複雜動態網路的模糊同步化設計

Fuzzy Synchronization Design of Complex Dynamical Networks

楊書偉(Shu-Wei Yang)

指導教授 ： 練光祐 ； 張政元

中原大學/工學院/電機工程研究所/碩士(2008年)

https://doi.org/10.6840/cycu200800441

摘要

摘要
在黑暗中的螢火蟲，其數目難以估計，卻同步地發出閃光；在演講的末了，觀眾的
鼓掌，卻同步的起落，這些同步行為的背後，到底是蘊含了什麼機制呢？混沌的圖形是
很美麗的，如同蝴蝶翅膀上不規則，卻又有著一絲條理。在本篇論文中，吾人藉著T-S
模糊模型的方法，去探討混沌系統的同步性，吾人將混沌系統的同步化問題，轉成一個
線性矩陣是否有解的問題。
首先，吾人將介紹這有趣的小世界、混沌系統與耦合動態網路的同步化。這所謂的
小世界理論，簡單的說就是－這世界是何等之小啊。接著，研究動機，研究貢獻與相關
書籍的介紹，將會被一一介紹。然後，圖形理論與如何藉由圖形理論來討論同步化的問
題也會一一說明。在了解上述的內容之後，吾人將介紹T-S 模糊模型的方法。
然後我們開始探討非線性系統的同步化問題，我們呈現了一個一般的複雜動態網
路，我們探討它的耦合結構與內部的耦合結構。藉著T-S 模糊模型的方法，我們獲得了
線性矩陣不等式，並且得到了不錯的同步化效果。完全連結的網路、正規的網路與小世
界網路藉著吾人的設計，都完美的表現出同步的狀態。吾人簡單的介紹MATLAB 的LMI
工具盒，並且提出幾點要注意的地方。更有甚者，我們提出了更方便去找出同步化問題
關鍵的方法。在一般的論文中，都藉著在平衡點線性化討論非線性的問題，但是多少都
會有些許的誤差，導致某些符合線性分析所得的條件，實際上無法使系統同步．而本論
文中，是採用非線性的方法處理非線性的問題，所以不會出現上述的問題．
最後吾人呈現了模擬的結果，並且可以看到效果是不錯的。並在最後，得到了一些
重要的結論，與規劃未來此研究的後續發展。

關鍵字

線性化 ； 混沌系統 ； 複雜網路 ； T-S 模糊模型 ； 同步化

並列摘要

Abstract
In the dark, the fireflies whose sparks are synchronous. In the end of speech, the sound
of claps are synchronous, too. What is the secret behind above? In the Internet, the
configuration of the nexus can even make the merchandise sold well or very bad. The
figure of Chaos is very beautiful like the butterfly’s wings. In this thesis, we discuss the
synchronization of the chaotic system by T-S fuzzy model approach, and we make the
synchronous problem transform into linear matrix inequalities. We discuss how to let
the chaotic system synchronous, and find out the key point in our case.
Firstly, the interesting small-world network, chaotic system and the synchronization
of coupled dynamic systems are introduced. The small-world theory is just that the
world is so small. Secondly, the motivation, contribution, and review of books is shown.
Next, the graph theory, and the synchronization about it are illustrated. After above is
shown, the T-S fuzzy model approach is described to continue the following derivation.
Then, the general complex network is shown, and we discuss the coupling configuration
matrix C and inner coupling matrix H. Based on the T-S fuzzy model approach,
we derive LMIs by Lyapunov’s theory, and we obtain the well results of synchronization.
The full connected network, regular network and small-world network all can
achieve synchronous by our design. We design the inner coupling matrix H to obtain
the synchronization of the general complex network. Afterwards, we briefly introduce
the MATLAB LMI toolbox. We also give some remarks for solving the LMIs. Moreover,
we propose the K and L to analysis the general complex network which is composed
of N Lorenz attractors. After that, the simulation of synchronization is shown, and we
can see that the response of it is very good. In the common papers, researchers utilize
the linearization at the equilibrium. But, this way must have some errors. That is, the
solution of constraints may not make systems synchronous. In this thesis, we utilize
the nonlinear method to address the nonlinear problem. Hence, we do not have above
problem. Then, the simulation results are shown in Chapter 5. Finally, we make some
conclusions and describe the future work.

並列關鍵字

T-S Fuzzy Model ； Synchronization ； Linearization ； Complex Network ； Chaotic System

參考文獻

[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of ’small-world’ networks,”

[3] M. E. J. Newman and D. J. Watts, “Scaling and percolation in the small-world

[4] S. A. Pandit and R. E. Amritkar, “Characterization and control of small-world

[6] M. Marchiori and V. Latora, “Harmony in the small-world,” Physica A, issue 3-4,

pp. 539-546, October 2000.