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.

up vote 2 down vote accepted

The Poincaré map $f(r_0)$ is the radius of the first intersection with $S$ of the solution starting at $(0,r_0)$ : Poincaré map 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 $$ f(r_0)= \frac{1}{1-\frac{r_0 - 1}{r_0} e^{-2\pi} }. $$

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.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.