Constructing a Poincare Map

Question

I need to construct a Poincare Map of the following dynamical system:

$\dot x = x-(x+y)(x^2+y^2)$

and

$\dot y = y + (x-y)(x^2+y^2)$

I changed the system to polar coordinates which gives me:

$\dot r = -r(r^2-1)$

and

$\dot\theta = r^2$

Next, I found $dr \over d\theta$

$\dot r \over \dot\theta$ = $-r(r^2-1) \over r^2$ = $-(r^2-1) \over r$ = $dr \over d\theta$

My approach is then to solve for $d\theta$ and integrate the right hand side from $r_n$ to $r_{n+1}$ as such (The left hand side is just $2\pi$):

$\int d\theta= -\int {rdr \over (r^2-1)}$

After this, I plan to solve for $r_{n+1}$ in order to get the Poincare map. $r_{n+1}$ is the next intersection of the transverse curve through the trajectories of the dynamical system. Is this a correct approach?

Answer 1

Yes, it is the correct approach (notice only the wrong sign of $\dot r$).

Given $r_0>0$, the equation involving $dr/d\theta$ lets you compute $r=r(\theta,r_0)$ with $r(0,r_0)=r_0$. The Poincaré map is then given by $T(r_0)=r(2\pi,r_0)$, and computing $r(\theta,r_0)$ explicitly lets you compute $T$ explicitly.