Cusp bifurcation ($\dot{x}={\beta }_1+{\beta }_{2}x-x^3$)

A cusp is a two-parameter degeneracy at which two saddle-node bifurcation curves meet and the reduced steady equation has a triple root. In the displayed scalar normal form, simultaneous vanishing of the polynomial and its derivative gives ${\beta }_2=3x^2$, ${\beta }_1=-2x^3$. Thus the folds satisfy $4{\beta }_2^3=27{\beta }_1^2$, with ${\beta }_2\ge 0$. Between the folds there are three real equilibria, and outside them one. A transverse stable direction may be present in a higher-dimensional realization.