I have the system (it is time dependent, this is a simplified notation): \begin{cases} x' = x - y - x^3 \\ y' = x + y - y^3 \\ \end{cases}
I can't seem to solve it for r, $\theta$. (The change of variables is: $x = r\cos \theta$ and $y = r\sin \theta$, I used the fact that $x^2 + y^2 = r^2 $ and then $rr' = xx' + yy'$, but I stumble it the middle of the calculation). All feedback is welcomed.
Edit: Here are my calculations: $$rr' = xx' + yy' \Leftrightarrow rr' = x(x-y-x^3) + y(x+y-y^3)$$ $$\Leftrightarrow rr' = r^2 - (x^4 + 2x^2y^2 + y^4) + 2x^2y^2 $$
$$\Leftrightarrow rr' = r^2 - r^4 + r^4(\cos^2 \theta \sin^2 \theta).$$ I was actually wondering whether it was possible to solve for $r$ without having $\theta$ in the result.