Given the differential equation $$\begin{cases}x'=P_n(x,y)\\y'=Q_n(x,y)\end{cases}$$ Where $P_n$ and $Q_n$ are homogeneous polynomials of order $n$, I have shown that if the differential equation has an isolated critical point then it is the origin $(0,0)$ and it is unique. Now, how can I prove that if $n$ is even, then the origin can't either be a center of a node?
2026-03-29 15:41:34.1774798894
Proving a dynamical system can't have a node or a center.
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