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
A_{2} 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 (A_{3}) |
u^{4} + v^{2} | u^{4} + v^{2} + t_{1} u^{2} + t_{2} u |
Swallowtail (A_{4}) |
u^{5} + v^{2} | u^{5} + v^{2} + t_{1} u^{3} + t_{2} u^{2} + t_{3} u |
Elliptic Umbilic (D_{4}^{-}) |
u^{2} v - v^{3} | u^{2} v - v^{3} + t_{1} u^{2} + t_{2} u + t_{3} v |
Hyperbolic Umbilic (D_{4}^{+}) |
u^{2} v + v^{3} | u^{2} v + v^{3} + t_{1} u^{2} + t_{2} u + t_{3} v |
Cusp | Swallowtail |
Hyperbolic Umbilic | Elliptic Umbilic |
Bifurcation sets of the four singularities