is not always sufficient, and we combine several additional tests to create a single *pruning operator* (see Section \[sec:Prune\]). One problem is that $y \equiv 0$ is always a trivial periodic solution to . To avoid this pitfall, we use Lemma \[prop:zeroneighborhood2\] which rules out small periodic solutions [@BergJaquette]. A further difficulty is that at the Hopf bifurcation, the principal branch of periodic solutions is pinched to a point as their amplitudes approach zero. To handle this case, we use Lemma \[prop:BifNbd\] which explicitly gives a neighborhood about the Hopf bifurcation within which the only solutions that could exist are on the principal branch [@BergJaquette]. Lastly, and most simply, if we can directly show that $\|F\|$ is bounded away from zero on a cube $X_n$, then $(b)$ holds.

Algorithm \[alg:BranchAndPrune\] follows the standard format of a global branch and bound method. In short, for a collection of cubes we successively prune each of its cubes. If $(a)$ holds for a given cube, then it is set aside and added to a list of solutions. If $(b)$ holds for a given cube, then that cube is discarded. If the pruning operator significantly reduces the size of a cube, then the pruning 