Catastrophe Theory
Catastrophe theory studies how the qualitative nature of equation solutions depends on the parameters that appear in theequations. 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 ofcatastrophe ``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 withfewer than five control factors and fewer than three behavior axes, these are the only seven catastrophes possible. The followingtables 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 | ParabolicUmbilic |
References
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.
Catastrophe Theory
- Fundamental Theorem of Algebra
- Fundamental Theorem of Arithmetic
- Fundamental Theorem of Curves
- Fundamental Theorem of Directly Similar Figures
- Fundamental Theorem of Gaussian Quadrature
- Fundamental Theorem of Genera
- Fundamental Theorem of Plane Curves
- Fundamental Theorem of Projective Geometry
- Fundamental Theorem of Space Curves
- Fundamental Theorem of Symmetric Functions
- Fundamental Theorems of Calculus
- Fundamental Unit
- Funnel
- Fuss's Problem
- Futile Game
- Fuzzy Logic
- FWHM
- Gabriel's Horn
- Gabriel's Staircase
- Gadget
- Gale-Ryser Theorem
- Galilean Transformation
- Gallows
- Gall's Stereographic Projection
- Gallucci's Theorem