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I have been assigned a project in which I need to study the following system: $$\begin{cases}\dot{x} = x(ax^n + by^n + c)\\\dot{y} = y(dx^n + ey^n + f)\end{cases}$$ where $(a,b,c,d,e,f) \in \mathbb{R}^6$ and $n \in \mathbb{N}$. I'm being asked to find a relationship between the parameters of the system, $\phi(a,b,d,c,e,f,n)$ such that if $\phi(a,b,c,d,e,f,n) = 0$ the system may have periodic orbits but cannot contain limit cycles.

I've tried using Bendixson-Dulac 's Theorem in order to find a function $B$ and an open set $U$ homeomorphic to a crown (disc with a hole) around every critic point and, perhaps, move on from there knowing there might exist periodic orbits. However, I don't believe this is even close to a possible solution and don't really know how to move on.

Any hints on how I could be facing this problem are more than welcome. Thanks!

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