# CATASTROPHE TEACHER

## an introduction for experimentalists

# The seven elementary catastrophes

## Catastrophes in systems with only one state variable:

### The cusp catastrophe

#### Germ

#### Unfolding

V_{(a, b)} = x^{4} - ax^{2} - 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..

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Auteur: Lucien Dujardin

Faculté de Pharmacie BP 83 F 59006 Lille Cedex France

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