Take a system of equations:

$\displaystyle \dot{r}=r(1-r^2)$

$\displaystyle \dot{\theta}=1$

Let S be the positive x-axis and $\displaystyle r_0$ is an initial condition on S.

Since $\displaystyle \dot{\theta}=1$ the first return to S occurs at $\displaystyle t=2\pi$.

Then $\displaystyle r_1=P(r_0)$ where $\displaystyle r_1$ satisfies

$\displaystyle \int_{r_0}^{r_1}\frac{dr}{r(1-r^2)} = \int_0^{2\pi}dt = 2\pi \Rightarrow r_1$

Hence $\displaystyle P(r)=[1+e^{-4\pi}(r^{-2}-1)]^{-\frac{1}{2}}$

I don't understand how the "hence" bit is derived. I've spent a fair amount of time googling, and I believe partial fractions may hold part of the solution, but honestly I'm a little lost...

Many thanks!!!