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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 only one state variable:

The swallowtail catastrophe

Germ

V = x5

Unfolding

V(a, b, c) = x5 - ax3 - bx2 - cx

An applet

The swallowtail catastrophe properties will be studied by means of this applet.

  • Bottom : Control panel. The swallowtail catastrophe has three control parameters
  • Left : space control representation, control parameters values and bifurcation set on a plane (a = A) through the three dimensional control space
  • Right : state variable versus potential value; below, state variable equilibrium value on a color scale
    Stability of state variable equilibrium may be tested by clicking on a potential curve point.
  • Bifurcation set

    The control space is three-dimensional; bifurcation set is made of three surfaces of fold points (lines on a slice of control space) which meet on two lines of cusp points which themselves meet on a swallowtail point.
    When the parameters values go through a fold surface in control space, a minimum appears or disappears on the potential curve.
    When the parameters values cross a cusp line in control space, two minimums and one maximum meet together.
    When the parameters values are at swallowtail point in control space, the situation is more sophisticated; At this point in control space, potential curve (germ) has only one minimum which corresponds to the collapse of two minimums and two maximums.

    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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