水平視線圖方法於非線性動力學系統之不可逆過程探討

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黃岱珩(Dai-Heng Huang) 查詢紙本館藏

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物理學系

論文名稱

水平視線圖方法於非線性動力學系統之不可逆過程探討
(Studies on irreversible process in some non-linear dynamical systems using the Horizontal Visibility Graph approach)

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摘要(中)

近年來，視線演算法（ visibility algorithm）已被用在許多非線性科學的領域，因其可將序列內部的關聯轉成對應的網路結構。而藉由指向性的水平視線演算法（ directed horizontal visibility algorithm），時間序列可被轉成指向性的網路結構，使得往「未來」與「過去」的分布可被測量；並提議以 Kullback-Leibler 散度值度量兩個分布的差異，作為量化不可逆性的指標。

本文旨在利用指向性水平視線演算法，研究非線性動力學與混沌系統的不可逆性，包括 Tent map 和 Logistic map。

隨著混沌系統的參數增加，系統整體而言越來越不可逆。在 Tent map 中，散度值的變化隨著不同週期帶呈現碎形特徵，且平均散度值隨著週期帶合併而倍增；在 Logistic map 中，則以週期窗口表現碎形特徵，並且在全混沌帶內週期窗口會以 $[RL^{alpha}R^{infty},RL^{alpha+1}R^{infty}]$ 為區間顯現對稱性。

最後，由於在參數首次進入混沌前的倍週期區間，水平視線演算法是無法分辨不可逆性的，促使我們將差值的因素放入演算法中（ modified-Horizontal Visibility graph ），使得前述的倍週期不可逆性可以顯現，但差值的效果卻也破壞了原本演算法所顯現的碎形特徵。

摘要(英)

In recent years, the so-called visibility algorithm has been used in many areas in nonlinear sciences by capturing the correlations of the time series through constructing the corresponding network. Using the directed horizontal visibility algorithm, the time-series can be mapped to a directed network system, and the in-degree and out-degree distributions can be calculated, and it is proposed that the irreversibility of the dynamics can be measured quantitatively by the Kullback-Leibler divergences of these degree distributions.

Here, by using the directed horizontal visibility graph, we focus on the irreversible dynamics of several nonlinear dynamical and chaotic systems, including the Tent map and the Logistic map.

With the increasing of the relevant parameter in the chaotic system, the dynamics globally become more and more irreversible. For the Tent map, we observe that the divergence value doubling in each band′s merging point shows a fractal structure. In the Logistic map, the fractal features demonstrated by the intermittency can reveal the symmetric structure between interval $[RL^{alpha}R^{infty},RL^{alpha+1}R^{infty}]$ inside the fully chaotic band.

Furthermore, we found that the horizontal visibility algorithm fails to identify the irreversibility of the period-doubling region before the relevant parameter is varied into the chaotic regime for the Logistic map, we thus propose a modified-Horizontal Visibility Graph method, which then can reveal the irreversibility for these periodic dynamics, but the associated fractal structure from the former algorithm will be destroyed.

關鍵字(中)

★ 非線性
★ 水平視線圖
★ Kullback-Leibler散度
★ Lyapunov 指數

關鍵字(英)

★ non-linear
★ Horizontal Visibility Graph
★ Kullback-Leibler divergence
★ Lyapunov exponent

論文目次

1 緒論

1.1熱力學第二定律的近況.......................... 1
1.1.1 非平衡態的假設 ......................... 1
1.1.2 非平衡態之熵變化 ........................ 2
1.2視線演算法................................ 4
1.3不可逆性 ................................. 4
1.4非線性動力學............................... 5
1.4.1 不動點與週期運動 ........................ 5
1.4.2 週期帶運動............................ 5
2 模擬與方法 6
2.1 模擬對象——非線性系統 ........................ 6
2.1.1 Logisticmap ........................... 6
2.1.2 Tentmap ............................. 8
2.2 測量方法 ................................. 11
2.2.1 Directed-Horizontal Visibility Graph(DHVG) . . . . . . . . 11
2.2.2 Kullback-Leibler散度 ...................... 13
2.2.3 Jensen-Shannon散度....................... 14
2.3 DHVG在隨機系統中的結果....................... 15
2.3.1 無相關均勻分布(白噪音)—Uncorrelated Uniform Distribu-
tion ................................ 15
2.3.2 不偏隨機漫步(平衡態離散布朗運動)—Unbiased Random Walks 17
2.3.3 有偏隨機漫步(非平衡離散布朗運動)—Biased Random Walks 17
2.3.4 Logisticmap與Tentmap的全混沌 .............. 20
3 研究結果
3.1 DHVG在非平衡離散布朗運動的其他測量............... 24
3.2 DHVG在Tentmap的測量....................... 28
3.2.1 Tentmap的週期帶與帶合併點 ................. 28
3.2.2 帶合併點的Kullback-Leibler散度 ............... 29
3.2.3 不可逆量隨分歧增加的遞減性.................. 35
3.3 DHVG在Logisticmap的測量 ..................... 39
3.3.1 倍週期的Kullback-Leibler散度................. 39
3.3.2 單峰映射中的普適序列 ..................... 40
3.3.3 Logisticmap的帶合併點 .................... 52
3.4 modified-Directed Horizontal Visibility Graph (m-DHVG) 的測量 . . 59
3.4.1 m-DHVG在白噪音中的測試 .................. 60
3.4.2 m-DHVG在週期–4的結果 ................... 62
3.4.3 m-DHVG在Logisticmap中的結果 .............. 64
3.4.4 m-DHVG在Tentmap中的結果 ................ 67
4 總結與討論 73
4.1 DHVG與非平衡系統........................... 73
4.2 DHVG與混沌系統............................ 73
4.3 m-DHVG與混沌系統 .......................... 74
4.4 未來展望 ................................. 76
參考文獻 ................................. 78

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指導教授

黎璧賢

審核日期

2019-7-19

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