I have a dynamical system defined by :
$ \dot x = {(x+iy)^n + (x-iy)^n \over2}$ and $\dot y = {(x+iy)^n - (x-iy)^n \over2i}$
Converting the system to polar coordinates gives the system:
$\dot r = r^ncos((n-1)\theta)$ and $ \dot \theta = r^{n-1} sin ((n-1)\theta)$
Now the problem I have is finding an equation for the trajectory of the system that passes through the point (x,y) = (0,-1) given n =4.
Computing $ \dot x $ and $\dot y $ at the point (0,-1) gives 0 and -i respectively. This shows that the there is no x motion for the system at (0,-1). How do I begin to find the equation for the trajectory? My thoughts are to use $\dot r$ and solve for it explicitly but I can not make the differential equation $\dot r = r^ncos((n-1)\theta)$ separable.
Attempt
$ dr \over d\theta$ = $ \dot r \over \dot \theta$ = $rcot(3\theta)$
Now the differential equation is separable and after solving gives:
$ r = C (sin(3\theta))^{1/3}$
Using relation $r^2 = x^2+y^2$ and $tan^{-1}(y/x)$ gives r = 1 and $ \theta = -pi/4$ This yields a C = -1.122