Almost all Liouville torus is preserved for small oscillation problems even if we don't use second-order approximation to potential energy, right? In small oscillation problems, we use a second-order approximation to the potential energy function (suppose the oscillation is around the point $(0,\cdots, 0)$),
$$
V(x) = V(0) + \frac{\partial^2 V(0)}{\partial x_i\partial x_j}x_ix_j + o(|x|^3),
$$
and by ignoring the $o(|x|^3)$ term, we get a harmonic system, which is completely integrable.
Now I am considering what if we do not ignore the $o(|x|^3)$ term. I read a little bit about KAM theory, and came up with this idea: we can think of the $o(|x|^3)$ term as the small perturbation of the approximated harmonic system, and according to KAM theory, almost all the Liouville torus close to the origin are preserved with a small change in shape, therefore the motion of the non-perturbed system near the origin can still be viewed as the composition of $n$ periodic motions, with slightly different periods than the ones of approximated harmonic system.
However, since I am not really familiar with KAM theory, I am not sure whether I have used KAM theory in the correct way. So, is my idea correct? And why is it correct or not correct?
 A: Yes, your idea is correct.
Near-integrable Hamiltonian systems such as you describe are the most important application of the KAM theory. So, yes, if the technical conditions are respected (non-degeneracy, smoothness of the perturbation, etc.), most regular behavior is preserved.
From Scholarpedia:

A nearly-integrable Hamiltonian system is a Hamiltonian system governed by a Hamiltonian function of the form $H_\epsilon(y,x)=K(y)+\epsilon P(y,x)$ with $y=(y_1,...,y_n)$ (action variables) varying in a domain $B\subset \mathbb{R}^n$ and
$x=(x_1,...,x_n)$ (angle variables) varying in the standard $n$-dimensional torus $\mathbb{T}^n\ .$
For $\epsilon=0\ ,$ equations (1) [Hamilton equations] give
$\dot y=0$
and
$\dot x=\partial_y K(y)\ ,$
hence
$y=y_0=$ constant
and $x = x_0 + \omega_0\, t$ (mod $2\pi$),
with $\omega_0:= \partial_y K\big(y_0\big)\ .$ Thus the torus $\{y_0\}\times \mathbb{T}^n$ is invariant for the flow $\phi^t_{K}\ ,$ and if $\omega_0$ is Diophantine and $\partial_y^2 K(y_0)$ is invertible, then such a torus is  a non-degenerate KAM torus for $H_0=K\ .$
Since $K(y)$ can be expanded by Taylor's formula as
$K=K(y_0)+\omega_0\cdot (y-y_0)+ \frac{1}{2} \partial_y^2 K(y_0) (y-y_0) \cdot(y-y_0)+ O(|y-y_0|^3|),$
it follows from Kolmogorov's Theorem that
for $\epsilon$ small enough such tori persist, giving rise to non-degenerate KAM tori for $H_\epsilon$.

