We present a detailed mathematical study of a truncated normal form relevant to the bifurcations observed in wake flow past axisymmetric bodies, with and without thermal stratification. We employ abstract normal form analysis to identify possible bifurcations and the corresponding bifurcation diagrams in parameter space. The bifurcations and the bifurcation diagrams are interpreted in terms of symmetry considerations. Particular emphasis is placed on the presence of attracting robust heteroclinic cycles in certain parameter regimes. The normal form coefficients are computed for several examples of wake flows behind buoyant disks and spheres, and the resulting predictions compared with the results of direct numerical flow simulations. In general, satisfactory agreement is obtained.