Bifurcation Sets

Given a smooth function f: Rⁿ -> R, a k-unfolding of f is a smooth function F: R^n+k -> R, where F(p, 0) = f(p). We consider f as one element in a family of functions, and we consider all other functions in the family as slight perturbations of f. An unfolding is versal if every other unfolding can be induced from it (see, for instance, [BrG] or [G]) . A versal folding is universal if k is as small as possible.

Given a k-unfolding of a function f, the bifurcation set is the collection of points q in R^k such that F( ^. , q) has a degenerate singularity. A useful fact is that if F is a k-versal unfolding, the bifurcation set is an invariant of the singularity type of f. This fact is useful for studying objects like evolutes ([P2]), Gauss maps ([BaGM1], [BrGT1]), and binormal vectors.

If we have a versal 3-unfolding, we can actually see the bifurcation set. The following singularities can be expressed using two variables u and v, and they are versally unfolded using three (or less) parameters (the t_i's). The pictures of the bifurcation sets follow. (The fold, or A₂ singularity, is unfolded by one parameter, and so technically belongs on this list. But its bifurcation set is just a regular surface, and so it does not look very interesting. This list is a subset of the list of all singularities types of codimension less than 6, found by Thom. This result can be found in many sources; I would suggest [G].)

Singularity	Normal Form	Unfolding
Cusp (A₃)	u⁴ + v²	u⁴ + v² + t₁ u² + t₂ u
Swallowtail (A₄)	u⁵ + v²	u⁵ + v² + t₁ u³ + t₂ u² + t₃ u
Elliptic Umbilic (D₄^-)	u² v - v³	u² v - v³ + t₁ u² + t₂ u + t₃ v
Hyperbolic Umbilic (D₄⁺)	u² v + v³	u² v + v³ + t₁ u² + t₂ u + t₃ v


Cusp	Swallowtail

Hyperbolic Umbilic	Elliptic Umbilic

Bifurcation sets of the four singularities

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Last updated 2/26/00