Consider the following system given in polar coordinates $r'=r(r^2-5r\cos\theta - 6)$ and $\theta ' = 1$. Prove that there is a periodic orbit for the system.
I know that Poincaré-Bendixson theorem says that if there is an annular region R in which the vector field points towards the interior and there are no critical points in the region R then a periodic orbit exists in that region. I am trying to find an annulus of the form $a < r < b$ such that this annulus contains periodic orbit. Letting $r=1$ gives $r' \le 0$ and $r=6$ gives $r' \ge 0$ which is exactly opposite of what I want.
So I am not able to find values for $a, b$ for the annulus region $a < r <b$ such that $r=a$ will give $r'>0$ and $r=b$ will give $r'<0$ so that the field is pointing towards the region. Any suggestions?
You just want to prove the existence of a periodic orbit, it makes no difference if it is stable or unstable. So just consider the time-reversed system where the vector field points in the opposite direction at every point. Then the formulation of Poincaré-Bendixson that you have in mind applies, there is a periodic orbit, and it stays periodic if you reverse the time reversion.