The geometry of bifurcation surfaces in parameter space. I. A walk through the pitchfork
Rowena Ball
TL;DR
This work investigates how a pitchfork bifurcation organizes the geometry of limit-point shells in 3D parameter spaces across multiple dynamical problems. It develops the theory from Whitney's cusp to the prototypic pitchfork via a universal unfolding $G(x, heta,eta)=x^3- heta x+eta x^2$, and constructs the limit-point shell $L_p$ by unfurling along the extra parameter, highlighting the symmetry of the shell. The CSTR problem yields an asymmetric, convex shell $L_c$ with embedded bifurcations $H$, $T$, $I$, and $A$, and shows that a locally equivalent, simpler unfolding $P_{ TI}$ fails to predict global multiplicity boundaries. In the L–H transition context, the partially unfolded model $P_{LH}$ produces two realizable pitchforks $P_{1 LH}$ and $P_{3 LH}$ connected by hysteresis and an $E_2$ singularity, underscoring the global topological complexity of limit-point shells and the value of 3D visualization for design and control of dynamical systems.
Abstract
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.