Catastrophe theory

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Hyperbolic umbilic catastrophe

A catastrophe which can occur for three control factors and two behavior axes. The hyperbolic umbilic is the universal unfolding of the function germ . The codimension of is 3, and therefore the universal unfolding of has three unfolding parameters.

Catastrophe

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 axes1fold2cusp3swallowtail hyperbolic umbilic, elliptic umbilic4butterflyparabolic umbilicThe following table gives prototypical examples for equations showing each type of catastrophe.equationcatastrophefold catastrophecusp catastrophe, Riemann-Hugoniot catastropheswallowtail..

Butterfly catastrophe

A catastrophe which can occur for four control factors and one behavior axis. The butterfly catastrophe is the universal unfolding of the singularity of codimension 4, i.e., with four unfolding parameters. It has the form .The equations(1)(2)display such a catastrophe (von Seggern 1993).

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