走進混沌世界

第1 章 混沌基礎理論淺析··············································································.1
1.1 杜芬振子如何演化出混沌 ····································································.1
1.1.1 線性系統特性 ··········································································.1
1.1.2 杜芬方程及其混沌····································································.4
1.1.3 杜芬振子的應用·······································································.9
1.2 蝴蝶效應與洛侖茲方程 ·······································································.9
1.2.1 混沌的定義 ············································································.10
1.2.2 洛倫茲方程 ············································································.11
1.2.3 應用實例 ···············································································.14
1.2.4 日常生活中的蝴蝶效應·····························································.15
1.3 邏輯斯蒂方程與人口模型 ···································································.16
1.3.1 邏輯斯蒂（logistic）連續方程 ····················································.16
1.3.2 邏輯斯蒂映射方程···································································.18
1.3.3 倍周期分岔中的費根鮑姆常數和多尺度現象··································.22
1.4 符號動力學初步 ···············································································.23
1.4.1 符號動力學的“字提升法”·······················································.23
1.4.2 邏輯斯蒂映射的超穩定軌道參數求解···········································.24
1.5 隨機共振 ························································································.28
1.5.1 非線性郎之萬方程···································································.28
1.5.2 隨機共振系統 ·········································································.29
1.6 圓映射與阿諾德舌 ············································································.31
1.7 李雅普諾夫指數 ···············································································.35
1.7.1 李雅普諾夫指數的數學描述·······················································.35
1.7.2 幾種典型映射的李雅普諾夫指數 ···············································.36
第 2 章 分形理論與重整化群···········································································43
2.1 分形維數 ························································································.43
2.1.1 自相似維數 ············································································.44
2.1.2 豪斯多夫（Hausdorff）維數·······················································.44
2.1.3 容量維（capacity dimension）和盒維數（box dimension）·················.46
2.2 朱利亞集和曼德勃羅集 ······································································.47
2.2.1 朱利亞（Julia）集 ···································································.47
2.2.2 曼德勃羅（Mandelbrot）集························································.49
2.3 日常生活中的分形現象 ······································································.50
2.4 分形在設備故障診斷中的應用 ·····························································.51
2.5 自相似行為的重整化群 ······································································.51
第 3 章 混沌控制與同步·················································································56
3.1 混沌控制 ························································································.56
3.2 外加正弦驅動力控制方法 ···································································.57
3.2.1 外加正弦驅動力控制原理··························································.57
3.2.2 仿真實例 ···············································································.59
3.3 時間延遲反饋控制方法 ······································································.62
3.3.1 時間延遲反饋控制原理·····························································.62
3.3.2 數值仿真 ···············································································.63
3.4 反饋控制實驗 ··················································································.65
3.5 自適應控制方法 ···············································································.67
3.6 混沌同步 ························································································.69
3.6.1 混沌同步的類型······································································.70
第 4 章 混沌保密通信與光孤子通信··································································74
4.1 信號的載波通信方式 ·········································································.74
4.2 信號混沌加密的通信方式 ···································································.76
4.2.1 混沌掩蓋 ···············································································.78
4.2.2 混沌鍵控 ···············································································.78
4.2.3 混沌調制 ···············································································.79
4.2.4 混沌加密的一般步驟································································.81
4.3 孤波及光孤子通信 ············································································.81
4.3.1 一個奇特的水波······································································.81
4.3.2 色散效應與 KdV 方程·······························································.82
4.4 光孤子通信 ·····················································································.85
4.4.1 全光型孤立子通信···································································.86
4.4.2 非線性薛定諤光學孤立波方程····················································.86
第 5 章 元胞自動機導論·················································································89
5.1 元胞自動機的起源與發展 ···································································.89
5.2 元胞自動機的基本概念和定義 ·····························································.89
5.3 元胞自動機的構成及演化規則 ·····························································.91
5.3.1 元胞 ·····················································································.91
5.3.2 元胞空間 ···············································································.91
5.3.3 鄰居 ·····················································································.93
5.3.4 元胞自動機的演化規則·····························································.94
5.4 元胞自動機的特征 ············································································.94
5.5 幾種典型的元胞自動機 ······································································.95
5.5.1 斯蒂芬·沃爾夫勒姆和初等元胞自動機········································.96
5.5.2 元胞自動機分類····································································.100
5.5.3 基於 184 號規則的交通仿真應用···············································.104
5.6 康威和“生命遊戲” ·······································································.106
5.7 凝聚擴散模型 ················································································.109
5.8 元胞自動機的仿真實現 ······································································112
5.9 元胞自動機的應用 ············································································115
第 6 章 自組織理論····················································································.117
6.1 自組織基本概念 ···············································································117
6.1.1 渦旋 ·····················································································118
6.1.2 激光 ·····················································································118
6.1.3 孕育生命的溫床······································································119
6.1.4 自組織臨界 ············································································119
6.1.5 事件發生的頻率與大小的關系····················································119
6.1.6 被打斷了的平衡····································································.120
6.2 自組織理論的歷史及特征 ·································································.120
6.3 自組織理論的建立與發展 ·································································.121
6.3.1 耗散結構論 ··········································································.122
6.3.2 協同學 ················································································.122
6.3.3 突變論 ················································································.123
6.3.4 混沌論 ················································································.123
第 7 章 自然界中的渦旋現象·········································································127
7.1 柏努利方程 ···················································································.127
7.1.1 空吸原理 ·············································································.127
7.1.2 文特利流速計 ·······································································.128
7.1.3 空氣流動的上舉力·································································.129
7.2 層流、湍流和渦旋 ··········································································.130
7.3 流體渦旋是怎樣形成的 ····································································.130
7.4 渦旋的分形結構 ·············································································.132
7.5 自然界中的渦旋現象 ·······································································.132
7.5.1 渦旋星系 ·············································································.132
7.5.2 幸運的地球 ··········································································.135
7.5.3 臺風 ···················································································.135
7.5.4 龍卷風··················································································.136
7.5.5 沙塵暴 ················································································.137
7.5.6 機翼翼尖的渦旋····································································.138
7.5.7 日常生活中的渦旋·································································.140
第 8 章 混沌的展望·····················································································141
8.1 牛頓的經典力學 ·············································································.141
8.2 愛因斯坦的相對論力學 ····································································.143
8.2.1 狹義相對論 ··········································································.143
8.2.2 廣義相對論與混沌·································································.145
8.3 量子混沌 ······················································································.149
8.3.1 什麼是量子 ··········································································.149
8.3.2 量子疊加態和糾纏態······························································.150
8.3.3 量子隧穿效應 ·······································································.155
8.3.4 量子混沌 ·············································································.156
參考文獻 ····································································································162