Butterfly catastrophe

Example

The butterfly catastrophe is the family of functions of one variable $x \in \R$ $$V(x; a, b, c, d) = x^6 + a x^4 + b x^3 + c x^2 + d x,$$ with four control parameters $(a, b, c, d) \in \R^4$. For $a = b = c = d = 0$ the function is $x^6$, whose critical point at $0$ is degenerate: the first five derivatives vanish there.

The derivative $V'(x) = 6x^5 + 4a x^3 + 3b x^2 + 2c x + d$ has degree $5$, so $V$ has at most five critical points. Since $V(x) \to +\infty$ as $x \to \pm\infty$, nondegenerate critical points alternate minimum, maximum, minimum, maximum, minimum. So $V$ can have three local minima, which the fold, cusp and swallowtail cannot.

Check with $a = -15/2$, $b = 0$, $c = 12$, $d = 0$. Then $V(x) = x^6 - \tfrac{15}{2} x^4 + 12 x^2$ and $$V'(x) = 6x^5 - 30x^3 + 24x = 6x(x^2 - 1)(x^2 - 4).$$ The critical points are $-2, -1, 0, 1, 2$. The sign of $V'$ is negative on $(-\infty, -2)$ and changes at each of the five simple roots. So $-2, 0, 2$ are local minima and $-1, 1$ are local maxima. The region of parameters with three minima is the "pocket" of the butterfly, bounded by the bifurcation set.