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CATASTROPHE TEACHER

an introduction for experimentalists

 


The seven elementary catastrophes

the fold |  the cusp |  the swallowtail |  the butterfly |  the hyperbolic umbilic |  the elliptic umbilic |  the parabolic umbilic |

Catastrophes in systems with two state variables:

The elliptic umbilic

Germ

V = x3 - 3xy2

Unfolding

V(a, b, c) = x3 - 3xy2 + a(x2y2) + bx + cy

An applet

The elliptic umbilic properties will be studied by means of this applet.

  • Bottom : Control panel. The elliptic umbilic has three control parameters;
  • Left : space control representation, control parameters values and bifurcation set on a plane (a = A value) through the three dimensional control space.
  • Middle : surface of potential as a "geographical map". Grey levels convey information about the potential value, coordinates on the plane are state variable values. Darkest greys are minimums.
  • Right : state variables equilibrium values on a two-dimensional color scale
    Stability of state variable equilibrium may be tested by clicking on a potential map point.
  • Bifurcation set

    Hyperbolic umbilic germ may be thought as the collapse of one minimum or one maximum and three saddles.

    the fold |  the cusp |  the swallowtail |  the butterfly |  the hyperbolic umbilic |  the elliptic umbilic |  the parabolic umbilic |

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    Author: Lucien Dujardin

    Faculté de Pharmacie BP 83 F 59006 Lille Cedex France

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