Computing the Poincare Map of a Dynamical System

Question

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.

Answer 1

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 $$ f(r_0)= \frac{1}{1-\frac{r_0 - 1}{r_0} e^{-2\pi} }. $$

Answer 2

Hints:

• This is in polar coordinates, because it would not make sense otherwise.
• The Poincaré map gives the relation between one intersection of $S$ and the next.
• $\dot{r}=r(1-r)$ can be solved analytically.
• The two differential equations are uncoupled, thus allowing you to easily determine the time points of intersections.