Consider the system $$x' = −y + x(r^4 − 3r^2 + 1)$$ $$y' = x + y(r^4 − 3r^2 +1)$$ where $$r^2=x^2 + y^2$$ Question: Show that $r' < 0$ on the circle $r = 1$ and $r' > 0$ on the circle $r = 2$. Use the Poincare Bendixon theorem and the fact that the only equilibrium point of this system is at the origin to show that there is a periodic orbit in the annular region $$A = {x ∈ R^2 : 1 < |x| < 2}$$ Why does the periodic orbit not have any point common to the boundary of A?
I was able to do the first part and found that $r' < 0$ on the circle $r = 1$ and $r' > 0$ on the circle $r = 2$. I am currently stuck on how to use the Poincare Bendixon Theorem to show that there is an orbit in that given annular region.
Poincare-Bendixon Theorem- Suppose that $f ∈ C^1 (E)$ where $E$ is an open subset of $R^2$ and that $x'=f(x)$ has a trajectory $Γ$ with $Γ^+$ contained in a compact subset $F$ of $E$. Then if $ω(Γ)$ contains no critical points of $x'=f(x)$, $ω(Γ)$ is a periodic orbit of $x'=f(x)$