1999/10/31 by Rowena Ball, Ball, Rowena
Computer Science · Physics and Astronomy · #58Fxx #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.math/9910176
openalex publication_date 1999/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical pitchfork of singularity theory is a twice-degenerate bifurcation that typically occurs in dynamical system models exhibiting Z2 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.