Importance of finitude in the Poincaré-Bendixson theorem I have a question regarding the Poincaré-Bendixson theorem:

I understand everything, but I'm looking for an example that shows the importance of the finitude of the singular points within the theorem. What happens when we have an infinite number (countable or uncountable) of singular points?
Until now I have only been able to find examples that teach me or show me how this theorem works but I would like to see an example about the importance of the finitude of said singular points.
 A: You may start out with this vector-field:
$$ v = \pmatrix{ -y\\x}+(1-r^2)\pmatrix{x\\y}.$$
It has one critical point at the origin. Any other initial point will have the unit circle as its $\omega$-limit (so a typical periodic case in the P-B theorem).
Now multiply this vector field with $(1-r^2)^2$. Then the unit circle becomes a continuum of critical points since
the vector field vanishes there. On the other hand any non-critical point will still have the full unit-circle as $\omega$-limit.
The above is of course rather trivial as the factor $(1-r^2)^2$ simply slows the orbit down when approaching its $\omega$-limit. It may, however, be used in a more general construction that relies on some complex analysis: Take any Jordan curve $\gamma$ (topologically a circle) in the plane. I claim that you may realize that curve as an $\omega$-limit of a $C^1$ (or even $C^\infty$) vector field which has precisely  $\gamma$ and one more point as critical sets.
To simplify the topology complete the plane with a point at infinity so it becomes the Riemann sphere.
The curve $\gamma$ separates the sphere  into two simply-connected components $D_1$ and $D_2$ (suppose $D_2$ contains infinity).
The Riemann mapping theorem tells you that there is a conformal map $\phi_1$ taking the unit disc ${\Bbb D}$  conformally onto $D_1$ and similarly a map $\phi_2$ from the exterior of $\overline{\Bbb D}$ to $D_2$ (taking infinity to infinity). Now transport the vector field defined above from the unit disk to $D_1$ by $\phi_1$ and to $D_2$ by $\phi_2$.  The derivative of the Riemann mappings typically blows up when approaching the boundary so a priori the vector-field will be singular on $\gamma$. However, by multiplying with a suitable scalar function going to zero fast enough (some subtle variation of the type $\exp(-1/(1-r^2)^2)$ you may kill the blow-up and assure that the resulting vector field becomes even $C^\infty$ and has precisely $\gamma$ plus one more point (the image of zero by $\phi_1$) as critical points. There are more general constructions in which for example any connected Julia set of a polynomial  may be realized as $\omega$-limit and critical set (plus possibly some discrete set of critical points).
