I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the angular position, $v$ is the velocity, $b>0$ measures the damping force and $k\geq0$ the applied torque.
The question ask, Suppose $k>1$ Prove that there exist a periodic solution for this system.
My original idea was to use the Poincaré-Bendixon theorem in our book which says that if $\Omega$ is a closed, and bounded limit set of a planar system of differential equations that contains no equilibrium point. Then $\Omega$ is a closed orbit.
If we look at the $v$ nullcline, we can see that the vector field in the region above (say when $v=v_2$) points downwards. Above (say when $v=v_1$) it points upwards.
We also know that when $k>1$ there are no longer any equilibrium points.
But...this version of the Poincaré-Bendixon theorem is used for planar systems, where I guess in this problem we are working on a cylinder. Is there another way to look at this question, or a different theorem to use. Any help would be great!