Suppose we have the following system $$ \theta'=v $$ $$ v'= -bv - \sin\theta + k. $$ I need to prove that there exists a periodic solution in the region where $k>1$.
The hint that was given is to look at a strip $$ v_1 < (k - \sin\theta)/b < v_2. $$ I think I need to use the Poincaré-Bendixon Theorem that states 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.
Any ideas how I can work this out properly?