Cusp catastrophe

Example

The cusp catastrophe is the two-parameter family of functions $$V_{a,b}(x) = x^4 + a x^2 + b x, \qquad x, a, b \in \R.$$ Its critical points solve $4x^3 + 2a x + b = 0$, and such a point is degenerate when also $12 x^2 + 2a = 0$.

Bifurcation set: a degenerate critical point $x$ has $a = -6x^2$ and then $b = -4x^3 - 2ax = 8x^3$. Eliminating $x$ gives the curve $$8 a^3 + 27 b^2 = 0,$$ a semicubical parabola with a cusp at the origin, lying in the half-plane $a \le 0$. Check: with $a = -6x^2$ and $b = 8x^3$, one has $8a^3 = -1728 x^6$ and $27 b^2 = 1728 x^6$.

Number of critical points: the cubic $4x^3 + 2ax + b$ has discriminant $-16(8a^3 + 27b^2)$. So for $8a^3 + 27b^2 < 0$, inside the cusp, there are three critical points, two local minima and one local maximum between them. For $8a^3 + 27b^2 > 0$, outside the cusp, there is one critical point, a minimum. The family is the universal unfolding of the germ $x^4$, whose codimension is $2$.