Lattices from Morse-Bott and attractor structures provide sufficient topological data to describe bifurcations. Equivariant cohomology can recover information about orbits and local fixed point behaviour, linking to an equivariant Morse description. Bifurcations of fixed points can be captured by topological data in these reduced systems. Deformations of Lie algebras can be tied to the influence of controls, finding classes of infinitesimal symmetry breaking perturbations. Analysing the effects of deformation cocycles through topological data can contribute to a higher dimensional understanding of bifurcations, which have natural extensions to more general classes of systems.