I'm looking at the system of ODEs:
$$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$
I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first order equation so that I can then find an approximation for the limit cycle.
So far I have that $\dot{z} = (k+i)z + xy^2 - ix^2$ and I'm having difficulty with the last two terms, in particular they are different order in x and y so I'm struggling to see how I could replace them with a simple function of z.
Thanks