I'm trying to understand applications of the Poincaré-Bendixson theorem. The Poincaré-Bendixson theorem goes as follows:
Poincaré-Bendixson Theorem: Consider the equation $\dot{x} = f(x)$ in $\mathbb{R}^2$ and assume that $\gamma^+$ is a bounded, positive orbit and that $\omega(\gamma^+)$ contains ordinary points only. Then $\omega(\gamma^+)$ is a periodic orbit. If $\omega(\gamma^+)\neq \gamma^+$ the periodic orbit is called a limit cycle. An analogous result is valid for a bounded, negative orbit.
(Here $\omega(\gamma^+)$ is called the $\omega$-limitset and denotes the set of positive limitpoints of $\gamma^+$.)
In the example below the Bendixson criterium is used as well:
Bendixson's Criterium: Suppose that the domain $D\subset \mathbb{R}^2$ is simply connected; $(f,g)$ is continuously differentiable in $D$. The equation $\dot{x} = f(x)$ can only have periodic solutions if $\nabla .(f,g)$ changes sign in $D$ or if $\nabla . (f,g) = 0$ in $D$.
The Poincaré-Bendixson theorem is illustrated in the following example:
Example: Consider the following system $$ \begin{align} \dot{x} &= x(x^2 + y^2 -2x - 3) - y\\ \dot{y} & = y(x^2 + y^2 - 2x -3) + x \end{align} $$ The only critical point is $(0,0)$; this is a spiral point with positive attraction. To see whether closed orbits are possible apply the Bendixson criterium. We find for the divergence of vector function on the righthand-side $$ 4x^2 + 4y^2 -6x - 6 = 4[(x - \frac{3}{4})^2 + y^2 -\frac{33}{16}] $$ Inside the Bendixson circle with centre $(\frac{3}{4}, 0)$ and radius $\sqrt{33}/4$ the expression is sign definite and no closed orbits can be contained in the interior of this circle.
Closed orbits are possible which enclose or which intersect the Bendixson-circle. We transform the system to polar coordinates by $x = r\cos\theta$ and $y = r\sin\theta$ to find $$ \begin{align} \dot{r} &= r(r^2 - 2r\cos\theta -3)\\ \dot{\theta} &= 1 \end{align} $$ If $r<1$ we have $\dot{r} < 0$, if $r > 3$ we have $\dot{r} > 0$. According to the Poincaré-Bendixson theorem the annulus $1<r<3$ must contain one or more limit cycles.
Question:
Why does the Poincaré-Bendixson theorem show that there have to be limit cycles inside the annulus $1 < r< 3$? I understand how the Bendixson criterium is used and why there cannot be any closed orbits inside the circle with centre $(\frac{3}{4}, 0)$ and radius $\sqrt{33}/4$, but I don't get why the fact that $\dot{r} < 0$ when $r > 1$ and $\dot{r} > 0$ when $r<3$ insinuates that the are limit cycles inside the annulus.
