Poincare Map for Polar ODEI am currently trying to obtain a Poincare Map for the ODE system originally given by{x˙=(1−x2−y2)x−yy˙=x+(1−x2−y2)yon the region&nbsp;x∈(12,32)&nbsp;and&nbsp;y=0. Since&nbsp;x2+y2=r2&nbsp;and&nbsp;tan⁡(θ)=yx, we obtain that{r˙=xx˙+yy˙rsec2⁡(θ)θ˙=xy˙−yx˙x2&nbsp;⟹&nbsp;{r˙=r−r3θ˙=1However, 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&nbsp;(x,y)=(1,0)&nbsp;(or in polar coordinates&nbsp;(r,θ)=(1,0)?

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Poincare Map for Polar ODEI am currently trying to obtain a Poincare Map for the ODE system originally given by{x˙=(1−x2−y2)x−yy˙=x+(1−x2−y2)yon the region&amp;nbsp;x∈(12,32)&amp;nbsp;and&amp;nbsp;y=0. Since&amp;nbsp;x2+y2=r2&amp;nbsp;and&amp;nbsp;tan⁡(θ)=yx, we obtain that{r˙=xx˙+yy˙rsec2⁡(θ)θ˙=xy˙−yx˙x2&amp;nbsp;⟹&amp;nbsp;{r˙=r−r3θ˙=1However, 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&amp;nbsp;(x,y)=(1,0)&amp;nbsp;(or in polar coordinates&amp;nbsp;(r,θ)=(1,0)?

elvis0217t2x · Accepted Answer

Explanation:Obviously you return to y=0 with a positive x after a full rotation, t=θ=2π. Now solve the Bernoulli equation for the radius(r−2−1)′=−2(r−2−1)⇒(r(2π)−2−1)=e−4π(r(0)−2−1).

Poincare Map for Polar ODE I am currently trying to obtain a Poincare Map for the ODE system ori

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