Bifurcation set

Remarks

Example: the cusp family $V(x, a, b) = x^4/4 + a x^2/2 + b x$ with $n = 1$, $k = 2$. Clause 1 reads $x^3 + a x + b = 0$ and clause 2 reads $3x^2 + a = 0$. Then $a = -3x^2$ and $b = 2x^3$, and eliminating $x$ gives $B_V = \set{(a, b) : 4a^3 + 27b^2 = 0}$, a curve with a cusp at the origin.

Non-example: for $V(x, c) = x^2 + c x$ the only critical point is $x = -c/2$, and $V'' = 2 \ne 0$, so clause 2 always fails and $B_V$ is empty.

Let $M = \set{(x, c) : \nabla_x V(x, c) = 0}$ and let $\chi \colon M \to \R^k$ be the projection $(x, c) \mapsto c$. When $M$ is a smooth submanifold of dimension $k$, which holds for generic families, $B_V$ is the set of critical values of $\chi$. Some authors also add the Maxwell set, where two critical values of $V(\cdot, c)$ are equal; this node uses the narrower meaning.