常微分方程 ODE -- Autonomous Equations and Stability

ODE – Autonomous Equations and Stability

Autonomous Equation

is an equation of the special form
$x'=f(x) \qquad (1)$x′=f(x)(1)
Note: 可以看出这个方程于函数$x(t)$x(t)的自变量$t$t无关

Direction field

因为与自变量无关，所以对于任意的t值，v相同处的方向场都是一样的

如下图

Equilibrium Points and Solutions

If $f(x_0)=0$f(x0​)=0, then the constant function $x(t)=x_0$x(t)=x0​ satisfies
$x'(t)=0=f(x_0)=f(x(t))$x′(t)=0=f(x0​)=f(x(t))
This constant function is a particular solution to (1).

We call a point $x_0$x0​ such that $f(x_0)=0$f(x0​)=0 an equilibrium point.

The constant function $x(t)=x_0$x(t)=x0​ is called an equilibrium solution.

Phase Line

简单来说，对于一个autonomous equation $y'=f(y)$y′=f(y), $y(t)$y(t)就是phase line

Stability

两种Equilibrium points

Asymptotically stable: Solution curves 在 $t \rightarrow \infty$t→∞ 时逼近 equilibrium point

Unstable: Solution curves 远离 equilibrium point

在phase line上，stable用实心点，unstable用空心点，根据f(x)的正负标记箭头

如下图

Theorem

Suppose that $x_0$x0​ is an equilibrium point for the differential equation $x'=f(x)$x′=f(x), where $f$f is a differentiable function

1. If $f'(x_0) \lt 0$f′(x0​)<0, then $f$f is decreasing at $x_0$x0​ and $x_0$x0​ is asymptotically stable.
2. If $f'(x_0) \gt 0$f′(x0​)>0, then $f$f is increasing at $x_0$x0​ and $x_0$x0​ is unstable.
3. If $f'(x_0)=0$f′(x0​)=0, no conclusion can be drawn.

如下图