Find all limit cycles of a system of differential equations

Question

I am currently trying to study for an exam and ran into a problem regarding limit cycles.

The question is to find all limit cycles of the following system of the differential equations:

$$\dot{x}=-y-\frac{x(x^2+y^2-2)}{\sqrt{x^2+y^2}}$$ $$\dot{y}=x-\frac{y(x^2+y^2-2)}{\sqrt{x^2+y^2}}$$

The problem also gave a hint, which is to compute $$\frac{d(x^2+y^2)}{dt}$$ and observe that a limit cycle C must be the orbit of a periodic solution to the given system if it contains no equilibrium points.

So, I followed the hint and computed $$\frac{d(x^2+y^2)}{dt}=2x\dot{x}+2y\dot{y}$$but, I got something messy and I am not sure what to do afterwards...

Any help at all would be appreciated!

Thank you.

Answer 1

Hint.

$$ \frac 12\frac{d}{dt}(x^2+y^2-2) = -\sqrt{x^2+y^2}(x^2+y^2-2) $$

hence

$$ \frac 12 \frac{d}{dt}\ln(x^2+y^2-2) = -\sqrt{x^2+y^2} $$

Here $x^2+y^2 = r^2$