Browsing by Author "Li, Chin-Lung"

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Eigenvalue problems and their application to the wavelet method of chaotic control
(2006-07-01)
Juang, Jonq
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Li, Chin-Lung
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應用數學系
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Department of Applied Mathematics
Controlling chaos via wavelet transform was recently proposed by Wei, Zhan, and Lai [Phys. Rev. Lett. 89, 284103 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue lambda(1)(alpha,beta) of the (wavelet) transformed coupling matrix C(alpha,beta) for each alpha and beta. Here beta is a mixed boundary constant and alpha is a scalar factor. In particular, beta=1 (respectively, 0) gives the nearest neighbor coupling with periodic (respectively, Neumann) boundary conditions. The first, rigorous work to understand the eigenvalues of C(alpha,1) was provided by Shieh [J. Math. Phys. (to be published)]. The purpose of this paper is twofold. First, we apply a different approach to obtain the explicit formulas for the eigenvalues of C(alpha,1) and C(alpha,0). This, in turn, yields some new information concerning lambda(1)(alpha,1). Second, we shed some light on the question whether the wavelet method works for general coupling schemes. In particular, we show that the wavelet method is also good for the nearest neighbor coupling with Neumann boundary conditions.
Global synchronization in lattices of coupled chaotic systems
(2007-09-01)
Juang, Jonq
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Li, Chin-Lung
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Liang, Yu-Hao
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應用數學系
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Department of Applied Mathematics
Based on the concept of matrix measures, we study global stability of synchronization in networks. Our results apply to quite general connectivity topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized.
Perturbed block circulant matrices and their application to the wavelet method of chaotic control
(2006-12-01)
Juang, Jonq
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Li, Chin-Lung
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Chang, Jing-Wei
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應用數學系
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Department of Applied Mathematics
Controlling chaos via wavelet transform was proposed by Wei [Phys. Rev. Lett. 89, 284103.1-284103.4 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue lambda(2)(alpha,beta) of the (wavelet) transformed coupling matrix C(alpha,beta) for each alpha and beta. Here beta is a mixed boundary constant and alpha is a scalar factor. In particular, beta=1 (0) gives the nearest neighbor coupling with periodic (Neumann) boundary conditions. In this paper, we obtain two main results. First, the reduced eigenvalue problem for C(alpha,0) is completely solved. Some partial results for the reduced eigenvalue problem of C(alpha,beta) are also obtained. Second, we are then able to understand behavior of lambda(2)(alpha,0) and lambda(2)(alpha,1) for any wavelet dimension j is an element of N and block dimension n is an element of N. Our results complete and strengthen the work of Shieh [J. Math. Phys. 47, 082701.1-082701.10 (2006)] and Juang and Li [J. Math. Phys. 47, 072704.1-072704.16 (2006)]. (c) 2006 American Institute of Physics.
Snap-back repellers and chaotic traveling waves in one-dimensional cellular neural networks
(2007-06-01)
Chang, Ya-Wen
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Juang, Jonq
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Li, Chin-Lung
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應用數學系
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Department of Applied Mathematics
In 1998, Chen et al. [1998] found an error in Marotto's paper [1978]. It was pointed out by them that the existence of an expanding fixed point z of a map F in Br(z), the ball of radius r with center at z does not necessarily imply that F is expanding in Br( z). Subsequent efforts (see e. g. [Chen et al., 1998; Lin et al., 2002; Li & Chen, 2003]) in. xing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give suficient conditions so that F is "globally" expanding. This, in turn, gives more satisfying definitions of a snap-back repeller. We then use those results to show the existence of chaotic backward traveling waves in a discrete time analogy of one-dimensional Cellular Neural Networks (CNNs). Some computer evidence of chaotic traveling waves is also given.
The theory of wavelet transform method on chaotic synchronization of coupled map lattices
(2011-01-01)
Juang, Jonq
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Li, Chin-Lung
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應用數學系
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Department of Applied Mathematics
The wavelet transform method originated by Wei et al. [Phys. Rev. Lett. 89, 284103.4 (2002)] was proved [Juang and Li, J. Math. Phys. 47, 072704.16 (2006); Juang et al., J. Math. Phys. 47, 122702.11 (2006); Shieh et al., J. Math. Phys. 47, 082701.10 (2006)] to be an effective tool to reduce the order of coupling strength for coupled chaotic systems to acquire the synchrony regardless the size of oscillators. In Juang et al., [IEEE Trans. Circuits Syst., I: Regul. Pap. 56, 840 (2009)] such method was applied to coupled map lattices (CMLs). It was demonstrated that by adjusting the wavelet constant of the method can greatly increase the applicable range of coupling strengths, the parameters, range of the individual oscillator, and the number of nodes for local synchronization of CMLs. No analytical proof is given there. In this paper, the optimal or near optimal wavelet constant can be explicitly identified. As a result, the above described scenario can be rigorously verified. (C) 2011 American Institute of Physics. [doi:10.1063/1.3525802]
The wavelet transform method applied to a coupled chaotic system with asymmetric coupling schemes
(2013-10-01)
Juang, Jong
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Li, Chin-Lung
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應用數學系
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Department of Applied Mathematics
The wavelet transform method originated by Wei et al. (2002) [19] is an effective tool for enhancing the transverse stability of the synchronous manifold of a coupled chaotic system. Much of the theoretical study on this matter is centered on networks that are symmetrically coupled. However, in real applications, the coupling topology of a network is often asymmetric; see Belykh et al. (2006) [23,24], Chavez et al. (2005) [25], Hwang et al. (2005) [26], Juang et al. (2007) [17], and Wu (2003) [13]. In this work, a certain type of asymmetric sparse connection topology for networks of coupled chaotic systems is presented. Moreover, our work here represents the first step in understanding how to actually control the stability of global synchronization from dynamical chaos for asymmetrically connected networks of coupled chaotic systems via the wavelet transform method. In particular, we obtain the following results. First, it is shown that the lower bound for achieving synchrony of the coupled chaotic system with the wavelet transform method is independent of the number of nodes. Second, we demonstrate that the wavelet transform method as applied to networks of coupled chaotic systems is even more effective and controllable for asymmetric coupling schemes as compared to the symmetric cases. (C) 2013 Elsevier Ltd. All rights reserved.
Wavelet Transform Method for Coupled Map Lattices
(2009-04-01)
Juang, Cheng
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Li, Chin-Lung
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Liang, Y. H.
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Juang, Jonq
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應用數學系
The purpose of this paper is twofold. First, we derive a necessary and sufficient condition for local synchronization in coupled map lattices (CMLs) with symmetric coupling. In particular, we also identify the best choice of coupling strength in the sense that such a coupling strength gives the fastest convergence rate of initial values toward the synchronous manifold. Furthermore, such a coupling strength is independent of the choice of the individual chaotic map. In addition, it is demonstrated that the wavelet transform method, which is proposed by Wei et al., can greatly increase the applicable ranges of coupling strengths, the parameters of the individual chaotic map, and the number of nodes for local synchronization of CMLs.