Compound matrices and symmetric groups are not only two ofthe most important concepts in algebra, but also are widely applied in the research ofstability of dynamic systems. Compound matrices are the wonderful tools to investi-gate the stability of matrices, and are the bridge between low and high dimensionaldynamic systems, which has the unique charm in dealing with some problems aboutthe global stability, the existence of periodic orbits and so on. Symmetric groups arethe theoretic foundation for studying symmetric dynamic systems. Based on the grouptheory, M. Golubitsky et al. obtained the Hopf bifurcation theory for the symmetricordinary differential equation (ODE). And J. Wu extended it to the symmetric delayeddifferential equation (DDE).
The bifurcation problem is one of the best important subjects in dynamic sys-tems. Motivated by M. Li et al. who used compound matrices to judge the stability ofmatrices and the existence of Hopf bifurcations in continuous dynamic systems, weobtained the effective method to judge the Schur stability of matrices on the base ofspectral property of compound matrices, which can be used to judge the asymptoticalstability and the existence of Hopf bifurcation of discrete dynamic systems.
The DDE describes the evolution systems depending on both the present stateand the past state, which has wide applications in ecology, physics, chemistry, engi-neering, information science, economics and physical science and so on. Since it isa very difficult task to research the infinite dimensional nature caused by the delays,the deep investigation of the bifurcation problem in DDE needs not only the theoryof the classical differential equations, but also the knowledge of algebra, functional,topology one. Some general theorems are available about the Hopf bifurcation forDDE, for example, the n dimensional Bendixson theorem, the symmetric Hopf bifur-cation theorem for DDE and so on. However,applications of these general theoremsto concrete dynamic models often involve the following difficult tasks: (I) calculationof the second additive compound matrices, the case of 3×3 and 4×4 matrices is easily resolved; (II) distribution of zeros in characteristic equations which are usu-ally transcendental and depend on parameters; (III) discussion on certain generalizedeigenspaces of the infinitesimal generator of continuous semigroups for a linearizedsystem; (IV) analysis on the symmetry of systems. To resolve some of the aboveproblems, the following issues are organized:
(1) The simple and direct method to calculate the second additive compoundmatrix of any n×n matrix is obtained, which can be used to calculate the compoundequation of any n dimensional dynamic systems. Applying the calculation methodcoupled with n dimensional Bendixson theorem of Muldowney et al. and global Hopfbifurcation of J. Wu, a class of BAM neural network models with delays are discussed.The conclusion that there is no nonconstant periodic solutions with some special pe-riod is got. And global existence of periodic solutions are established.
(2) The eigenvalues and eigenvectors of circle block matrix are exhibited, whichsolve the problem on generalized eigenspaces of linearized systems for all Dn-symmetric systems. Applying it coupled with symmetric Hopf bifurcation theoremof J. Wu, the stability and Hopf problem of a class of n-coupled BVP oscillators mod-els with delays and a class of coupled neural oscillators network models with delaysare investigated. Common Hopf bifurcations occurring at the zero equilibrium as thedelay increases are exhibited. The equivariant Hopf bifurcations are obtained andanalyzed, and their spatio-temporal patterns are demonstrated.
(3) Using the presentation theory of symmetric groups, the symmetry of concretemodels discussed is investigated. Depicting the fixed point subspaces of isotropicsubgroups of symmetric groups, the corresponding periodic solutions patterns are de-scribed.
(4) By means of space decomposition, the distribution of zeros of the character-istic equations is subtly discussed. Hence, sufficient conditions are derived to ensurethat the zero solution of models are asymptotically stable.
分支问题是动力系统中最重要的研究课题之一,本论文通过复合矩阵的谱性质给出了利用复合矩阵有效地判断矩阵Schur稳定的方法,进而可以判断离散动力系统的平衡解的渐近稳定性及Hopf分支的存在性.
时滞微分方程用于描述既依赖于当前状态也依赖于过去状态的发展系统.毫无疑问,时滞所引起的无限维空间性质的研究是一项很困难的任务,所以对时滞微分方程分支问题的研究既要用到微分方程理论,又要用到代数、泛函、拓扑等相关知识.虽然基于复合矩阵和对称群对研究时滞微分方程中的Hopf分支问题已有一些一般性的理论,如n维Bendixson准则、对称性泛函微分方程Hopf分支定理等.然而将这些一般性的结果应用到具体的动力系统模型中,却可能遇到其中以下困难:(I)计算系统的线性化矩阵的二阶可加复合矩阵和复合方程,目前只能计算3维、4维动力系统的复合方程; (II)特征方程通常是超越的而且是依赖于参数的,很难分析它的零点分布; (III)讨论线性化系统所产生的连续半群的无穷小生成元的广义特征空间; (IV)系统的对称性分析.本论文解决了上述部分问题,内容如下:
(1)给出了任意n阶矩阵的二阶可加复合矩阵的简单、直接算法.用此算法可以计算任意n维动力系统的复合方程.作为此算法的应用,再结合Muldowney等人的n维Bendixson准则和J. Wu的全局Hopf分支定理,讨论了一类具时滞的BAM神经网络模型的Hopf分支,得出该系统不存在以某一特殊常数为周期的非常数周期解,讨论了其Hopf分支周期解的全局存在性.
(2)给出了分块循环矩阵的特征值与特征向量,解决了所有以二面体群Dn为对称群的动力系统的广义特征空间的计算问题.作为此算法的应用,结合J. Wu的对称性时滞微分方程Hopf分支定理,论述了一类具时滞的n个BVP振子模型和一类具时滞的带耦合振子的神经元模型的稳定性和Hopf分支.讨论了系统零解的一般Hopf分支,给出了出现等变Hopf分支的充分条件.
(3)利用对称群及其表示论,研究了具体动力系统模型的对称性.通过刻画对称群的迷向子群的不动点子空间,描述了上述两类模型的Hopf分支的时空存在模式.
(4)借助空间分解,讨论了对称模型相应的超越方程的零点分布,从而给出系统平衡解渐近稳定的条件.
The focus of this dissertation is to study issues related to stability of dynamicsystems on the base of compound matrices and symmetric groups, especially the bi-furcation problem. Compound matrices and symmetric groups are not only two ofthe most important concepts in algebra, but also are widely applied in the research ofstability of dynamic systems. Compound matrices are the wonderful tools to investi-gate the stability of matrices, and are the bridge between low and high dimensionaldynamic systems, which has the unique charm in dealing with some problems aboutthe global stability, the existence of periodic orbits and so on. Symmetric groups arethe theoretic foundation for studying symmetric dynamic systems. Based on the grouptheory, M. Golubitsky et al. obtained the Hopf bifurcation theory for the symmetricordinary differential equation (ODE). And J. Wu extended it to the symmetric delayeddifferential equation (DDE).
The bifurcation problem is one of the best important subjects in dynamic sys-tems. Motivated by M. Li et al. who used compound matrices to judge the stability ofmatrices and the existence of Hopf bifurcations in continuous dynamic systems, weobtained the effective method to judge the Schur stability of matrices on the base ofspectral property of compound matrices, which can be used to judge the asymptoticalstability and the existence of Hopf bifurcation of discrete dynamic systems.
The DDE describes the evolution systems depending on both the present stateand the past state, which has wide applications in ecology, physics, chemistry, engi-neering, information science, economics and physical science and so on. Since it isa very difficult task to research the infinite dimensional nature caused by the delays,the deep investigation of the bifurcation problem in DDE needs not only the theoryof the classical differential equations, but also the knowledge of algebra, functional,topology one. Some general theorems are available about the Hopf bifurcation forDDE, for example, the n dimensional Bendixson theorem, the symmetric Hopf bifur-cation theorem for DDE and so on. However,applications of these general theoremsto concrete dynamic models often involve the following difficult tasks: (I) calculationof the second additive compound matrices, the case of 3×3 and 4×4 matrices is easily resolved; (II) distribution of zeros in characteristic equations which are usu-ally transcendental and depend on parameters; (III) discussion on certain generalizedeigenspaces of the infinitesimal generator of continuous semigroups for a linearizedsystem; (IV) analysis on the symmetry of systems. To resolve some of the aboveproblems, the following issues are organized:
(1) The simple and direct method to calculate the second additive compoundmatrix of any n×n matrix is obtained, which can be used to calculate the compoundequation of any n dimensional dynamic systems. Applying the calculation methodcoupled with n dimensional Bendixson theorem of Muldowney et al. and global Hopfbifurcation of J. Wu, a class of BAM neural network models with delays are discussed.The conclusion that there is no nonconstant periodic solutions with some special pe-riod is got. And global existence of periodic solutions are established.
(2) The eigenvalues and eigenvectors of circle block matrix are exhibited, whichsolve the problem on generalized eigenspaces of linearized systems for all Dn-symmetric systems. Applying it coupled with symmetric Hopf bifurcation theoremof J. Wu, the stability and Hopf problem of a class of n-coupled BVP oscillators mod-els with delays and a class of coupled neural oscillators network models with delaysare investigated. Common Hopf bifurcations occurring at the zero equilibrium as thedelay increases are exhibited. The equivariant Hopf bifurcations are obtained andanalyzed, and their spatio-temporal patterns are demonstrated.
(3) Using the presentation theory of symmetric groups, the symmetry of concretemodels discussed is investigated. Depicting the fixed point subspaces of isotropicsubgroups of symmetric groups, the corresponding periodic solutions patterns are de-scribed.
(4) By means of space decomposition, the distribution of zeros of the character-istic equations is subtly discussed. Hence, sufficient conditions are derived to ensurethat the zero solution of models are asymptotically stable.
引文
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94 R. A. Horn, C. R. Johnson. Matrix Analysis. Cambridge University Press. 2005:12-65
95 J. Hopfield. Neurons with Graded Response Have Collective ComputationalProperties like those of Two-state Neurons. Proc. Nat. Acad. Sci. U.S.A. 1984,81: 3088-3092
96 C. M. Marcus, R. M. Westervel. Stability of Analog Neural Network with Delay.Phys. Rev. A 1989, 39: 347-359
97 J. Belair. Stability in a Model of a Delayed Neural Network. J. Dyn. Diff. Equs.1993, 5: 607-623
98 J. Belair, S. A. Campbell, P. VandenDriesche. Frustration, Stability and Delay-Induced Oscillations in a Neural Network Model. SIAM J. Appl. Math. 1996,56: 245-255
99 K. Gopalsamy, I. Leung. Delay Induced Periodicity in a Neural Network ofExcitation and Inhibition. Phys. D 1996, 89: 395-426
100 J. Wu, T. Faria, Y. S. Huang. Synchronization and Stable Phase-Locking inNetwork of Neurons with Memory. Math. Comp. Mode. 1999, 30: 117-138
101 B. Kosko. Adaptive bi-directional Associative Memories. Appl Opt. 1987,26(23): 4947-4960
102 B. Kosko. Bi-directional Associative Memories. IEEE Trans Syst Man Cyber.1988 18(1): 49-60
103 B. Kosko. Neural Networks and Fuzzy Systems-a Dynamical System ApproachMachine Intelligence. Englewood Cli.s (NJ), Prentice-Hall. 1992: 2-17
104 K. Gopalsamy, X. He. Delay-independent Stability in bi-directional AssociativeMemory Networks, IEEE Trans. Neural Network. 1994, 5: 998-1002
105 X. Huang, J. Cao. LMI-based Approach for Delay-dependent Exponential Sta-bility Analysis of BAM Neural Networks. Chaos Soliton Fract. 2005, 24(3):885-898
106 B. D. Zheng, Y. Zhang, C. R. Zhang. Stability and Bifurcation of a DiscreteBAM Neural Network Model with Delays. Chaos Soliton Fract. 2008, 36: 612-
616107 C. R. Zhang, B. D. Zheng, L. C. Wang. Multiple Hopf Bifurcation of SymmetricBAM Neural Network with Delay. Appl. Math. Lett. 2009, 22: 616-622
108 J. H. Park. A Novel Criterion for Global Asymptotic Stability of BAM NeuralNetworks with Time Delays. Chaos Soliton Fract. 2006, 29(2): 446-453
109 Y. Song, M. Han, J. Wei. Stability and Hopf Bifurcation Analysis on a Simpli-fied BAM Neural Network with Delays, Phys. D 2005, 200: 185-204
110 J. Wei, M. Velarde. Bifurcation Analysis and Existence of Periodic Solutions ina Simple Neural Network with Delays. Chaos. 2004, 14: 940-953
111 S. Ruan, J. Wei. On the Zeros of a Third Degree Exponential Polynomial withApplication to a Delayed Model for the Control of Testosterone Secretion.SIAM Math Appl Medic Biol. 2001, 18: 41-52
112 S. A. Campabell, S. Ruan, J. Wei. Qualitive Analysis of a Neural NetworkModel with Multlple Time Deleys.Internat. J. Bifur. Chaos. 1999, 9: 1585-1595.
113 A. L. Hodgkin, A. F. Huxley. Currents Carried by Sodium and Potassiumionsthrough the Membrane of the Giant Axon of Loligo. J. Physiol. 1952, 116: 44972114 A. L. Hodgkin, A. F. Huxley. The Components of Membrane Conductance inthe Giant Axon of Loligo. J. Physiol. 1952, 116: 473-496
115 A. L. Hodgkin, A. F. Huxley. The Components of Membrane Conductance inthe Giant Axon of Loligo. J. Physiol. 1952, 116: 497-506
116 A. L. Hodgkin, A. F. Huxley. The Components of Membrane Conductance inthe Giant Axon of Loligo. J. Physiol. 1952, 116: 506-544
117 R. FitzHugh. Impulses and Physiological States in the Oretical Models of NerveMembrane, J. Biophys. 1961, 1: 445-466
118 J. Nagumo, S. Arimoto, S. Yoshizawa. An Active Pulse Transmission Line Sim-ulating Nerveaxon. Proc. IRE. 1962, 50: 2061-2070
119 T. Ueta, H. Kawakami. Chaos in Asymmetrically Coupled BVP Oscillators.IEEE Circuits Systs. 2002, 2:544-547
120 A. N. Bautin. Qualitative Investigation of a Particular Nonlinear System. J.Appl. Math. Mech. 1975, 39:606-615
121 H. Kitajima, Y. Katsuta. Bifurcations of Periodic Solutions in a Coupled Oscil-lator with Voltage Ports. IEICE Trans Fund. 1991, 11:476-412
122 T. Ueta, H. Kawakami. Chaos in Cross-coupled BVP oscillators. IEEE CircuitsSysts. 2003, 3:61-71
123 T. Ueta, H. Miyazaki, T. Kousaka, H. Kawakami. Bifurcation and Chaos inCoupled BVP Oscillators. Int. J. Bifurcation Chaos. 2004, 14(4): 1305-1324
124 H. Zhang, A. V. Holdem, Chaotic Meander of Spiral Waves in theFitzHugh–Nagumo System. Chaos Soliton Fract. 1995, 1: 303-334
125 B. Braaksma, J. Grasman. Critical Dynamic of the Bonhoeffer–VanderpolEquation and its Chaotic Response to Periodic Stimulation. Phys. D 1993, 68:265-280
126 T. Mashima, H. Kawakami. Bifurcation of Equilibrium Points and Synchro-nized Periodic Solution in Four Coupled Oscillators, 1997 International Sym-posium on nonlinear theory and its application, 1997, 29: 141-144
127王东生,曹磊.混沌分形及其应用.中国科学技术大学出版社. 1995:34-55
128刘秉正.非线性动力学与混沌基础.东北师范大学出版社. 1994: 29-45
129方锦清.非线性系统中混沌的控制与同步及其应用前景.物理学进展.1996, 16(1): 1-3
130 X. S. Yang, Q. Li. Horseshoe Chaos in Cellular Neural Networks. Int. J. Bifur-cation Chaos. 2006, 16(1): 147-151
131 R. Singh, V. M. Maru, P. S. Moharir. Complex Chaotic Systems and EmergentPhenomena. J. Nonlin. Sci. 1998, 8: 235-259
132 W. Lu, T. Chen. Synchronization Analysis of Linearly Coupled Systems De-scribed by Differential Equations with a Coupling Delay. Phys. D 2006, 19:118-134
133 H. Gao, J. Lam, G. Chen. New Criteria for Synchronization Stability of GeneralComplex Dynamical Networks with Coupling Delays. Phys. Lett. A 2006, 360:263-273
134 D. Aronson, G. B. Ermentrout, N. Kopell. Amplitude Response of CoupledOscillators. Phys. D 1990, 41: 403-449
135 M. Kawato, M. Sokabe, R. Suzuki. Synergism and Antagonism of NeuronsCaused by an Electrical Synapse. Biol. Cybern. 1979, 34: 81-89
136 D. Hansel, G. Mato, C. Meunier. Phase Dynamics for Weakly CoupledHodgkin-Huxley Neurons. Europhys. Lett. 1993, 23: 367-372
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93 B. D. Zheng, L. J. Liang, C. R. Zhang. Extended Jury Criterion. Sci. China. SerA. 2010, 53: 1133-1150
94 R. A. Horn, C. R. Johnson. Matrix Analysis. Cambridge University Press. 2005:12-65
95 J. Hopfield. Neurons with Graded Response Have Collective ComputationalProperties like those of Two-state Neurons. Proc. Nat. Acad. Sci. U.S.A. 1984,81: 3088-3092
96 C. M. Marcus, R. M. Westervel. Stability of Analog Neural Network with Delay.Phys. Rev. A 1989, 39: 347-359
97 J. Belair. Stability in a Model of a Delayed Neural Network. J. Dyn. Diff. Equs.1993, 5: 607-623
98 J. Belair, S. A. Campbell, P. VandenDriesche. Frustration, Stability and Delay-Induced Oscillations in a Neural Network Model. SIAM J. Appl. Math. 1996,56: 245-255
99 K. Gopalsamy, I. Leung. Delay Induced Periodicity in a Neural Network ofExcitation and Inhibition. Phys. D 1996, 89: 395-426
100 J. Wu, T. Faria, Y. S. Huang. Synchronization and Stable Phase-Locking inNetwork of Neurons with Memory. Math. Comp. Mode. 1999, 30: 117-138
101 B. Kosko. Adaptive bi-directional Associative Memories. Appl Opt. 1987,26(23): 4947-4960
102 B. Kosko. Bi-directional Associative Memories. IEEE Trans Syst Man Cyber.1988 18(1): 49-60
103 B. Kosko. Neural Networks and Fuzzy Systems-a Dynamical System ApproachMachine Intelligence. Englewood Cli.s (NJ), Prentice-Hall. 1992: 2-17
104 K. Gopalsamy, X. He. Delay-independent Stability in bi-directional AssociativeMemory Networks, IEEE Trans. Neural Network. 1994, 5: 998-1002
105 X. Huang, J. Cao. LMI-based Approach for Delay-dependent Exponential Sta-bility Analysis of BAM Neural Networks. Chaos Soliton Fract. 2005, 24(3):885-898
106 B. D. Zheng, Y. Zhang, C. R. Zhang. Stability and Bifurcation of a DiscreteBAM Neural Network Model with Delays. Chaos Soliton Fract. 2008, 36: 612-
616107 C. R. Zhang, B. D. Zheng, L. C. Wang. Multiple Hopf Bifurcation of SymmetricBAM Neural Network with Delay. Appl. Math. Lett. 2009, 22: 616-622
108 J. H. Park. A Novel Criterion for Global Asymptotic Stability of BAM NeuralNetworks with Time Delays. Chaos Soliton Fract. 2006, 29(2): 446-453
109 Y. Song, M. Han, J. Wei. Stability and Hopf Bifurcation Analysis on a Simpli-fied BAM Neural Network with Delays, Phys. D 2005, 200: 185-204
110 J. Wei, M. Velarde. Bifurcation Analysis and Existence of Periodic Solutions ina Simple Neural Network with Delays. Chaos. 2004, 14: 940-953
111 S. Ruan, J. Wei. On the Zeros of a Third Degree Exponential Polynomial withApplication to a Delayed Model for the Control of Testosterone Secretion.SIAM Math Appl Medic Biol. 2001, 18: 41-52
112 S. A. Campabell, S. Ruan, J. Wei. Qualitive Analysis of a Neural NetworkModel with Multlple Time Deleys.Internat. J. Bifur. Chaos. 1999, 9: 1585-1595.
113 A. L. Hodgkin, A. F. Huxley. Currents Carried by Sodium and Potassiumionsthrough the Membrane of the Giant Axon of Loligo. J. Physiol. 1952, 116: 44972114 A. L. Hodgkin, A. F. Huxley. The Components of Membrane Conductance inthe Giant Axon of Loligo. J. Physiol. 1952, 116: 473-496
115 A. L. Hodgkin, A. F. Huxley. The Components of Membrane Conductance inthe Giant Axon of Loligo. J. Physiol. 1952, 116: 497-506
116 A. L. Hodgkin, A. F. Huxley. The Components of Membrane Conductance inthe Giant Axon of Loligo. J. Physiol. 1952, 116: 506-544
117 R. FitzHugh. Impulses and Physiological States in the Oretical Models of NerveMembrane, J. Biophys. 1961, 1: 445-466
118 J. Nagumo, S. Arimoto, S. Yoshizawa. An Active Pulse Transmission Line Sim-ulating Nerveaxon. Proc. IRE. 1962, 50: 2061-2070
119 T. Ueta, H. Kawakami. Chaos in Asymmetrically Coupled BVP Oscillators.IEEE Circuits Systs. 2002, 2:544-547
120 A. N. Bautin. Qualitative Investigation of a Particular Nonlinear System. J.Appl. Math. Mech. 1975, 39:606-615
121 H. Kitajima, Y. Katsuta. Bifurcations of Periodic Solutions in a Coupled Oscil-lator with Voltage Ports. IEICE Trans Fund. 1991, 11:476-412
122 T. Ueta, H. Kawakami. Chaos in Cross-coupled BVP oscillators. IEEE CircuitsSysts. 2003, 3:61-71
123 T. Ueta, H. Miyazaki, T. Kousaka, H. Kawakami. Bifurcation and Chaos inCoupled BVP Oscillators. Int. J. Bifurcation Chaos. 2004, 14(4): 1305-1324
124 H. Zhang, A. V. Holdem, Chaotic Meander of Spiral Waves in theFitzHugh–Nagumo System. Chaos Soliton Fract. 1995, 1: 303-334
125 B. Braaksma, J. Grasman. Critical Dynamic of the Bonhoeffer–VanderpolEquation and its Chaotic Response to Periodic Stimulation. Phys. D 1993, 68:265-280
126 T. Mashima, H. Kawakami. Bifurcation of Equilibrium Points and Synchro-nized Periodic Solution in Four Coupled Oscillators, 1997 International Sym-posium on nonlinear theory and its application, 1997, 29: 141-144
127王东生,曹磊.混沌分形及其应用.中国科学技术大学出版社. 1995:34-55
128刘秉正.非线性动力学与混沌基础.东北师范大学出版社. 1994: 29-45
129方锦清.非线性系统中混沌的控制与同步及其应用前景.物理学进展.1996, 16(1): 1-3
130 X. S. Yang, Q. Li. Horseshoe Chaos in Cellular Neural Networks. Int. J. Bifur-cation Chaos. 2006, 16(1): 147-151
131 R. Singh, V. M. Maru, P. S. Moharir. Complex Chaotic Systems and EmergentPhenomena. J. Nonlin. Sci. 1998, 8: 235-259
132 W. Lu, T. Chen. Synchronization Analysis of Linearly Coupled Systems De-scribed by Differential Equations with a Coupling Delay. Phys. D 2006, 19:118-134
133 H. Gao, J. Lam, G. Chen. New Criteria for Synchronization Stability of GeneralComplex Dynamical Networks with Coupling Delays. Phys. Lett. A 2006, 360:263-273
134 D. Aronson, G. B. Ermentrout, N. Kopell. Amplitude Response of CoupledOscillators. Phys. D 1990, 41: 403-449
135 M. Kawato, M. Sokabe, R. Suzuki. Synergism and Antagonism of NeuronsCaused by an Electrical Synapse. Biol. Cybern. 1979, 34: 81-89
136 D. Hansel, G. Mato, C. Meunier. Phase Dynamics for Weakly CoupledHodgkin-Huxley Neurons. Europhys. Lett. 1993, 23: 367-372
137 H. R. Wilson, J. D. Cowan. Excitatory and Inhibitory Interactions in LocalizedPopulations of Model Neurons. J. Biophys. 1972, 12: 1-24