 Solution

For the unforced undamped pendulum, the upper homoclinic separatrix has energy $E=p^2/2-\cos \theta =1$ and $p_0(\theta )=2\cos (\theta /2)$ for $-\pi \le \theta \le \pi$. With forcing and damping,The first-order energy change along this separatrix isThe zero of this first-order stable/unstable-manifold splitting is simple in $F$. Smooth dependence of the saddle manifolds and the implicit function theorem therefore give a nearby homoclinic branch withThe asymptotic formula is the leading-order stable/unstable-manifold balance. On a lower homoclinic orbit, $p\le 0$ and the lifted angle decreases by $2\pi$, so the exact integrated energy balance would bewhen $F>0$. Hence no lower homoclinic orbit exists for positive forcing.