Consider new variable $z=x+iy$. In this variable your systems reads $ \dot z=(1+i)z-z|z|^2. $ Passing to the polar coordinates $z(t)=\rho(t)e^{i\varphi(t)}$ you will find \begin{align} \dot \rho&=\rho(1-\rho^2)\\ \dot \varphi&=1. \end{align} Note that the equations are decoupled and conclude that 1) the first equation has a solution $\rho=1$ -- which means this is a closed trajectory on the phase plane 2) there are no other closed solutions in a small neighborhood of this orbit -- which means that your closed trajectory is a limit cycle.
Unfortunately there are no general methods to detect limit cycles. What you wrote concerns the conditions of non-existence of limit cycles. In textbook problems it is often useful to try polar coordinates. Another important thing to know is the Poincare--Bendixson theory. An extremely good book addressing the issues of finding periodic solutions to systems of ODE is Periodic Motions by Miklos Farkas.