I have a system of ordinary differential equations
$ \theta' = v$
$ v' = -bv - \sin \theta + k$
These are the equations for a pendulum with $\theta$ being angular position, and $v$ being angular velocity. The constant $b > 0$ is a damping constant, and $k \geq 0$ is a constant torque in the counterclockwise direction. I am asked to prove that for $k > 1$ there exists a unique periodic solution for this system. There are two hints:
The first hint, to prove existence, says to consider a vector field in a strip of the form $0 < v_1 < (k - \sin \theta)/b < v_2$.
The second hint, to prove uniqueness, says to consider $E(\theta, v) = 1/2 v^2 - \cos \theta + 1$ and use the fact that the total change of E along any periodic solution must be $0$.
So far I've noticed that the if we set $v' = 0$ we get that $v = (k - \sin \theta) / b$ is a v-nullcline. Therefore the strip $0 < v_1 < (k - \sin \theta)/b < v_2$ is divided by this nullcline into two regions. I'm thinking I should consider some solution with initial condition in this strip and show that this is periodic but I'm not sure how to go about doing that exactly.