# CATASTROPHE TEACHER

## an introduction for experimentalists

# The seven elementary catastrophes

## Catastrophes in systems with two state variables:

### The parabolic umbilic

#### Germ

#### Unfolding

V_{(a, b, c)} = x^{2}y + y^{4} + ax^{2} + by^{2} + 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 = -6a^{2}, c = 0, d = -8a^{3} 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 a^{2} + b^{2} = 6.25
Left, parameters values on the {a, b} plane; the blue line is umbilic line b = -6a^{2}
Right, section of bifurcation set through the {c, d} plane

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Author: Lucien Dujardin

Faculté de Pharmacie BP 83 F 59006 Lille Cedex France

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