# Poincare Map for Polar ODE

I am currently trying to obtain a Poincare Map for the ODE system originally given by $$\begin{cases}\dot{x} = (1-x^2-y^2)x-y\\\dot{y} = x+(1-x^2-y^2)y\end{cases}$$ on the region $$x \in (1/2, 3/2)$$ and $$y = 0$$. Since $$x^2 + y^2 = r^2$$ and $$\tan(\theta) = \frac{y}{x}$$, we obtain that $$\begin{cases}\dot{r} = \frac{x\dot{x} + y\dot{y}}{r}\\\sec^2({\theta}) \dot{\theta} = \frac{x\dot{y} - y\dot{x}}{x^2}\end{cases} \implies \begin{cases}\dot{r} = r - r^3\\\dot{\theta} = 1\end{cases}$$ with $$r \in (1/2, 3/2)$$ and $$\theta = 0$$.

However, I am stuck here with trying to identify the Poincare Map for the given system. Are there any recommendations for how to proceed? Moreover, how can I linearize this system at the point $$(x, y) = (1, 0)$$ (or in polar coordinates $$(r, \theta) = (1, 0)$$?

Obviously you return to $$y=0$$ with a positive $$x$$ after a full rotation, $$t=θ=2\pi$$. Now solve the Bernoulli equation for the radius $$(r^{-2}-1)'=-2(r^{-2}-1)\implies (r(2\pi)^{-2}-1)=e^{-4\pi}(r(0)^{-2}-1).$$