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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 cusp catastrophe

Germ

V = x4

Unfolding

V(a, b) = x4 - ax2 - bx

An applet

The cusp catastrophe properties will be studied by means of this applet

  • Bottom : Control panel. The cusp catastrophe has two control parameters
  • Left : space control representation, control parameters values and bifurcation set
  • 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 two-dimensional; bifurcation set is made of two lines of fold points and one cusp point which is at the union of the two lines.
    When the parameters values go through a fold line in control space, a minimum appears or disappears on the potential curve.
    At the cusp point the goemetry is more sophisticated. At this point in control space, potential curve (germ) has only one minimum which corresponds to the collapse of two minimum and one maximum..

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

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    Auteur: Lucien Dujardin

    Faculté de Pharmacie BP 83 F 59006 Lille Cedex France

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