Show that a system of differential equation has a limit cycle.

Question

WTS that the system

$$x'= -x -y +x(3x^2+y^2)\\ y'= x - y +y(3x^2+y^2)$$

has a limit cycle.

I derived the equation

$$r'(t) = -r(t)(-1+2r^2(t)cos^2\theta(t) +r^2)\\ \theta'(t) = 1$$

but I can't draw the phase plane diagram. How can I do this?

Answer 1

You got $$ r(1-3r^2)\le \dot r\le r(1-r^2) $$ which should give you bounds on the time evolution of the radius, or at least and first qualitative information on that evolution.