  There is widespread agreement that ice sheets flowed into the ocean in
tropical latitudes at sea level during the Earth's past. Whether these extreme
ice ages were snowball Earth events, with the entire surface covered in ice, or
whether ocean water remained ice free in regions about the equator, continues
to be controversial. For the latter situation to occur, the effect of positive
ice albedo feedback would have to be damped to stabilize an advancing ice sheet
shy of the equator. In this paper we analyze a conceptual model comprised of a
zonally averaged surface temperature equation coupled to a dynamic ice line
equation. This difference equation model is aligned with the cold world of
these great glacial episodes through an appropriately chosen albedo function.
Using the spectral method, the analysis leads to a nonsmooth singular
perturbation problem. The Hadamard graph transform method is applied to prove
the persistence of an invariant manifold, thereby providing insight into model
behavior. A stable climate state with the ice line resting in tropical
latitudes, but with open water about the equator, is shown to exist. Also
presented are local smooth and nonsmooth bifurcations as parameters related to
atmospheric CO$_2$ concentrations and the efficiency of meridional heat
transport, respectively, are varied.
