在文獻中[15,16,17,18,28,33]中，為了提升混沌系統中多種同步化的橫向穩定性，因此引用了小波變換方法。很多的理論在這問題上都集中討論是在耦合對稱的狀況。然而，在實際上的運用 [3,4,5,12,19,37]，卻是幾乎都是不對稱的。在這篇論文，耦合混沌系統在對稱的稀疏拓譜連結中是我們已經知道的，此外，這代表著我們不知道如何去踏出第一步去控制其穩定性在藉由小波變換方法去處理全域同步的非對稱動態混沌系統。但是我們有一些規則可循。第一，我們可以找出下界，然後用小波變換方法取的一些節點。第二，我們可以顯示用小波變換方法在比起對稱的情況下，在非對稱的耦合混沌系統裡是更有效用也更容易去控制的。

The wavelet transform method [15,16,17,18,28,33] is an effective tool to enhance the transverse stability of the synchronous manifold of a coupled chaotic system. Much of the theoretical study on this matter is centered on the networks being symmetrically coupled. However, in real applications, the coupling topology of a network is often asymmetric [3,4,5,12,19,37]. In this paper, 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 acquiring the synchrony of the coupled chaotic system with the wavelet transform method is independent of the number of nodes. Second, we deminstrate that the wavelet transform method on networks of coupled chaotic systems is even more effective and controllable for asymmetric coupling schemes as compared to those of the symmetric cases.

[1] Ashwin, P., Synchronization from chaos, Nature, 422, 384-385 (2003).
[2] Barahona, M., and Pecora, L. M., Synchronization in small-world systems, Phys. Rev. Lett.,
89, 054101 (2002).
[3] I. Belykh, V. Belykh, and M. Hasler, Generalized connection graph method for synchronization
in asymmetrical networks, Physica D, 224, pp. 42-51 (2006).
[4] V. Belykh, I. Belykh, and M. Hasler, Synchronization in asymmetrically coupled networks with
node balance, Chaos, 16, 015102 (2006).
[5] M. Chavez, D. U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Synchronization
is enhanced in weighted complex networks, Phys. Rev. Lett., 94, 218701, (2005).
[6] M. Y. Chen, Some simple synchronization criteria for complex dynamical networks, IEEE
Trans. Circuits Syst., II: Analog Digital Signal Process., 53 (11), pp. 1185-1189 (2006).
[7] Chen, Y., Rangarajan, G., and Ding, M., General stability analysis of synchronized dynamics
in coupled systems, Phys. Rev. E (3), 67, 026209 (2003).
[8] Daido, H., Lower critical dimension for populations of oscillators with randomly distributed
frequencies: A renormalization-group analysis, Phys. Rev. Lett., 61, pp. 231234 (1988).
[9] Daubechies, I., Ten lectures on wavelets, CBMS-NSF Series in Applied Mathematics, SIAM,
Philadelphia (1992).
[10] Fink, K. S., Johnson, G., Carroll, T., Mar, D., and Pecora, L., Three coupled oscillators as a
universal probe of synchronization stability in coupled oscillator arrays, Phys. Rev. E (3), 61,
pp. 5080-5090 (2000).
[11] Gade, P. M., Synchronization of oscillators with random nonlocal connectivity, Phys. Rev. E
(3), 54, pp. 6470 (1996).
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