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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 parabolic umbilic

Germ

V = x2y + y4

Unfolding

V(a, b, c) = x2y + y4 + ax2 + by2 + cx + dy

An applet

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

  • Bottom : Control panel. The parabolic umbilic has four control parameters;
  • Left : space control representation, control parameters values and bifurcation set on a plane (a = A value b = B value) through the four 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

    Parabolic umbilic germ may be thought as the collapse of

    The parabolic umbilic is the most complicated among the 7 elementary catastrophes.
    Here are two applets to visualize its bifurcation set:

    The umbilic line

    The line of which equation is b = -6a2, c = 0, d = -8a3 est named "umbilic line".

  • By using the control cursor, control parameters evolve along the umbilic line.
  • Green lines are the intersection between the potential surface and the plane through the umbilic point
  • If a > 0, the potential surface has an elliptic umbilic
  • If a < 0, the potential surface has an hyperbolic umbilic
  • The applet shows how an elliptic umbilic change to an hyperbolic one through a parabolic umbilic (if a = 0).
  • Slices of bifurcation set

  • By using the control cursor, the a and b parameters values are choosed such that a2 + b2 = 6.25
  • Left, parameters values on the {a, b} plane; the blue line is umbilic line b = -6a2
  • Right, section of bifurcation set through the {c, d} plane
  • 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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