Catastrophe Theory

Catastrophe theory studies how the qualitative nature of equation solutions depends on the parameters that appear in the equations. Subspecializations include bifurcation theory, nonequilibrium thermodynamics, singularity theory, synergetics, and topological dynamics. 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, (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 (Goetz).

Control Factors	1 Behavior Axis	2 Behavior Axes
1	Fold
2	Cusp
3	Swallowtail	Hyperbolic Umbilic, Elliptic Umbilic
4	Butterfly	Parabolic Umbilic

References

Catastrophe Theory

Arnold, V. I. Catastrophe Theory, 3rd ed. Berlin: Springer-Verlag, 1992.

Gilmore, R. Catastrophe Theory for Scientists and Engineers. New York: Dover, 1993.

Goetz, P. ``Phil's Good Enough Complexity Dictionary.'' http://www.cs.buffalo.edu/~goetz/dict.html.

Saunders, P. T. An Introduction to Catastrophe Theory. Cambridge, England: Cambridge University Press, 1980.

Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, p. 211, 1987.

Thom, R. Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Reading, MA: Reading, MA: Addison-Wesley, 1993.

Thompson, J. M. T. Instabilities and Catastrophes in Science and Engineering. New York: Wiley, 1982.

Woodcock, A. E. R. and Davis, M. Catastrophe Theory. New York: E. P. Dutton, 1978.

Zeeman, E. C. Catastrophe Theory--Selected Papers 1972-1977. Reading, MA: Addison-Wesley, 1977.