Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\end{pmatrix},
$$
where $x\in \mathbb R^n$ and $f_k$​ is a **multivariate polynomial** of degree at most $m$. Assume $x_0$ is a singular point of the equation, i.e., $f(x_0)=0$.

My question is **whether there is a standardized procedure to determine the (asymptotic) stability of $x_0$​ using the coefficients of $(f_k​)$.**

I know that linearization around $x_0$​ can be used to assess stability: if the real parts of the eigenvalues of the corresponding Jacobian matrix are strictly less than zero, the singular point is asymptotically stable; if there exists an eigenvalue with a strictly positive real part, it is unstable. However, this method fails in other cases.

I am aware that constructing a Lyapunov function can be employed to determine stability, but this approach heavily relies on intuition and lacks a universal construction method.

There is a center manifold theorem that I do not fully understand, which seems to allow for stability assessment through **center manifold reduction**. I have found some examples online, but I am skeptical about whether this can be generalized into a universal algorithm.

I suspect that algebraic geometry, which deals with multivariate polynomial-related problems, might provide some insights. I would (rather boldly) guess whether algebraic geometry methods could be applied to this problem. (I also added the tag "algebraic geometry", and **feel free to remove it if it is irrelevant**.)

I understand that for general nonlinear equations, there is no standard procedure to determine the stability of singular points. However, multivariate polynomials are the simplest cases of nonlinearity, and I hope there might be relevant results, especially when the dimension $n$ and degree $m$ are low.