For any system that seeks to minimize a function, only seven different local forms of catastrophe "typically" occur for four or fewer variables:
1. Fold catastrophe,
2. Cusp catastrophe,
5. Elliptic umbilic catastrophe,
6. Hyperbolic umbilic catastrophe, and
7. Parabolic umbilic catastrophe.
More specifically, for any system with fewer than five control factors and fewer than three behavior axes, these are the only seven catastrophes possible. The following tables gives the possible catastrophes as a function of control factors and behavior axes.
|control factors||1 behavior axis||2 behavior axes|
|3||swallowtail||hyperbolic umbilic, elliptic umbilic|
The following table gives prototypical examples for equations showing each type of catastrophe.
|cusp catastrophe, Riemann-Hugoniot catastrophe|
|hyperbolic umbilic catastrophe|
|elliptic umbilic catastrophe|
|parabolic umbilic catastrophe|