A fold catastrophe is a catastrophe which can occur for one control factor and one behavior axis. It is the universal unfolding
of the singularity and has the equation .
For a second-order dynamical system, a generic slow transverse passage through a fold can be reduced at leading order by translations
and scalings to
which is the equation defining the first Painlevé transcendent (Haberman 1979, Ribes Metidieri et al. 2026). This reduction
gives a universal description of the local dynamics near the fold.
Haberman, R. "Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems." SIAM J. Appl. Math.37,
69-106, 1979. https://doi.org/10.1137/0137006.Ribes
Metidieri, A.; Bonga, B.; Krishnan, B.; and Jaramillo, J. L. "Universality
in the Transition from Inspiral to Plunge: High-Accuracy Analytic Solutions and Catastrophe
Theory." 11 Jun 2026. https://arxiv.org/abs/2606.13786.Sanns,
W. Catastrophe
Theory with Mathematica: A Geometric Approach. Germany: DAV, 2000.
and has the equation 
.
