小波變換是個很有用的方法在控制混沌非線性動態系統上。本篇論文是在具有週期性邊界條件的廣義複雜延遲動態系統上做小波變換。並且在Duffing oscillator 和 Lorenz system 上，利用小波參數α 來控制動態系統的同步。

Wavelet Transform method is auseful tool to control chaos for nonlinear dynamical systems. Synchronization dynamics for a general complex delayed dynamical network with periodic boundary conditions via the wavelet transform method is investigated in this thesis. We discuss that the influence of the wavelet constant α on synchronization intervals for delayed dynamical networks, including Duffing oscillator and Lorenz system.

References
[1] A.-L. Barabási and R. Albert. Emergence of scaling in random networks. science,
286(5439):509–512, 1999.
[2] J. Cao, H. Li, and D. W. Ho. Synchronization criteria of lur¡¦e systems with time-delay feedback
control. Chaos, Solitons & Fractals, 23(4):1285–1298, 2005.
[3] J. Cao and J. Liang. Boundedness and stability for cohen–grossberg neural network with timevarying
delays. Journal of Mathematical Analysis and Applications, 296(2):665–685, 2004.
[4] J. Cao and J. Wang. Absolute exponential stability of recurrent neural networks with lipschitzcontinuous
activation functions and time delays. Neural networks, 17(3):379–390, 2004.
[5] J. Cao and J. Wang. Global asymptotic and robust stability of recurrent neural networks with
time delays. Circuits and Systems I: Regular Papers, IEEE Transactions on, 52(2):417–426, 2005.
[6] J. Cao, W. Yu, and Y. Qu. A new complex network model and convergence dynamics for reputation
computation in virtual organizations. Physics Letters A, 356(6):414–425, 2006.
[7] G. Chen, J. Zhou, and Z. Liu. Global synchronization of coupled delayed neural networks
and applications to chaotic cnn models. International Journal of Bifurcation and Chaos,
14(07):2229–2240, 2004.
[8] J. Chen and X. Chen. Special matrices. Special Matrices, 2001.
[9] L. O. Chua. The genesis of Chua’s circuit. Electronics Research Laboratory, College of Engineering,
University of California, 1992.
[10] I. Daubechies et al. Ten lectures on wavelets, volume 61. SIAM, 1992.
[11] J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of
vector fields. 1983.
[12] S. Hu and J. Wang. Global exponential stability of continuous-time interval neural networks.
Physical Review E, 65(3):036133, 2002.
[13] C. Juang, C.-L. Li, Y.-H. Liang, and J. Juang. Wavelet transform method for coupled map
lattices. Circuits and Systems I: Regular Papers, IEEE Transactions on, 56(4):840–845, 2009.
[14] J. Juang and C.-L. Li. Eigenvalue problems and their application to the wavelet method of chaotic
control. Journal of mathematical physics, 47(7):072704, 2006.
[15] J. Juang and C.-L. Li. The theory of wavelet transform method on chaotic synchronization of
coupled map lattices. Journal of Mathematical Physics, 52(1):012701, 2011.
[16] J. Juang, C.-L. Li, and J.-W. Chang. Perturbed block circulant matrices and their application to
the wavelet method of chaotic control. Journal of mathematical physics, 47(12):122702, 2006.
[17] C. Li and G. Chen. Stability of a neural network model with small-world connections. Physical
Review E, 68(5):052901, 2003.
27
[18] C. Li and G. Chen. Synchronization in general complex dynamical networks with coupling delays.
Physica A: Statistical Mechanics and its Applications, 343:263–278, 2004.
[19] Z. Li and G. Chen. Robust adaptive synchronization of uncertain dynamical networks. Physics
Letters A, 324(2):166–178, 2004.
[20] X. Liao, K.-W. Wong, Z. Wu, and G. Chen. Novel robust stability criteria for interval-delayed
hopfield neural networks. Circuits and Systems I: Fundamental Theory and Applications, IEEE
Transactions on, 48(11):1355–1359, 2001.
[21] H. Lu. Chaotic attractors in delayed neural networks. Physics Letters A, 298(2):109–116, 2002.
[22] J. Lü and G. Chen. A time-varying complex dynamical network model and its controlled synchronization
criteria. Automatic Control, IEEE Transactions on, 50(6):841–846, 2005.
[23] J. Lü, X. Yu, and G. Chen. Chaos synchronization of general complex dynamical networks.
Physica A: Statistical Mechanics and its Applications, 334(1):281–302, 2004.
[24] J. Lü, X. Yu, G. Chen, and D. Cheng. Characterizing the synchronizability of small-world dynamical
networks. Circuits and Systems I: Regular Papers, IEEE Transactions on, 51(4):787–796,
2004.
[25] W. Lu and T. Chen. Synchronization of coupled connected neural networks with delays. IEEE
Transactions on Circuits and Systems Part 1: Regular Papers, 51(12):2491–2503, 2004.
[26] S.-F. Shieh, Y. Wang, G. Wei, and C.-H. Lai. Mathematical analysis of the wavelet method of
chaos control. Journal of mathematical physics, 47(8):082701, 2006.
[27] S. H. Strogatz. Exploring complex networks. Nature, 410(6825):268–276, 2001.
[28] W. Wang and J. Cao. Synchronization in an array of linearly coupled networks with time-varying
delay. Physica A: Statistical Mechanics and its Applications, 366:197–211, 2006.
[29] X. F. Wang. Complex networks: topology, dynamics and synchronization. International Journal
of Bifurcation and Chaos, 12(05):885–916, 2002.
[30] X. F. Wang and G. Chen. Synchronization in scale-free dynamical networks: robustness and
fragility. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on,
49(1):54–62, 2002.
[31] X. F. Wang and G. Chen. Synchronization in small-world dynamical networks. International
Journal of Bifurcation and Chaos, 12(01):187–192, 2002.
[32] D. J. Watts. Small worlds: the dynamics of networks between order and randomness. Princeton
university press, 1999.
[33] D. J. Watts and S. H. Strogatz. Collective dynamics of ¡¥small-world¡¦networks. nature,
393(6684):440–442, 1998.
[34] G. Wei, M. Zhan, and C.-H. Lai. Tailoring wavelets for chaos control. Physical review letters,
89(28):284103, 2002.
28
[35] C. W. Wu. Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal
time-varying coupling. Circuits and Systems II: Express Briefs, IEEE Transactions on,
52(5):282–286, 2005.
[36] C. W. Wu and L. O. Chua. Synchronization in an array of linearly coupled dynamical systems.
Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on,
42(8):430–447, 1995.
[37] W. Yu and J. Cao. Adaptive qs (lag, anticipated, and complete) time-varying synchronization
and parameters identification of uncertain delayed neural networks. Chaos: An Interdisciplinary
Journal of Nonlinear Science, 16(2):023119, 2006.
[38] W. Yu and J. Cao. Stability and hopf bifurcation analysis on a four-neuron bam neural network
with time delays. Physics Letters A, 351(1):64–78, 2006.
[39] W. Yu and J. Cao. Adaptive synchronization and lag synchronization of uncertain dynamical
system with time delay based on parameter identification. Physica A: Statistical Mechanics and
its Applications, 375(2):467–482, 2007.
[40] W. Yu and J. Cao. Synchronization control of stochastic delayed neural networks. Physica A:
Statistical Mechanics and its Applications, 373:252–260, 2007.
[41] W. Yu, J. Cao, and G. Chen. Robust adaptive control of unknown modified cohen–grossberg
neural networks with delays. Circuits and Systems II: Express Briefs, IEEE Transactions on,
54(6):502–506, 2007.
[42] W. Yu, J. Cao, K.-W. Wong, and J. Lü. New communication schemes based on adaptive synchronization.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 17(3):033114, 2007.
[43] W. Yu, G. Chen, J. Cao, J. Lü, and U. Parlitz. Parameter identification of dynamical systems
from time series. Physical Review E, 75(6):067201, 2007.
[44] J. Zhou and T. Chen. Synchronization in general complex delayed dynamical networks. Circuits
and Systems I: Regular Papers, IEEE Transactions on, 53(3):733–744, 2006.
[45] J. Zhou, J. Lu, and J. Lu. Adaptive synchronization of an uncertain complex dynamical network.
arXiv preprint nlin/0512031, 2005.
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