Finding a periodic solution using polar coordinates

[rande09]

So I have been given a problem describing a system in R^2 using polar coordinates r,theta by

r' = (1 - (1/2)*cos(theta))*(r - r^2) = f(r,theta)

(theta)' = 1 = g(r,theta)

and I'm asked to find a periodic solution r(t), and theta(t) for the system. (for this purpose theta = 2*k*pi , with k is an integer, is identified with theta=0)
I really don't know how to solve it, any help please?

 

[rande09]

before I get started I want to ensure we're talking about the same thing.

do you mean

\(\displaystyle \frac{dr}{dt}[t]= f[r[t],\theta[t]]=(1-\frac{1}{2}cos[\theta[t]])*(r[t]-r^2[t])\)

\(\displaystyle \frac{d\theta}{dt}[t]=g[r[t],\theta[t]]=1\)

yes that looks like that is it