The geometry of bifurcation surfaces in parameter space. I. A walk through the pitchfork

The classical pitchfork of singularity theory is a twice-degenerate bifurcation that typically occurs in dynamical system models exhibiting Z_2 symmetry. Non-classical pitchfork singularities also occur in many non-symmetric systems, where the total bifurcation environment is usually more complex. In this paper three-dimensional manifolds of critical points, or limit-point shells, are introduced by examining several bifurcation problems that contain a pitchfork as an organizing centre. Comparison of these surfaces shows that notionally equivalent problems can have significant positional differences in their bifurcation behaviour. As a consequence, the parameter range of jump, hysteresis, or phase transition phenomena in dynamical models (and the physical systems they purport to represent) is determined by other singularities that shape the limit-point shell.