Planar system:

x' = x-y-x^3

y'=x+y-y^3

only has a critical point at origin.

The question asks to show that this system has a periodic orbit.

The hint for this says to convert to polar coordinates and show that for all

>0, r'<0 on the circle r=sqrt(2)+

and r'>0 on the circle r=1-

. Then show that this implies that there is a limit cycle in the annulus: {x

R^2 : 1<= |x| <= sqrt(2)}.

I'm not sure how to even convert this to polar coordinates. I know that x=rcos(

) and y=rsin(

). But after i plug it in, I just get a equation and I don't know how to get r' or

' from it.