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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 hyperbolic umbilic

Germ

V = x3 + y3

Unfolding

V(a, b, c) = x3 + y3 + axy + bx + cy

An applet

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

  • Bottom : Control panel. The hyperbolic 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, one maximum and two 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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