# 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

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

• two minimums or two maximums or one maximum and one minimum,
• and three saddles.

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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