Show that the system has a limit cycle

Question

Show that the system has a limit cycle $$\left\{\begin{array}{l}\dot{x}=-y+\dfrac{x}{\sqrt{x^{2}+y^{2}}}\big(1-(x^{2}+y^{2})\big)\\\dot{y}=x+\dfrac{y}{x^{2}+y^{2}}\big(1-(x^{2}+y^{2})\big)\end{array}\right.$$

Answer 1

Consider the level sets of $V=x^2+y^2$. Then $$ \dot V=2\left(\frac{x^2}{\sqrt{V}}+\frac{y^2}V\right)(1-V) $$ The middle factor is always positive for $V>0$, so that the dynamic of the radius $r=\sqrt V$ is determined by the factor $(1-V)$.

Answer 2

Suppose, it did.

This means, there exists some $c$, s.t. $$x^2 + y^2 = c$$

Then, by differentiating both sides, we get $$x\dot{x} + y\dot{y} = 0$$

Plug in given equations (presumably the right ones) and see if you can find $c$.