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!
Hint.
Regarding the differential equation
$$ M(x,y) dx+N(x,y) dy = 0 $$
if $N_x = M_y$ then there exist $V(x,y)= C$ such that
$$ V_x dx + V_y dy = 0 $$
in our case if
$$ \cases{c+f=0\\ (n+1)e+b=0\\ (n+1)a+d=0} $$
then
$$ V(x,y) = a x^n y + \frac bn x y^n+c x y $$
is a solution. If $V(x,y)$ with $n$ even has relative minima/maxima then the associated differential system will have periodic orbits. Example
$$ V(x,y) = 3 x y+x^2 y-x y^2 $$
has a relative minimum at $x = -1, y = 1$. Attached the level curves around this point.
The associated differential system
$$ \cases{ \dot x = -x(3 + x - 2 y)\\ \dot y = y(3 + 2 x - y) } $$
with the near stream plot
NOTE
Limit cycles occur only in dissipative systems.