Q:

Computing the Poincare Map of a Dynamical System

Consider the two-dimensional system $\dot{r} = r(1 − r)$ and $\dot{\theta} = 1$ Compute the
Poincare map for the local section $S = \{(r, θ) | θ = 0, r ∈ (0, 2)\}$. 
I'm not really sure how to proceed. I know what a Poincare map is (definitionally), but I'm not sure how to compute it for this particular system.

A:

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ :

Note that this intersection is at $\theta=t=2\pi$.
The solution of the initial value problem
$$
\dot r= r(1-r),\quad r(0)=r_0
$$
is
equal to
$$r(t)= \frac{1}{1-\frac{r_0 - 1}{r_0} e^{-t} },$$
thus, the radius of the next intersection is
$$
r(2\pi)= \frac{1}{1-\frac{r_0 - 1}{r_0} e^{-2\pi} }.
$$
Finally, the Poincaré map is
$$
f(r_0)= \frac{1}{1-\frac{r_0 - 1}{r_0} e^{-2\pi} }.
$$

