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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 butterfly catastrophe

Germ

V = x6

Unfolding

V(a, b, c, d) = x6 - ax4 - bx3 - cx2 - dx

An applet

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

  • Bottom : Control panel. The butterfly catastrophe has four control parameters; the a parameter is named "butterfly factor", the b one "bias factor", the c one "splitting factor, and the d one "normal factor".
  • Left : space control representation, control parameters values and bifurcation set on a plane (a = butterfly value, b = bias value) through the four 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 four-dimensional; bifurcation set is made of fold hypersurfaces (lines on a two-dimensional slice of control space) which meet on surface of cusp points which themselves meet on lines of swallowtail points. Finally, this lines of swalowtail points meet together at a butterfly point. At this point in control space, potential curve (germ) has only one minimum which corresponds to the collapse of three 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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