For the given system: $$x'=-y+x(1-x^2-y^2),~y'=x+y(1-x^2-y^2),$$ four parts have been asked. (a) Show that $p(t)=(\cos t , \sin t)^T$ is a periodic solution. (b) Rewrite interms of polar coordinates $(r,\theta).$ (c) Let $\Sigma$ be the ray $\theta=\theta_{0}$ through the origin. Show that the Poincare map $P:\Sigma \rightarrow \Sigma$ is given by $$P(r_0)=\left[ 1+ \left( \frac{1}{r_0^2}-1 \right)e^{-4 \pi} \right]^{-1/2}$$ and that $P(1)=1.$ (d) Use the Poincare map $P$ to show that the periodic solution $p(t)$ is orbitally asymptotically stable.
Part (a) is pretty straight forward and part (b) gives $$r \cdot \left( \frac{1-r}{1+r} \right)^{1/2}=e^t~~~~~~\theta=t.$$
Can someone help me with parts (c) and (d) ? We haven't covered about Poincare maps in class. Thank you for your time.