# 蝴蝶效應

想搵奧地利電影嘅話，請睇蝴蝶效應 (電影)

## 數學模型

{\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x)\\[6pt]{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y\\[6pt]{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z\end{aligned}}}

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}}$

 當 t = 0, x = ..., y = ..., z = ...
當 t = 1, x = ..., y = ..., z = ...
當 t = 2, x = ..., y = ..., z = ...
...


• 藍色線係設 ${\displaystyle (x,y,z)=(0,0,1)}$ ${\displaystyle (\sigma ,\rho ,\beta )=(10,28,8/3)}$  得出嘅；
• 黃色線啲參數數值一樣，但係 ${\displaystyle (x,y,z)=(0,0,1+\epsilon )}$ ，當中 ${\displaystyle \epsilon =10^{-5}}$

—對比藍色線黃色線，睇得出變數嘅初始值係噉咦改咗少少，出嗰條軌跡已經有明顯差異。

## 媒體描述

For want of a nail the shoe was lost.

For want of a shoe the horse was lost.

For want of a horse the rider was lost.

For want of a rider the message was lost.

For want of a message the battle was lost.

For want of a battle the kingdom was lost.

And all for the want of a horseshoe nail.

## 註解

1. 有關呢啲式嘅數學細節，可以睇吓數學分析微積分嘅概念。
2. 不過亦有人指，社會科學預測未來嘅能力遠遠弱過自然科學。
3. 但有專業啲嘅研究者提出，呢種理解唔完全正確—蝴蝶效應講嘅係一個系統有一啲特性，對初始狀態極敏感，細變化好多時會引起大變化，但就算喺呢啲系統當中，細變化都唔一定會引致大變化。

## 參考

1. butterfly effect
2. sensitive dependence of solutions on initial conditions，SDIC
3. Lorenz system
4. Edward Norton Lorenz
5. differential equations
6. dynamical system
7. deterministic system
8. theoretical model
10. A Sound of Thunder
11. For Want of a Nail，留意 want 喺廿一世紀初英文裡便通常指想要，但喺中古英文入便係指缺乏

1. Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press.
2. Gaspard, P. (2005). Chaos, scattering and statistical mechanics (No. 9). Cambridge University Press.
3. Patrick Young - "The trouble with weather forecasting is that it's right too often for us to ignore it and wrong too often for us to rely on it."
4. Lorenz Attractor. Wolfram MathWorld.
5. Alves, J., & Soufi, M. (2014). Statistical stability of geometric Lorenz attractors. Fundamenta Mathematicae, 3(224), 219-231.
6. Gleick, James (1987). Chaos: Making a New Science. Viking. p. 16.
7. Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130-141.
8. Mandl, C. E. (2023). Prediction, Butterfly Effect, and Decision-Making. In Managing Complexity in Social Systems: Leverage Points for Policy and Strategy (pp. 35-45). Cham: Springer International Publishing，對於一個模型要點先算係有用：
1. All the causal loops of the structure must be plausible and evidence-based.
2. The simulated patterns must not be sensitive to external influences.
3. The model must be able to reproduce past patterns based on earlier initial conditions.
9. Arthur, W. B. (1989). Competing technologies, increasing returns, and lock-in by historical events. The economic journal, 99(394), 116-131.
10. Burstein, F., W Holsapple, C., Bennet, A., & Bennet, D. (2008). The decision-making process in a complex situation. Handbook on Decision Support Systems 1: Basic Themes, 3-20.
11. Shen, B.W., Sr Roger, & Zeng, X. (2024). Special Issue Theme Topic: "Advances in Understanding the Butterfly Effect, Chaos, and Multiscale Dynamics in the AI Era": Reframing Predictability Through AI and Chaos Theory.
12. Ray Bradbury (1952). A Sound of Thunder.
13. Holmes, Neil (2004). "Fateful butterfly". New Scientist. 182 (2443): 31.
14. （英文） The Butterfly Effect Movies，有列出一啲英文電影作品。
15. （英文） 蝴蝶效應，The Decision Lab（決策實驗室