Fold catastrophe

Example

The fold catastrophe is the one-parameter family of functions $$V_a(x) = x^3 + a x, \qquad x, a \in \R.$$ Its critical points solve $V_a'(x) = 3x^2 + a = 0$.

1. For $a < 0$ there are two critical points $x = \pm \sqrt{-a/3}$; since $V_a''(x) = 6x$, the point $-\sqrt{-a/3}$ is a local maximum and $\sqrt{-a/3}$ is a local minimum.
2. For $a = 0$ there is one critical point, $x = 0$, and it is degenerate, since $V_0''(0) = 0$.
3. For $a > 0$ there are no critical points.

So as $a$ increases through $0$, the minimum and the maximum approach each other, merge at $x = 0$ and disappear. The bifurcation set, the set of parameters where a degenerate critical point occurs, is the single point $a = 0$. The family is the universal unfolding of the germ $x^3$. Indeed, the derivative of $V_a$ in $a$ is $x$, and $1$ and $x$ span $\mathcal{E}_1$ modulo the ideal $(x^2)$.