operator is applied again. If none of these are the case, then the cube is split in half, and both pieces are added back to the collection of cubes to inspect. This process repeats until all of the cubes have been removed or reduced to a sufficiently small size.

The output of Algorithm \[alg:BranchAndPrune\] is three collections of cubes: ${\mathcal{A}}$, ${\mathcal{B}}$, and ${\mathcal{R}}$ (see Figure \[fig:Verified\]). In Theorem \[prop:BnB\] we show that these sets have the properties that, $(i)$ each cube in ${\mathcal{A}}$ has a unique solution with respect to $\alpha$, $(ii)$ the cubes in ${\mathcal{B}}$ are near the Hopf bifurcation, with any solutions contained therein residing on the principal branch, and $(iii)$ all solutions to $F=0$ are contained in $\bigcup {\mathcal{A}}\cup {\mathcal{B}}\cup {\mathcal{R}}$.

Ideally ${\mathcal{R}}= \emptyset$, and this will often be the case if the zeros of $F$ are simple and the algorithm is allowed to run a sufficiently long time. However we are trying to verify not just simple, isolated solutions, but a 1-parameter family of solutions. As such, sometimes when a cube is split in two this division will bisect the curve of solutions (see Figure \[fig:BranchANDBound\]). When this occurs the algorithm will be forced to subdivide 