I was looking for applications of the Poincaré-Bendixson theorem and on this site I have found several examples almost all similar to this post. So I tried to make a quite natural generalization $$ \left\{ \begin{array}{rcl} x'&=&ax+by-x(mx^2+ny^2)\\ y'&=&-bx+dy-y(px^2+qy^2) \end{array} \right. $$ when $(a-d)^2-4b^2<0$ holds, the Poincaré-Bendixson theorem ensures the existence of a limit cycle. But my question is this could this still achieve some kind of natural generalization? That is, for example, change $mx^2+ny^2$ for a homogeneous polynomial of degree $2$ or degree $n$ or I don't know... any ideas? (along with some hint on how to test it or some reference)

This question has already been asked here without much success, I would like to know your approach to this question, I know that there can be many approaches to a natural generalization.



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