 |
|
|
|
以作者查詢圖書館館藏 、以作者查詢臺灣博碩士 、以作者查詢全國書目 、勘誤回報 、線上人數：14 、訪客IP：18.118.226.105
姓名 李俊憲(Chun-Hsien Li) 查詢紙本館藏 畢業系所 數學系 論文名稱 非線性耦合動力網路的同步現象分析
(Analysis of synchronization in nonlinearly coupled dynamical networks)相關論文 檔案 [Endnote RIS 格式]
[相關文章]
[文章引用]
[完整記錄]
[館藏目錄]
[檢視]
- 本電子論文使用權限為同意立即開放。
- 已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
- 請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。
摘要(中) 在相互影響的大尺度複雜網路系統裡最簡單且著名的集體動力行為就是同步現象。本文主要的目的為探究非線性耦合動力網路中在有或無時間遲滯影響下的全局指數型同步化機制。在沒有時間遲滯影響下，我們引用Lyapunov函數結合三種相當不同的技巧取得數個能確保非線性耦合動力網路全局指數型同步的準則。在第一種方法中，我們利用線性矩陣不等式的技巧來研究內含合作-競爭型外耦合矩陣的非對稱非線性耦合動力網路同步現象。第二種方法使用某些圖形理論的技巧，我們改進了Belykh等人針對對稱線性耦合動力網路的同步現象分析而發展出所謂的連結圖形穩定法，使其適用於非對稱非線性耦合動力網路。在第三種方法中，我們提出一個以特徵值為基礎的架構，探討隨時間變化的非線性耦合動力網路的同步現象，並進一步以這個方法為基礎探索網路拓樸和網路同步性之間的關係，透過一系列詳盡的數值模擬計算，我們比較了三類不同的網路群在某些相同性質設定下的同步性差異因素。另一方面，針對離散型或連續型時間遲滯影響下的非線性耦合動力網路，我們使用Lyapunov泛函結合線性矩陣不等式技術推導出一個使動力網路能全局指數型同步的準則，此同步化準則不僅與時間遲滯大小無關且不受時間遲滯類型的影響。與現存所知文獻的結果相比較，本文主要的優勢在於耦合函數可以是線性或非線性的、狀態變數的分量可以是完全地連結或部分地連結、外耦合矩陣可以是對稱或非對稱的。我們同時提出數個數值模擬實例來驗證上述理論分析的正確性，其中包括了著名的Chua耦合電路網、FitzHugh-Nagumo耦合神經元網路與Hindmarsh-Rose耦合神經元網路；文中也特別記述了貓的腦皮層和獼猴的視覺皮層兩個真實神經元網路的數值實驗結果。
摘要(英) The simplest and most prominent collective behavior in large-scale complex networks of interacting systems is their synchronization. The purpose of this thesis is to investigate the mechanism for global exponential synchronization in nonlinearly coupled dynamical networks without or with coupling time delays. In the absence of coupling time delays, we apply the Lyapunov function method combined with three quite different approaches to derive several criteria that ensure the nonlinearly coupled dynamical networks to be globally exponentially synchronized. First, we study the synchronization in nonlinearly coupled dynamical networks with an asymmetrically cooperative-competitive outer-coupling matrix by utilizing the linear matrix inequality techniques. Secondly, employing some graph theory techniques, we improve the so-called connection graph stability method for the synchronization analysis, that was originally developed by Belykh et al. for symmetrically linear coupled dynamical systems, to fit the asymmetrically nonlinear coupled case. Thirdly, we propose a general framework based on an eigenvalue approach for studying the synchronization in time-varying complex networks of nonlinearly coupled dynamical systems. Based on this eigenvalue approach, we explore more deeply the connection between network topologies and network synchronizability. The synchronizability of three network ensembles with prescribed global network properties are compared through a series of numerical computations. On the other hand, for nonlinearly coupled dynamical networks with discrete or distributed time delays, we derive a criterion for the networks to be globally exponentially synchronized from the Lyapunov functional method combined with the linear matrix inequality techniques. This synchronization criterion is independent of not only the time delay but also the delay type.
Compared with the existing results in the literature, the primary strengths of this study are that the coupling function can be linear or nonlinear, the components of a state variable can be fully coupled or partially coupled, and the outer-coupling matrix can be symmetric or asymmetrical. Numerical experiments of several illustrative examples including the coupled Chua’’s circuits, the coupled FitzHugh-Nagumo neurons and the coupled Hindmarsh-Rose neurons are given to demonstrate the theoretical analysis. More interestingly, the numerical results of two real-world networks of the cat cortex and the macaque visual cortex both modeled by the asymmetrically linear coupled FitzHugh-Nagumo equations are also reported.
關鍵字(中) ★ 無尺度網路
★ 耦合動力網路
★ 連續型時間遲滯
★ 全局指數型同步
★ 離散型時間遲滯
★ Lyapunov函數
★ Lyapunov泛函
★ Chua耦合電路網
★ FitzHugh-Nagumo耦合神經元網路
★ 社群網路
★ Hindmarsh-Rose耦合神經元網路
★ 小世界網路關鍵字(英) ★ Chua's circuit
★ Lyapunov functional
★ Lyapunov function
★ FitzHugh-Nagumo neuron
★ distributed time delay
★ global exponential synchronization
★ coupled dynamical network
★ Hindmarsh-Rose neuron
★ discrete time delay
★ modular network
★ small-world network
★ scale-free netwo論文目次 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
1 Introduction and Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Global exponential synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Graphs and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Graphs and directed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Special matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 A classification of diffusive matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.4 The Kronecker product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Assumptions on the intrinsic function f and the nonlinear coupling function g . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Coupling topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 A Linear Matrix Inequality Approach to Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Synchronization analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 The nonlinearly coupled Hindmarsh-Rose neurons with an asymmetrical cooperative outer-coupling matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 The nonlinearly coupled Chua's circuits with an asymmetrical cooperative-competitive outer-coupling matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 A Graph Approach to Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 An improvement of the connection graph stability method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 Synchronization analysis of the nonlinearly coupled case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.2 Synchronization analysis of the linearly coupled case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Examples and simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1 An ideal regular network with a modular structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.2 The symmetrically nonlinear coupled Hindmarsh-Rose neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.3 The asymmetrically nonlinear coupled Chua's circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.4 A real-world network of the cat cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 An Eigenvalue Approach to Synchronization in Time-Varying Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1 Synchronization analysis of the linearly coupled case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Synchronization analysis of the nonlinearly coupled case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Synchronizability and coupling topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1 Structural properties versus spectral properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.2 Networks with the same order and the same number of edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3.3 Networks with the same order and almost the same characteristic path length . . . . . . . . . . . . . . . . . . . . . . 49
5.4 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4.1 The nonlinearly coupled Chua's circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4.2 The nonlinearly coupled Hindmarsh-Rose neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4.3 A real-world network of the macaque visual cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Synchronization in Coupled Dynamical Networks with Coupling Time Delays . . . . . . . . . . . . . . . . . . . . . . . . 57
6.1 Synchronization analysis of the discrete delay case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2 Synchronization analysis of the distributed delay case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3.1 The nonlinearly coupled FitzHugh-Nagumo neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3.2 The nonlinearly coupled Hindmarsh-Rose neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3.3 The linearly coupled Chua's circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A Some Properties in Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.1 Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.2 Ger·sgorin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.3 Nonnegative matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
參考文獻 [1] F. M. Atay, T. Biyikoglu, and J. Jost, Network synchronization: spectral versus statistical properties, Physica D, 224 (2006), pp. 35--41.
[2] A.-L. Barabasi and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), pp. 509--512.
[3] M. Barahona and L. M. Pecora, Synchronization in small-world systems, Phys. Rev. Lett., 89 (2002), 054101.
[4] V. Belykh, I. Belykh, and M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D, 195 (2004), pp. 159--187.
[5] I. Belykh, V. Belykh, and M. Hasler, Blinking model and synchronization in small-world networks with a time-varying coupling, Physica D, 195 (2004), pp. 188--206.
[6] I. Belykh, V. Belykh, and M. Hasler, Synchronization in asymmetrically coupled networks with node balance, Chaos, 16 (2006), 015102.
[7] I. Belykh, V. Belykh, and M. Hasler, Generalized connection graph method for synchronization in asymmetrical networks, Physica D, 224 (2006), pp. 42--51.
[8] I. Belykh, M. Hasler, M. Lauret, and H. Nijmeijer, Synchronization and graph topology, Int. J. Bifurcation and Chaos, 15 (2005), pp. 3423--3433.
[9] I. Belykh, E. de Lange, and M. Hasler, Synchronization of bursting neurons: what matters in the network topology, Phys. Rev. Lett., 94 (2005), 188101.
[10] J. Cao, P. Li, and W. Wang, Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Physics Letter A, 353 (2006), pp. 318--325.
[11] T. L. Carroll, Amplitude-independent chaotic synchronization, Phys. Rev. E, 53 (1996), 3117.
[12] M. Chen, Synchronization in time-varying networks: a matrix measure approach, Phys. Rev. E, 76 (2007), 016104.
[13] G. Chen, J. Zhou, and Z. Liu, Global synchronization of coupled delayed neural networks and applications to chaotic CNN models, Int. J. Bifurcation and Chaos, 14 (2004), pp. 2229--2240.
[14] T. Chen and Z. Zhu, Exponential synchronization of nonlinear coupled dynamical networks, Int. J. Bifurcation and Chaos, 17 (2007), pp. 999--1005.
[15] L. O. Chua, A zoo of strange attractors from the canonical Chua's circuits, Proceedings of the 35th Midwest Symposium on Circuits and Systems, IEEE, 2 (1992), pp. 916--926.
[16] L. O. Chua, M. Komuro, and T. Matsumoto, The double scroll family, IEEE Trans. on Circuits and Systems, vol. cas-33 (1986), pp. 1072--1118.
[17] M. Dhamala, V. K. Jirsa, and M. Ding, Transition to synchrony in coupled bursting neurons, Phys. Rev. Lett., 92 (2004), 028101.
[18] Z. Duan, G. Chen, and L. Huang, Complex network synchronizability: analysis and control, Phys. Rev. E, 76 (2007), 056103.
[19] K. S. Fink, G. Johnson, T. Carroll, D. Mar, and L. Pecora, Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays, Phys. Rev. E, 61 (2000), pp. 5080--5090.
[20] K. Gopalsamy and X.-Z. He, Stability in asymmetric Hopfield nets with transmission delays, Physcia D, 76(1994), pp. 344--358.
[21] K. Gu, An integral inequality in the stability problem of time-delay systems, Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000, pp. 2805--2810.
[22] S. Guan, X. Wang, K. Li, B. Wang, and C.-H. Lai, Synchronizability of network ensembles with prescribed statistical properties, Chaos, 18 (2008), 013120.
[23] H. Hong, B. J. Kim, M. Y. Choi, and H. Park, Factors that predict better synchronizability on complex networks, Phys. Rev. E, 69 (2004), 067105.
[24] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[25] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
[26] X. Huang and J. Cao, Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme, Nonlinearity, 19 (2006), pp. 2797--2811.
[27] L. Huang, Y.-C. Lai, and R. A. Gatenby, Optimization of synchronization in complex clustered networks, Chaos, 18 (2008), 013101.
[28] L. Huang, Y.-C. Lai, and R. A. Gatenby, Alternating synchronizability of complex clustered networks with regular local structure, Phys. Rev. E, 77 (2008), 016103.
[29] J. Juang, C.-L. Li, and Y.-H. Liang, Global synchronization in lattices of coupled chaotic systems,
Chaos, 17 (2007), 033111.
[30] J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 1998.
[31] L. Kocarev and U. Parlitz, General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett., 74 (1995), pp. 5028--5031.
[32] Z. Li, Exponential stability of synchronization in asymmetrically coupled dynamical networks, Chaos, 18 (2008), 023124.
[33] C. Li and G. Chen, Synchronization in general complex dynamical networks with coupling delays, Physcia A, 343 (2004), pp. 263--278.
[34] Z. Li and J. Lee, New eigenvalue based approach to synchronization in asymmetrically coupled networks, Chaos, 17 (2007), 043117.
[35] C.-H. Li and S.-Y. Yang, Exponential synchronization in drive-response systems of Hopfield-type neural networks with time delays, Int. J. Bifurcation and Chaos, 17 (2007), pp. 4167--4176.
[36] C.-H. Li and S.-Y. Yang, Synchronization in linearly coupled dynamical networks with distributed time delays, Int. J. Bifurcation and Chaos, 18 (2008), pp. 2039--2047.
[37] C.-H. Li and S.-Y. Yang, Synchronization in delayed Cohen-Grossberg neural networks with bounded external inputs, IMA J. Appl. Math., 74 (2009), pp. 178--200.
[38] X. Liu and T. Chen, Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling, Physica A, 381 (2007), pp. 82--92.
[39] X. Liu and T. Chen, Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix, Physica A, 387 (2008), pp. 4429--4439.
[40] J. LÄofberg, YALMIP: A toolbox for modeling and optimization in Matlab, Proceedings of the 2004 IEEE International Symposium on Computer and Control Systems Design, pp. 284--289, Taipei, Taiwan, 2004.
[41] H. Lu and G. Chen, Global synchronization in an array of linearly coupled delayed neural networks with an arbitrary coupling matrix, Int. J. Bifurcation and Chaos, 16 (2006), pp. 3357--3368.
[42] W. Lu and T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D, 213 (2006), pp. 214--230.
[43] W. Lu, T. Chen, and G. Chen, Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay, Physica D, 221 (2006), pp. 118--134.
[44] J. Lu and D. W. C. Ho, Local and global synchronization in general complex dynamical networks with delay coupling, Chaos, Solitons and Fractals, 37 (2008), pp. 1497--1510.
[45] J. Lu, D. W. C. Ho, and M. Liu, Globally exponential synchronization in an array of asymmetric coupled neural networks, Phys. Lett. A, 369 (2007), pp. 444--451.
[46] J. Lu, X. Yu, and G. Chen, Chaos synchronization of general complex dynamical networks, Physica A, 334 (2004), pp. 281--302.
[47] J. Lu, X. Yu, G. Chen, and D. Cheng, Characterizing the synchronizability of small-world dynamical networks, IEEE Trans. Circuits Syst. I, 51 (2004), pp. 787--796.
[48] Z. Ma, Z. Liu, and G. Zhang, A new method to realize cluster synchronization in connected chaotic networks, Chaos, 16 (2006), 023103.
[49] G. S. Medvedev and N. Kopell, Synchronization and transient dynamics in the chains of electrically coupled FitzHugh-Nagumo oscillators, SIAM J. Appl. Math., 61 (2001), pp. 1762--1801.
[50] R. Merris, Laplacian graph eigenvectors, Linear Algebra Appl., 278 (1998), pp. 221--236.
[51] R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), pp. 1645--1662.
[52] B. Mohar, Eigenvalues, diameter, and mean distance in graphs, Graphs and Combinatorics, 7 (1991), pp. 53--64.
[53] A. E. Motter, C. Zhou, and J. Kurths, Network synchronization, diffusion, and the paradox of heterogeneity, Phys. Rev. E, 71 (2005), 016116.
[54] M. E. J. Newman and D. J. Watts, Scaling and percolation in the small-world network model, Phys. Rev. E, 60 (1999), pp. 7332--7342.
[55] M. E. J. Newman and D. J. Watts, Renormalization group analysis of the small-world network model, Phys. Lett. A, 263 (1999), pp. 341--346.
[56] T. Nishikawa, A. E. Motter, Y.-C. Lai, and F. C. Hoppensteadt, Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? Phys. Rev. Lett., 91 (2003), 014101.
[57] K. Park, Y.-C. Lai, S. Gupte, and J.-W. Kim, Synchronization in complex networks with a modular structure, Chaos, 16 (2006), 015105.
[58] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), pp. 821--824.
[59] L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), pp. 2109--2112.
[60] X. Shi and Q.-S. Lu, Complete synchronization of coupled Hindmarsh-Rose neurons with ring structure, Chin. Phys. Lett., 21 (2004), pp. 1695--1698.
[61] X. Shi and Q.-S. Lu, Firing patterns and complete synchronization of coupled Hindmarsh-Rose neurons, Chin. Phys., 14 (2005), pp. 77--85.
[62] R. Toral, C. Masoller, C. R. Mirasso, M. Ciszak, and O. Calvo, Characterization of the anticipated synchronization regime in the coupled FitzHugh-Nagumo model for neurons, Physica A, 325 (2003), pp. 192--198.
[63] A. Ucar, K. E. Lonngren, and E.-W. Bai, Synchronization of the coupled FitzHugh-Nagumo systems,
Chaos, Solitons and Fractals, 20 (2004), pp. 1085--1090.
[64] H. U. Voss, Anticipating chaotic synchronization, Phys. Rev. E, 61 (2000), pp. 5115--5119.
[65] W. Wang and J. Cao, Synchronization in an array of linearly coupled networks with time-varying delay, Physica A, 366 (2006), pp. 197--211.
[66] X. F.Wang and G. Chen, Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits Syst. I, 49 (2002), pp. 54--62.
[67] X. F. Wang and G. Chen, Synchronization in small-world dynamical networks, Int. J. Bifurcation and Chaos, 12 (2002), pp. 187--192.
[68] D. J. Watts and S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), pp. 440--442.
[69] C. W. Wu, Perturbation of coupling matrices and its e®ect on the synchronizability in arrays of coupled chaotic systems, Phys. Lett. A, 319 (2003), pp. 495--503.
[70] C. W. Wu, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18 (2005), pp. 1057--1064.
[71] C. W.Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific, Singapore, 2007.
[72] C. W. Wu and L. O. Chua, A unified framework for synchronization and control of dynamical systems, Int. J. Bifurcation and Chaos, 4 (1994), pp. 979--998.
[73] C. W. Wu and L. O. Chua, Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits Syst. I, 42 (1995), pp. 430--447.
[74] X. Wu, B. Wang, T. Zhou, W, Wang, M. Zhao, and H. Yang, Synchronizability of highly clustered scale-free networks, Chin. Phys. Lett., 23 (2006), 1046.
[75] W. Yu, J. Cao, and J. Lu, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 108--133.
指導教授 楊肅煜(Suh-Yuh Yang) 審核日期 2009-6-26 推文 plurk
funp
live
udn
HD
myshare
netvibes
friend
youpush
delicious
baidu
網路書籤 Google bookmarks
del.icio.us
hemidemi
facebook
twitter
google
reddit