課程資訊
課程名稱
混沌力學導論
Introduction to Chaotic Dynamics 
開課學期
110-1 
授課對象
工學院  機械工程學研究所  
授課教師
伍次寅 
課號
ME5134 
課程識別碼
522 U1660 
班次
 
學分
3.0 
全/半年
半年 
必/選修
選修 
上課時間
星期四7,8,9(14:20~17:20) 
上課地點
工綜209 
備註
限本系所學生(含輔系、雙修生)
總人數上限:20人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1101ME5134_ 
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課程概述

課號:522 U1660
班次:
主授教授:伍次寅
學分: 3
課程名稱:混沌力學導論 (Introduction to Chaotic Dynamics)
教科書:1. 'Chaos and Nonlinear Dynamics', 2nd ed., R. C. Hilborn, Oxford Univ. Press, 2000.
2. 'Nonlinear Dynamics and Chaos', S. H. Strogatz, Westview Press, 2000.

課程大綱:

INTRODUCTION TO CHAOS

1. Introduction:
The name of the new science and origin of ‘chaos’. Is chaos a generic or pathological phenomenon?

2. Phenomenology of chaos:
three physical examples (and many others) demonstrating chaotic motions, bifurcations, ‘strange attractors’, fractals, metaphor of ‘butterfly effect’, universality of chaos.

DYNAMICAL SYSTEMS AND ANALYSIS

3. Dynamical systems and state-space dynamics: (in which chaos is dwelt)
Topics include linear and nonlinear stabilities, bifurcations, phase portraits, qualitative theories of dynamical systems.

GENESIS OF CHAOS

4. Routes to chaos: (via which chaotic motions emerge)
Topics include period-doubling bifurcation, quasi-periodicity bifurcation, intermittency, crises, chaotic transient (homoclinic bifurcation).

QUANTITATIVE STUDY OF CHAOS

5. Measures of chaos: (identifying and quantifying chaos)
Fourier spectrum, correlation function, Lyapunov exponent, Poincare section, return-map method.

6. Iterated maps and their complicated dynamics: (a simple yet generic way to generate chaotic motions)
quadratic map, renormalization theory, tent map, Baker’s map, circle map, Henon map, Smale horseshoe map, mathematical definition of ‘strong chaos’, concept of ‘topological equivalency’, hyperbolic intersections and applications of symbolic dynamics, statistical description of chaotic trajectories.

FRACTALS AND APPLICATIONS

7. Fractals: (the most generic way the nature manifests its patterns)
examples of mathematical fractals and physical fractals, self-similarity, fractal dimensions, correlation dimension, generalized dimension of fractal, mono- and multi-fractal, fractal basin boundaries, fractal attractors, Cantor set, Mandelbrot set, Julia set, fractals on everything at everywhere.

8. Advanced topics:
embedding theory, state-space reconstruction technique, nonlinear time-series analysis, synchronization of chaotic motions, chaos control, multifractal, etc.

課程概述:

渾沌現象被一些科學家譽為20世紀物理界三大重要發現之一。渾沌到底是什麼東西?渾沌系統所呈現的行為有何特殊面貌?是什麼原因造成某些系統會產生渾沌運動?渾沌現象果真有令人嘆為觀止之處,還是科學家們溢美之詞?希望同學們修完本課程後會得到滿意的答案。
本課程將以深入淺出之方式介紹渾沌系統與運動。首先,我們以一些簡單的實例來展示及說明渾沌現象。接著,我們將定義所謂的渾沌運動及其所具備的物理特性。由於渾沌運動乃源自於古典力學當中之系統動力學,因此課堂當中會用相當比例的時間來探討線性及非線性動態系統之運動特性及其分析方式,並解釋渾沌運動與非線性動力學之間的關聯及形成渾沌運動的機制。除此之外,渾沌運動在狀態空間中所勾劃之軌跡往往具碎形之特徵,本課程也會對碎形理論及其應用做簡單的介紹。
在處理實際的問題中,系統之數學模式通常是未知的。在一般的情形下我們僅有少數甚或只有一組由實驗或觀察所取得之數據資料,此數據資料稱之為時間序列。因此分析時間序列是瞭解複雜系統行為的重要手段之一。本課程將另外介紹最近一些從系統動態學及渾沌理論所發展出的非線性時間序列分析數值方法及應用程式,俾使同學們除了吸收渾沌理論知識外,也能運用工具來分析處理一些與動態系統相關的實際問題。
近年來,拜電腦軟硬體快速發展之賜,渾沌運動及碎形得以藉由電腦模擬揭開其神秘的面貌。若時間允許,課堂中將會運用簡單的電腦模擬程式在螢幕上以動態方式實際展現出渾沌運動之軌跡及多種碎形幾何形體,以加深同學們的印象並增進對渾沌與碎形的認知。
若要嚴謹剖析渾沌系統及其運動行為需要應用到高深的數學知識及解析技巧。本課程將不會碰觸到這些高深的學問,僅傳授其中一些簡單的分析方法及其應用,目的是讓學生對解析渾沌系統的方法有粗淺的涉獵。渾沌理論應用的範圍非常廣泛。舉凡工程、數學、自然科學、生命科學、醫學、社會科學甚至人文藝術上,只要所欲探討的系統行為是由一群隨時間變化的變數及一些控制參數藉由彼此之間非線性的交互作用所掌控,就有可能會發現渾沌的影子。期待同學們修完本課程後會有所啟發,將課堂上的知識靈活應用至各個領域。
修完本課程後,同學若有興趣可繼續修習高等動力學課程,以獲取更進一步有關系統動力學方面的知識。
 

課程目標
教導學生明瞭渾沌系統之特性及渾沌運動所呈現之物理現象與形成機制;傳授學生描述與分析渾沌系統的方法與技巧,期使學生能將所學的知識應用於解釋與解決各個不同科技領域當中與渾沌相關的課題。 
課程要求
須先行修過微積分與工程數學。 
預期每週課後學習時數
 
Office Hours
每週四 12:00~14:00 備註: 工綜#618 
參考書目
請至本平台文件項目下的list of references資料夾內下載。 
指定閱讀
1. 'Chaos and Nonlinear Dynamics', 2nd ed., R. C. Hilborn, Oxford Univ. Press, 2000.
2. 'Nonlinear Dynamics and Chaos', S. H. Strogatz, Westview Press, 2000. 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
作業報告(short report) 
25% 
2~3次,佔 25% 
2. 
小考(quiz) 
40% 
1次,佔 40%,日期暫定為第15週12/29(星期三)晚上19:00~21:30 
3. 
期末計劃(final term project) 
35% 
佔 35%,報告繳交時間原則上是在第18週的星期四(1/20 2022),至於該日是否須出席做口頭報告則視疫情而定。會在學期進行到後半段的時候再說明期末報告形式。 
 
課程進度
週次
日期
單元主題
第1週
  • Introduction -
What is chaos? Definition of the 'deterministic chaos', chaos and nonlinear dynamical system, brief history of chaos 
第2週
  • Phenomenology of chaos -
three examples of chaotic systems, period doubling phenomenon, bifurcation diagram, universal features of chaos 
第3週
  • Phenomenology of chaos -
other chaotic examples, summary of analytic tools
• State-space dynamics of dynamical systems -
state space, standard form of dynamical system, autonomous and non-autonomous systems, no-intersection rule, attractor 
第4週
  • 1-D state-space dynamics -
fixed point and stability, linear stability analysis of fixed point, structural stability, dissipative system
• 2-D state-space dynamics -
linear stability analysis of fixed point, eigenvalues & eigenvectors of the Jacobian matrix, dynamics of fixed point 
第5週
  • 2-D state-space dynamics -
limit cycle, Poincare-Bendixson theorem, stability of limit cycle, Poincare map, Floquet multiplier, Lyapunov exponent, dissipative system in 2-D state space
• Trajectories in 2-D state space -
phase plane methods, phase portraits, conservative system 
第6週
  • Trajectories in 2-D state space -
index theorem, gradient system, Poincare-Bendixson theorem, trapping of limit cycle, applications in Biology
• Bifurcation -
normal form of bifurcation in 1-D & 2-D systems, saddle-node bifurcation, transcritical bifurcation, pitch-fork bifurcation, Hopf bifurcation 
第7週
  • 3-D state-space dynamics -
linear stability analysis of fixed point, Poincare plane, stability of limit cycle, quasi-periodic motion, torus
• Nonlinear stability -
hyperbolic fixed point, persistence of hyperbolicity, Hartman & Grobman theorem, Lyapunov function, Lyapunov stability theorem 
第8週
  • Routes to chaos (through bifurcation) -
via period-doubling, via quasi-periodicity, via intermittency, via crises, via chaotic transient (homoclinic bifurcation), homoclinic intersection, tangles and horseshoe map 
第9週
  • Diagnostic tools for chaos -
Fourier spectrum, auto-correlation function, Lyapunov exponent for trajectories 
第10週
  • Diagnostic tools for chaos -
Lyapunov exponent for trajectories, return map 
第11週
  • 1-D iterated maps -
Quadratic map, Feigenbaum constant, Li-Yorke theorem, Sarkovskii theorem, critical point and supercycle, boundaries of attracting regions in bifurcation diagram
• 1-D iterated maps -
size-scaling law, derivations of Feigenbaum's universal constants, composition law, intermittency and crises revisited 
第12週
  • 1-D iterated maps -
intermittency and crises revisited
• 1-D iterated maps -
Tent map, symbolic dynamics, Baker's map and Bernoulli shift, concept of topological equivalency, Bernoulli shift and chaotic trajectory, definition of 'chaos' 
第13週
  • 1-D iterated maps -
statistical description of deterministic chaos
• 2-D iterated maps -
Henon map, fractal attractor, Smale's Horseshoe map, symbolic dynamics 
第14週
  • 2-D iterated maps -
topological equivalency between Horseshoe map and Bernoulli shift operation, Horseshoe map and homoclinic intersection
• Fractals -
fractals in Nature, self-similarity, fractal dimension 
第15週
  • quiz (暫定) 12/29 (星期三) 晚上19:00~21:30
• Fractals -
capacity dimension, Cantor set, mathematical fractal sets, Hausdorff dimension, correlation dimension, Lyapunov dimension 
第16週
  • Fractal basin boundaries -
fractal basin boundaries for pendulum system, fractal basin boundaries for Henon map, Mandelbrot set, Julia set
• Nonlinear time series analysis -
embedding theorem, state-space reconstruction technique 
第17週
  • Nonlinear time series analysis -
implementation and application
• Software and demonstration of computer simulation - 
第18週
  • Final term project presentation (報告繳交時間原則上是在第18週的星期四(1/20 2022),至於該日是否須出席做口頭報告則視疫情而定。會在學期進行到後半段的時候再說明期末報告形式。)