動態系統

動態系統（粵拼：dung6 taai3 hai6 tung2；英文：dynamical system）係應用數學成日會用到嘅一個概念，屬系統嘅一種。如果話一個（由一粒或者多粒粒子組成嘅）系統係個「動態系統」，意思即係指個系統包含描述一個點會點樣隨時間（ $t$ ）變化嘅函數，簡單講就係指「有一大柞會隨時間互相影響嘅部份」嘅系統。用數學方程式嚟表達嘅話，基本嘅動態系統可以噉樣定義^[1]：

y_{t+1}=f(y_{t})

呢條式意思係話，「每一點時間嘅狀態數值（ $y_{t+1}$ ；可以係個多元組）都係打前嗰點（ $y_{t}$ ）嘅函數（ $f$ ）」，而用微分方程嚟諗嘅話^{[註 1]}：

{\frac {\Delta y}{\Delta t}}=f(y)

當中 $f$ 係一個主宰個系統點演變嘅函數（演化法則；evolution rule）^[2]。順帶一提， $f$ 可以係完全決定性嘅或者係帶有隨機性嘅都得^[3]。

喺實際嘅科學應用上，有好多嘅現象都可以想像成動態系統，即係將研究緊嗰個現象想像成一大柞變數一路隨住時間變化一路互相影響：比較簡單嘅例子有物理學會研究嘅鐘擺^[4]、生物學會研究嘅生態系統（一個生物種群嘅個體數量因為繁殖等嘅原因隨時間變化）^[5]、工程學會研究嘅機械運作（部機械啲機件會郁，即係機件嘅位置隨時間變化）^[6]、經濟學會研究嘅經濟體上落（個經濟體唔同部份互動，令經濟指標隨時間變化）^[7]同埋係心理學研究嘅人類心理發育（例如一個細路嘅認知能力就有可能隨時間而變化）呀噉^[8]。

註釋

↑ 物理學上嘅研究主流係用微分方程嚟定義動態系統嘅。

睇埋

攷

↑ Weisstein, E. W. (1999). CRC concise encyclopedia of mathematics. Boca Raton, FL: Chapman & Hall/CRC.
↑ "Nature". Springer Nature.
↑ Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.
↑ Melby, P.; et al. (2005). "Dynamics of Self-Adjusting Systems With Noise". Chaos: An Interdisciplinary Journal of Nonlinear Science. 15 (3): 033902.
↑ Jackson, T.; Radunskaya, A. (2015). Applications of Dynamical Systems in Biology and Medicine. Springer.
↑ Kreyszig, Erwin (2011). Advanced Engineering Mathematics. Hoboken: Wiley.
↑ Gandolfo, Giancarlo (2009) [1971]. Economic Dynamics: Methods and Models (Fourth ed.). Berlin: Springer.
↑ Van Geert, P. (2009). Nonlinear complex dynamical systems in developmental psychology. In S. J. Guastello, M. Koopmans, & D. Pincus (Eds.), Chaos and complexity in psychology: The theory of nonlinear dynamical systems (pp. 242–281). Cambridge University Press.

[2] 物理學上嘅研究主流係用微分方程嚟定義動態系統嘅。

[1] Weisstein, E. W. (1999). CRC concise encyclopedia of mathematics. Boca Raton, FL: Chapman & Hall/CRC.

[3] "Nature". Springer Nature.

[4] Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.

[5] Melby, P.; et al. (2005). "Dynamics of Self-Adjusting Systems With Noise". Chaos: An Interdisciplinary Journal of Nonlinear Science. 15 (3): 033902.

[6] Jackson, T.; Radunskaya, A. (2015). Applications of Dynamical Systems in Biology and Medicine. Springer.

[7] Kreyszig, Erwin (2011). Advanced Engineering Mathematics. Hoboken: Wiley.

[8] Gandolfo, Giancarlo (2009) [1971]. Economic Dynamics: Methods and Models (Fourth ed.). Berlin: Springer.

[9] Van Geert, P. (2009). Nonlinear complex dynamical systems in developmental psychology. In S. J. Guastello, M. Koopmans, & D. Pincus (Eds.), Chaos and complexity in psychology: The theory of nonlinear dynamical systems (pp. 242–281). Cambridge University Press.

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