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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 $(r, \theta) = (1, 0)$?

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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 $$ (r^{-2}-1)'=-2(r^{-2}-1)\implies (r(2\pi)^{-2}-1)=e^{-4\pi}(r(0)^{-2}-1). $$

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