In polar coordinates, consider the system $$ \dot{r}=\sin\pi r,\quad \dot{\theta}=\cos\pi r. $$ Prove that the system has two limit cycles with one lying interior to the other and with no equilibrium points between them. Moreover, the two limit cycles are oriented in opposite sense.
I just learned the Poincaré-Bendixson theorem, but do not know how to use it here.
Actually, we can solve the ODE directly and obtain $$ \dfrac{1}{2}\ln\dfrac{1-\cos\pi r}{1+\cos\pi r}=\pi t+C, $$ but it also seems of little use.
Appreciate any help, and if you may, give a clear clafication of the theorem you use.
When does $\dot{r}=0$? For example, when $r=1,2$ (and for infinitely many other values actually). In these cases, $\dot{\theta}$ is constant and with opposite sign for each cycle (which gives the opposite orientation).
So your flows are two concentric circles, one flowing clockwise and one counterclockwise. These are indeed both limit cycles, and to show that there are no equilibrium points between them, note that $\dot{r}$ does not vanish in between them.