I need to the existence, or not, of limit cycles of the following ODE system:
$$ \dot{x}(\phi) =- y(\phi) + \frac{L x(\phi)}{(x(\phi)^{2} + y(\phi)^{2})^{2}} \,\, , $$
$$ \dot{y}(\phi) = x(\phi) + \frac{L y(\phi)}{(x(\phi)^{2} + y(\phi)^{2})^{2}} \,\, , $$
with $L \in \mathbb{R}$. I would like to prove this using the Poincaré-Bendixson theorem or Bendixson criterion, how could I do that? (My guess I that there is one in $r = 3$.)
The existence of limit cycles in the system
$$\dot{x}(\phi) =- y(\phi) + \frac{L x(\phi)}{(x(\phi)^{2} + y(\phi)^{2})^{2}} \,\, ,$$ $$\dot{y}(\phi) = x(\phi) + \frac{L y(\phi)}{(x(\phi)^{2} + y(\phi)^{2})^{2}} \,\, ,$$
depends on $L$.
Transfer the system to polar coordinate by: $$ x=rcos(\theta) ,\text { and } y=rsin(\theta) $$
Note that $$r^2 = x^2 + y^2 \implies r'= \frac {xx' +yy'}{r}$$
Upon simplification we come up with $$ r' = \frac {L}{(x^2 + y^2 )^{3/2}}$$
We also have $$ \theta ' =1$$
If $L>0$, $r$ is an increasing function which implies there in no limit cycle in this system.
If $L=0$, then every orbit is a periodic orbit.
If $ L<0$, then $r$ is a decreasing function and there is no loop.