Show that the system has a limit cycle. The full question reads:
Show that the system with $\dot{x} = x-y-(x^2 + \frac{3}{2} y^2)x$ and $\dot{y} = x+y-(x^2 + \frac{1}{2} y^2)y$ has a limit cycle.
I wanted to check the correctness of my method. My thought was to find the equilibrium points (If I am correct, the only one is $(0,0)$, then find the stability of that point. Then I attempted to show that if we go far enough out in the domain, that the flow of the direction field is going in the opposite direction of the direction field directly surrounding the equilibrium point. Then we can use Poincare-Bendixon and be done. Is this correct, or should I handle it differently? Thanks!

Yes, your approach looks good to me. If $r^2 = x^2 + y^2$ then we have
$r\dot{r} = x\dot{x} + y\dot{y} = x^2(1-x^2-\frac{3}{2}y^2) + y^2(1-x^2-\frac{1}{2}y^2)$
$\Rightarrow r(1-x^2-\frac{3}{2}y^2) \le \dot{r} \le r(1-x^2-\frac{1}{2}y^2)$
and since $1-\frac{3}{2}r^2 \le 1-x^2 - \frac{3}{2}y^2 \le 1-x^2-\frac{1}{2}y^2 \le 1 - \frac{1}{2}r^2$
which suggests that the circles $r=\sqrt{\frac{2}{3}}$ and $r=\sqrt{2}$ may be significant ...
