= 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 $-\pi\leq\theta\leq\pi$. With forcing and damping,
$$
\dot E=Fp-kp^2.
$$
The first-order energy change along this separatrix is
$$
\Delta E=F\int_{-\pi}^{\pi}d\theta-k\int_{-\pi}^{\pi}p_0(\theta)\,d\theta=2\pi F-8k.
$$
The 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 with
$$
\boxed{F_h(k)=\frac4\pi k+O(k^2).}
$$
The asymptotic formula is the leading-order stable/unstable-manifold balance. On a lower homoclinic orbit, $p\leq0$ and the lifted angle decreases by $2\pi$, so the exact integrated energy balance would be
$$
0=-2\pi F-k\int_{-\infty}^{\infty}p^2\,dt<0
$$
when $F>0$. Hence \b[no lower homoclinic orbit exists for positive forcing].
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