look at the fate of an arbitrary initial value $x_0$ after one period

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๋ณต์ˆ˜์ „๊ณตํ•˜๊ณ  ์žˆ๋Š” ์ˆ˜ํ•™๊ณผ์˜ ์กธ์—…์‹œํ—˜์„ ์œ„ํ•ด ํ•™๋ถ€ ์ˆ˜ํ•™ ๊ณผ๋ชฉ๋“ค์„ ๋‹ค์‹œ ๊ณต๋ถ€ํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค๋งŒโ€ฆ ๋ฏธ๋ถ„๋ฐฉ์ •์‹์€ ์กธ์—…์‹œํ—˜ ๋Œ€์ƒ ๊ณผ๋ชฉ์ด ์•„๋‹ˆ๋ผ๋Š” ๊ฑธ ๋‚˜์ค‘์— ์•Œ๊ฒŒ ๋˜์—ˆ์Šต๋‹ˆ๋‹คโ€ฆ OTLโ€ฆ ๊ทธ๋ž˜๋„ ์ด์™• ์‹œ์ž‘ํ•œ ๊ฑฐ ๋‹ค์‹œ ๋ณต์Šต ์ข€ ํ•ด๋ด…์‹œ๋‹ค! ๐Ÿƒ ๋ฏธ๋ถ„๋ฐฉ์ •์‹ ํฌ์ŠคํŠธ ์ „์ฒด ๋ณด๊ธฐ

์š” ๋‚ด์šฉ์€ ํ•™๋ถ€ 2ํ•™๋…„์˜ ๋ฏธ๋ถ„๋ฐฉ์ •์‹(Math2xx) ์ˆ˜์—…์ด ์•„๋‹ˆ๋ผ ํ•™๋ถ€ 4ํ•™๋…„์˜ ์ƒ๋ฏธ๋ถ„๋ฐฉ์ •์‹๋ก (Math4xx)์—์„œ ๋‹ค๋ฃฌ ๋‚ด์šฉ์ž…๋‹ˆ๋‹ค.

Express ODE with initial value

์‹œ๊ฐ„ $t$์— ์˜์กดํ•˜๋Š” ์–ด๋–ค ํ•จ์ˆ˜ $x(t)$๋ฅผ ์ƒ๊ฐํ•ด๋ณด์ž. ๊ทธ๋ฆฌ๊ณ  ์ด ํ•จ์ˆ˜ $x(t)$๋Š” ODE๋ฅผ ๋งŒ์กฑํ•œ๋‹ค. ์˜ˆ๋ฅผ ๋“ค์–ด ์•„๋ž˜์™€ ๊ฐ™์€ ํ˜•ํƒœ์ผ ์ˆ˜ ์žˆ๋‹ค.

  • $xโ€™ = a x$: exponential growth/shrinking
  • $xโ€™ = x(1-x)$: logistic population model

๋Œ€๋ถ€๋ถ„์˜ ODE์˜ ํ•ด $x(t)$๊ฐ€ ๊ทธ๋ ‡๋“ฏ ์ดˆ๊ธฐ๊ฐ’ $x_0$์— ๋”ฐ๋ผ์„œ ํ•จ์ˆ˜์˜ ์–‘์ƒ์ด ๋‹ฌ๋ผ์งˆ ์ˆ˜ ์žˆ๋‹ค.

Notes on Diffy Qs: Differential Equations for Engineers, Jiล™รญ Lebl

์˜ˆ๋ฅผ ๋“ค์–ด, logistic population model์—์„œ๋Š” ์ดˆ๊ธฐ ์ธ๊ตฌ ๊ฐ’์ด $0 < x_0 < x_1$๋ผ๋ฉด, ์ธ๊ตฌ ํ•œ๊ณ„ $x_1$๋ฅผ ํ–ฅํ•ด ์ฆ๊ฐ€ํ•˜๊ณ , $x > x_1$ ์˜€๋‹ค๋ฉด ์ธ๊ตฌ ํ•œ๊ณ„๋ฅผ ํ–ฅํ•ด ์ธ๊ตฌ๊ฐ€ ๊ฐ์†Œํ•œ๋‹ค.

์ง€๊ธˆ๊นŒ์ง€ ๋ฏธ๋ถ„๋ฐฉ์ •์‹์—์„œ๋Š” solution function์„ ํ‘œํ˜„ํ•  ๋•Œ, $x(t) = c_1 e^{\lambda t} + \cdots$์™€ ๊ฐ™์ด ํ‘œํ˜„ํ•˜๊ณ , $c_1$๊ณผ ๊ฐ™์€ ๊ณ„์ˆ˜๊ฐ’์€ ์ดˆ๊ธฐ๊ฐ’ $x_0$์„ ๋Œ€์ž…ํ•ด ๊ตฌํ•˜๋Š” ๋ฐฉ์‹์œผ๋กœ ๋™์ž‘ํ–ˆ๋‹ค.

๊ทธ๋Ÿฌ๋‚˜, ์ด์ œ๋Š” ์‹œ๊ฐ„ $t$์™€ ์ดˆ๊ธฐ๊ฐ’ $x_0$์„ ๋ชจ๋‘ ๊ณ ๋ คํ•œ solution function๋ฅผ ์•„๋ž˜์™€ ๊ฐ™์ด ์ •์˜ํ•ด ์‚ฌ์šฉํ•˜๊ณ ์ž ํ•œ๋‹ค. ์ด๊ฒƒ์€ ์šฐ๋ฆฌ๊ฐ€ ์œ„์—์„œ ์‚ดํŽด๋ณธ Logistic Model์˜ Solution Graph๋ฅผ ํ•จ์ˆ˜๋กœ ์˜ฎ๊ธด ๊ฒƒ์ด๋ผ๊ณ  ๋ณด๋ฉด ๋œ๋‹ค.

\[\phi(t, x_0): \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\]
  • It represents an point on solution graph.
  • It means the value of the system when it starts from $x_0$ and time $t$ flows.
  • Sometimes we write this function as $\phi_t (x_0)$.

Poincare Map

์œ„์˜ Population Model์€ ๊ทธ๋ ‡์ง€ ์•Š์•˜์ง€๋งŒ, ์–ด๋–ค ๊ฒฝ์šฐ์—๋Š” ODE ํ•จ์ˆ˜๊ฐ€ ์–ด๋–ค ์ฃผ๊ธฐ์„ฑ์„ ๊ฐ€์ง€๊ณ  ์žˆ์„ ์ˆ˜๋„ ์žˆ๋‹ค.

  • $xโ€™ = ax - h (1 - \sin (\omega t))$
  • $xโ€™ = x(1-x) - h (1 - \sin (\omega t))$

์ด๋•Œ, ์ดˆ๊ธฐ๊ฐ’ $x_0$์—์„œ ์ฃผ๊ธฐ $\omega$ ๋งŒํผ ์‹œ๊ฐ„์ด ์ง€๋‚œ ํ›„์˜ ํ•จ์ˆ˜๊ฐ’ $\phi(\omega, x_0)$๋ฅผ โ€œPoincare Mapโ€๋ผ๊ณ  ์ •์˜ํ•œ๋‹ค.

[Poincare Map]

\[P(x_0) = \phi(\omega, x_0)\]

the value of the solution when one period $\omega$ has left.

๋งŒ์•ฝ, ์ดˆ๊ธฐ๊ฐ’ $x_0$๊ฐ€ fixed point(or equilibrium point)๋ผ๋ฉด, ์•„๋ž˜์™€ ๊ฐ™์€ ์‹์ด ์„ฑ๋ฆฝํ•˜๊ฒŒ ๋œ๋‹ค.

\[P(x_0) = \phi(\omega, x_0) = x_0 = \phi(0, x_0)\]

The key idea is to look at the fate of an arbitrary initial value $x_0$ after one period.

Morris, Differential Equations, Dynamical Systems & An Introduction to Chaos 3rd Edition

์œ„์˜ ๊ทธ๋ฆผ์€ ์ฃผ๊ธฐ๋ฅผ $\omega = 1$๋ผ๊ณ  ํ–ˆ์„ ๋•Œ์˜ Poincare Map์„ ๊ทธ๋ฆฐ ๋ชจ์Šต์ด๋‹ค. ๊ฐ๊ฐ์˜ ์ดˆ๊ธฐ๊ฐ’์—์„œ ์‹œ์ž‘ํ•ด ์ฃผ๊ธฐ๊ฐ€ ์ง€๋‚  ๋•Œ๋งˆ๋‹ค ํ•จ์ˆ˜๊ฐ’์ด ์–ด๋–ป๊ฒŒ ๋ณ€ํ™”ํ•˜๋Š”์ง€๋ฅผ ํŒŒ์•…ํ•˜๊ธฐ ์‰ฝ๋‹ค.

Orbit

์œ ์˜ํ•  ๊ฒƒ์€ Poincare Map๋„ ์—ฌ์ „ํžˆ ํ•จ์ˆ˜์ด๋‹ค. ๊ทธ๋ž˜์„œ ์ด๊ฒƒ์„ ์–ด๋–ค ์ดˆ๊ธฐ๊ฐ’ $x_0$์— ๋Œ€ํ•ด ์—ฌ๋Ÿฌ๋ฒˆ ์ ์šฉํ•  ์ˆ˜ ์žˆ๋‹ค.

\[x_0 \rightarrow P(x_0) \rightarrow P(P(x_0)) \rightarrow P(P(P(x_0))) \rightarrow \cdots\]

์šฐ๋ฆฌ๋Š” ์ด๋ ‡๊ฒŒ ์–ด๋–ค ์  $x_0$์—์„œ ์‹œ์ž‘ํ•ด Poincare Map์„ ์ ์šฉํ•œ ๋ชจ๋“  ๊ฒฐ๊ณผ๋ฅผ ๋ชจ์•„์„œ ์ง‘ํ•ฉ์„ ์ •์˜ํ•  ์ˆ˜ ์žˆ๋Š”๋ฐ, ์ด๋ฅผ โ€œOrbitโ€๋ผ๊ณ  ๋ถ€๋ฅธ๋‹ค.

์ด๋•Œ, ์ง‘ํ•ฉ Orbit์€ ์œ ํ•œ ํฌ๊ธฐ์ผ ์ˆ˜๋„ ์žˆ๊ณ , ๋ฌดํ•œ ํฌ๊ธฐ๋ฅผ ๊ฐ€์งˆ ์ˆ˜๋„ ์žˆ๋‹ค. ๋งŒ์•ฝ Poincare Map์ด ์–ด๋–ค ์ดˆ๊ธฐ๊ฐ’์— ๋Œ€ํ•ด $P^n(x_0) = x_0$ํ•˜๋Š” ์„ฑ์งˆ์„ ๊ฐ€์ง€๊ณ  ์žˆ๋‹ค๋ฉด, ๊ทธ Orbit์€ ์œ ํ•œ ์ง‘ํ•ฉ์ผ ๊ฒƒ์ด๊ณ  ๊ทธ ํฌ๊ธฐ๋Š” $n$์ผ ๊ฒƒ์ด๋‹ค. ๋ฐ˜๋Œ€๋กœ $P^n(x_0) = x_0$๋ฅผ ๋งŒ์กฑํ•˜๋Š” ์ž์—ฐ์ˆ˜ $n$์ด ์—†๋‹ค๋ฉด, Orbit์˜ ์›์†Œ๋Š” ๋ฌดํ•œํžˆ ๋งŽ์„ ๊ฒƒ์ด๋‹ค.

์ง€๊ธˆ์€ ๊ทธ๋ƒฅ ์ง‘ํ•ฉ์— ๋Œ€ํ•œ ์ •์˜๋งŒ ํ•˜๊ณ  ๋„˜์–ด๊ฐ€์ง€๋งŒ, ๋’ค์— โ€œDynamical Systemsโ€์— ๋Œ€ํ•ด ์‚ดํŽด๋ณผ ๋•Œ ํ•œ๋ฒˆ๋” ๋งŒ๋‚˜๊ฒŒ ๋  ๊ฒƒ์ด๋‹ค ใ…Žใ…Ž ์ง€๊ธˆ์€ ๊ทธ๋ƒฅ ๋ถ„๋Ÿ‰ ์ฑ„์šฐ๊ธฐ ์œ„ํ•ด ์ ์€ ๊ฒƒ ใ…‹ใ…‹

On the view of trajectory


์š”๊ฑด Poincare Map์„ ์ดํ•ดํ•ด๋ณด๋ ค๊ณ , ์ธํ„ฐ๋„ท์„ ๋– ๋Œ๋‹ค๊ฐ€ ๋ฐœ๊ฒฌํ•œ ์˜์ƒ์ธ๋ฐ, ์—„๋ฐ€ํ•œ ์ •์˜ ์—†์ด ๊ทธ๋ฆผ์„ ํ†ตํ•ด Poincare Map์ด ๋ญ”์ง€ ์„ค๋ช…ํ•˜๊ณ  ์žˆ๋‹ค. ๋‹จ์ˆœํ•˜๊ฒŒ ์ƒ๊ฐํ•˜๋ฉด, ์–ด๋–ค Curve์™€ Trajectory๊ฐ€ ๋งŒ๋‚˜๋Š” ์ง€์  $x_k$๊ฐ€ ์žˆ๊ณ , ๊ทธ ์ ์—์„œ ์ถœ๋ฐœ ํ–ˆ์„ ๋•Œ ๋‹ค์Œ์— ๋งŒ๋‚˜๋Š” ์ง€์ ์„ $P(x_k) = x_{k+1}$๋กœ ์ •์˜ํ•œ๋‹ค๋Š” ์ปจ์…‰์ด๋‹ค.

์˜์ƒ์—์„œ๋Š” ์ฃผ๊ธฐ $\omega$์— ๋Œ€ํ•œ ์–ธ๊ธ‰์ด ์ „ํ˜€ ์—†๋Š”๋ฐ, ์•„๋งˆ Curve $\Sigma$๊ณผ ๊ถค์ ์ด ๋งŒ๋‚˜๋Š” ์ˆœ๊ฐ„์„ ํ•œ ์ฃผ๊ธฐ๋ฅผ ๋Œ์€ ๊ฒƒ์œผ๋กœ ํ•œ ๊ฒƒ ๊ฐ™๋‹ค. 4ํ•™๋…„ ๋ฏธ๋ฐฉ ์ˆ˜์—… ๋•Œ๋Š” ์ด๋Ÿฐ ๊ด€์ ์„ ์ฑ„ํƒํ•˜์ง€๋Š” ์•Š์ง€๋งŒ, ๊ถค์ ์„ ๋ถ„์„ํ•จ์— ์žˆ์–ด์„œ๋Š” ํฅ๋ฏธ๋กœ์šด ๊ด€์ ์ธ ๊ฒƒ ๊ฐ™๋‹ค.