Let $x' = f(x)$ be autonomous first–order equation differential with an equiliburiium point $x_0$.
Suppose $f'(x_0) = 0$ what can I say about the behavior of soluton near $x_0$?
If $f'(x_0) ≠ 0$ and $f''(x_0) = 0$ then what is the dynamical behavior near this point. And identically I have above question for this $f'(x_0) ≠ 0$ and $f''(x_0) ≠ 0$, but $f'''(x_0) ≠ 0$.