若給予渾沌系統適當頻率的驅策外力，外力與系統之間會發生相位鎖定現象。當系統的相位結構不簡單時，目前已有相位鎖定的判定方法不敷使用。於是我們觀察渾沌系統相空間中頻閃點的行為，發現相位鎖定的同時頻閃點會出現群聚的現象。利用訊息理論統計這些頻閃點的分布情況，企圖找出頻閃點行為與驅策力的振幅大小或頻率高低之間的關聯性，並證實頻閃點群聚行為就是相位鎖定的表徵現象。將此一理論應用於具備雙吸引子特性的蔡氏電路系統上，成功解決了當系統相位結構複雜而讓相位鎖定判定失效的窘境。另一方面將相同的相位鎖定判定法，應用至更複雜的超渾沌系統中，同樣也能成功\判定受到自身週期態或渾沌態訊號驅策的超渾沌系統中發生的弱相位鎖定現象。

The detection for phase locking in a forced oscillator with dual attractors and ill-defined phase structure is hard until a quantitative approach was constructed for detecting phase locking via stroboscopic method. We study the route to weak phase locking in a chaotic system “Chua oscillator” with complex attractor structure by analyzing the stroboscopic points. The onset of weak phase locking detected by using this statistical approach and the critical coupling strength calculated by conditional Lyapunov exponent are matched well. Detailed structure of phase locking intensity is described by the Arnold tongue diagram. Moreover, we apply this approach on three hyper-chaotic systems with multi-scroll attractor, including hyper-chaotic Rössler system, hyper-chaotic Lorenz system, and modified MCK oscillator. The weak phase locking between hyper-chaotic system and a periodic or a chaotic driving force is observable following the condition of stroboscopic point concentration.

摘要 1
Abstract 2

1.導論 6
1.1相位同步Phase synchronization 7
1.1.1相位角Phase angle 7
1.1.2同步Synchronization 12
1.1.3希爾伯特轉換Hilbert transform method 14
1.1.4濾波Band filter process 16
1.1.5相位曲率轉換法Curvature of phase track 17
1.1.6頻閃法Stroboscope 20
1.2相位鎖定Phase locking 23
1.2.1週期相位鎖定 Periodic phase locking 23
1.2.2高階鎖定 High order locking 28
1.2.3渾沌相位鎖定 Chaotic phase locking 29
1.2.4渾沌相位同步 Chaotic phase synchronization 31
1.2.5非完美相位鎖定 Imperfect phase lock 34
1.2.6頻閃點鎖定 Stroboscopic locking 39

2.弱相位鎖定Weak phase locking 43
2.1頻閃點群聚Stroboscopic point concentration 44
2.1.1 系統Model 44
2.1.2 受非線性週期力驅策的系統 53
2.1.3 頻閃點群聚行為 56
2.2頻閃熵比Entropy ratio of stroboscopic point distribution 63
2.2.1 頻閃點分布Stroboscopic point distribution 63
2.2.2 信息熵Information entropy 69
2.2.3 頻閃點統計 71
2.2.4 定義頻閃熵比 76
2.3穩定性分析 80
2.3.1 李雅普諾夫指數Lyapunov exponent 80
2.3.2 原本為零的李雅普諾夫指數Null Lyapunov exponent 84

3.超渾沌系統中的弱相位鎖定 87
3.1超渾沌Rössler系統 88
3.1.1 系統 88
3.1.2 頻閃點行為 95
3.1.3 分析結果 102
3.2超渾沌Lorenz系統 108
3.2.1 系統 108
3.2.2 頻閃點行為 116
3.2.3 分析結果 123
3.3修正型MCK系統 129
3.3.1 系統 129
3.3.2 頻閃點群聚 138
3.3.3 分析結果 145

4.結論 151
4.1成果討論 151
4.2未來展望 155

參考資料 156
作者資料 161
謝誌 163

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