# 洛倫茲系統

## 背景

{\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 = ...
當 t = 3, x = ..., y = ..., z = ...
...


## 圖像化

• 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 ${\displaystyle (x,y,z)=(0,0,1+\varepsilon )}$，當中 ${\displaystyle \varepsilon =10^{-5}}$

—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。

## 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。

## 參考

1. Lorenz attractor
2. Edward Norton Lorenz
3. differential equations
4. dynamical system

1. Lorenz Attractor. Wolfram MathWorld.
2. Alves, J., & Soufi, M. (2014). Statistical stability of geometric Lorenz attractors. Fundamenta Mathematicae, 3(224), 219-231.
3. Gleick, James (1987). Chaos: Making a New Science. Viking. p. 16.
4. Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130-141.