Let $$x'=1+y-x^2-y^2$$ $$y'=1-x-x^2-y^2$$
How do I use the Poincare map to show that this has a not asymptotically stable solution?
What I did:
I transformed the system to polar coordinates
$$r'=(\cos\theta+\sin\theta)(1-r^2)$$
$$\theta'=(\cos\theta-\sin\theta)(1/r-r)-1$$
which has a periodic solution $(r,\theta)=(1,-t)$ or $f(t)=(x,y)=(\cos(t),-\sin(t))$.
How do I find the Poincare map? And how do I show that $f(t)$ is not aymptotically stable?
Edit:
The original question gives the system as above and asks for an explicit periodic solution $f(t)$ (which I'm sure is the one I found above).
Show that $f(t)$ is stable (but not asymptotically stable). What can you say about the Poincare map?