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,

3. Swallowtail catastrophe,

4. Butterfly 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 factors1 behavior axis2 behavior axes
3swallowtail hyperbolic umbilic, elliptic umbilic
4butterflyparabolic umbilic

The following table gives prototypical examples for equations showing each type of catastrophe.

See also

Butterfly Catastrophe, Catastrophe Theory, Cusp Catastrophe, Elliptic Umbilic Catastrophe, Fold Catastrophe, Hyperbolic Umbilic Catastrophe, Parabolic Umbilic Catastrophe, Swallowtail Catastrophe


Sanns, W. Catastrophe Theory with Mathematica: A Geometric Approach. Germany: DAV, 2000.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Catastrophe." From MathWorld--A Wolfram Web Resource.