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Catastrophe

For any system that seeks to minimize a function, only seven different local forms of catastrophe "typically" occur for four or fewer variables:

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 1 fold 2 cusp 3 swallowtail hyperbolic umbilic, elliptic umbilic 4 butterfly parabolic umbilic

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

 equation catastrophe fold catastrophe cusp catastrophe, Riemann-Hugoniot catastrophe swallowtail catastrophe hyperbolic umbilic catastrophe elliptic umbilic catastrophe butterfly catastrophe parabolic umbilic catastrophe

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

References

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

Catastrophe

Cite this as:

Weisstein, Eric W. "Catastrophe." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Catastrophe.html