常微分方程 ODE -- Autonomous Equations and Stability

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ODE – Autonomous Equations and Stability

Autonomous Equation

is an equation of the special form
$x'=f(x) \qquad (1)$
Note: 可以看出这个方程与函数 $x(t)$的自变量 $t$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 $f(x_0)=0$, then the constant function $x(t)=x_0$ satisfies
$x'(t)=0=f(x_0)=f(x(t))$
This constant function is a particular solution to (1).

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

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

Phase Line

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

Stability

两种Equilibrium points

Asymptotically stable: Solution curves 在 $t \rightarrow \infty$ 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（勘误：图中横坐标应为x）

Theorem

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

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

如下图