01. Overview¶

ToC¶

$m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0 \tag{1}$

$\frac{\partial u}{\partial t} = \frac{\partial ^2 u}{\partial x^2}$

$\begin{align*} \dot{x}_1 &= f_1(x_1, \cdots , x_n) \\ & \vdots \\ \dot{x}_n &= f_n(x_1, \cdots , x_n) \end{align*} \tag{2}$

$\dot{x}_1 \equiv \frac{dx_i}{dt}$

$\begin{align*} \dot{x}_2 &= \ddot{x} = - \frac{b}{m} \dot{x} - \frac{k}{m} x \\ &= - \frac{b}{m} x_2 - \frac{k}{m} x_1 \end{align*}$

$\begin{cases} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= - \frac{b}{m} x_2 - \frac{k}{m} x_1 \end{cases}$

nonlinear

e.g., pendulum

$\begin{cases} \dot{x}_1 = x_2 \\ \dot{x}_2 = - \frac{g}{L} \sin(x_1) \end{cases}$

forced harmonic oscillator

$m\ddot{x} + b\dot{x} + kx = F \cos t$

let: - $x_1 = x$ - $x_2 = \dot{x}$ - $x_3 = t$ , then $\dot{x}_3 = 1$

$\begin{cases} \dot{x}_1 = x_2 \\ \dot{x}_2 = \frac{1}{m} (-kx_1 - bx_2 + F\cos x_3) \\ \dot{x}_3 = 1 \end{cases} \tag{3}$

growth of population

$\dot{x} = r x$

swinging of pendulum

$\ddot{x} + \frac{g}{L} \sin{x} = 0$

	$n=1$	$n=2$	$n\ge3$	$n\gg1$	Continuum
Linear	Growth, decay, equilibrilium Exponential growth, RC circuit, Radioactive decay	Oscillation Linear oscillator, Mass and spring, RLC circuit, 2-body problem		Collective phenomenon	Waves & pattern
Nonlinear	Fixed points, Bifurcation, Overdampled system, Logistic equation	Pendlum, Anharmonic oscillator, Limit cycle, Biological oscillator,	Chaos		Spatio-temporal complexity