Example of $\omega$-limit set with two singular points (at least). According to Poincaré-Bendixson Theorem, if an orbit in the plane contains only finitely many fixed points, then the $\omega$-limit set is either

*

*a fixed point,

*a periodic orbit, or

*a connected set composed of a finite number of fixed points together with orbits connecting them.

We can see examples of the first and the second cases with the vector field $F(x,y)=(-\sin(x),y)$.
What are some examples of the third case?

The following example is given on p. 224 in Gerald Teschl's book Ordinary Differential Equations and Dynamical Systems:
$$
\dot x = y+x^2 - \tfrac14 x (y-1+2x^2)
,\qquad
\dot y = -2(1+y)x
.
$$

