Let $f:\mathbb{R}\to\mathbb{R}$ be $C^\infty$. Suppose the following
(i) $f(0)=0$
(ii) there is a smallest $n\in\mathbb{N}$ so that $f^{(n)}(0)\neq0$
I want to examine the (Lyapunov) stability of the equilibrium at 0 for the differential equation $\dot{x}=f(x)$, in terms of $n$ and the sign of $f^{(n)}(0)$.
To do this I apply Taylor's theorem:
$\dot{x}=f(x)=\left(\frac{f^{(n)}(0)}{n!}+xh(x)\right)x^n$
with $h\in C^\infty$, so $Df(0)=0$ and hence it is a nonhyperbolic equibilibrium.
Any advice on how to get started and make a proper argumentation ?
(For reference this is Prop. 8.8 in Stability and Stabilization: An Introduction By William J. Terrell)