Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 30A / b

The Poincare-Bendixson theorem says that a nonempty compact omega-limit set of a smooth planar flow containing no equilibrium is a periodic orbit; use its allowed cylinder version here. For $F>1$, choose $p_{-}=(F-1)/(2k)$ and $p_{+}=2(F+1)/k$. At the lower boundary $\dot{p}>0$ and at the upper boundary $\dot{p}<0$, so the compact annulus $p_{-}\le p\le p_{+}$ is positively invariant. It has no equilibrium because $F-\sin \theta >0$. The theorem supplies a periodic orbit, and $\dot{\theta }=p>0$ makes it encircle the cylinder.

Every periodic orbit must lie in $p>0$: for $p\le 0$, $\dot{p}>0$, and a crossing of $p=0$ can only be upward. In $p>0$ a periodic orbit is a graph of a function satisfyingTwo distinct function graphs cannot cross by uniqueness of this scalar differential equation. If $p_1>p_2$, their difference obeysIt contracts by a factor strictly less than $1$ during one circuit, contradicting periodicity of both function graphs. The scalar return map is order preserving, so a multi-circuit periodic graph of a function also cannot have a period larger than one circuit. Therefore there is exactly one periodic orbit, and it encircles the cylinder. The standard auxiliary results used are existence/uniqueness of smooth differential equations and invariance at inward-pointing boundaries.