Stability of ODEs with polynomial nonlinearity

Question

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.

Answer 1

No, no such standardized procedure exists. Even the planar case still has open question. In particular, the problem of distinguishing center from focus is solved only for quadratic vector fields on the plane (see here). If the equilibrium is not monodromic (i.e., is not focus or center) then there are efficient algorithms to understand the structure of a neighborhood of an equilibrium (it consists of so-called Brower's sectors, and the types and order of sectors can be read off the Newton's diagram corresponding to the powers of polynomials), but I am not aware whether this approach was generalized to dimensions $n\geq 3$.