# CATASTROPHE TEACHER

## an introduction for experimentalists

# The seven elementary catastrophes

## Catastrophes in systems with only one state variable:

### The swallowtail catastrophe

#### Germ

#### Unfolding

V_{(a, b, c)} = x^{5} - ax^{3} - bx^{2} - cx

#### An applet

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

Bottom : Control panel. The swallowtail catastrophe has three control parameters
Left : space control representation, control parameters values and bifurcation set on a plane (a = A) through the three 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 three-dimensional; bifurcation set is made of three surfaces of fold points (lines on a slice of control space) which meet on two lines of cusp points
which themselves meet on a swallowtail point.

When the parameters values go through a fold surface in control space, a minimum appears or disappears on the potential curve.

When the parameters values cross a cusp line in control space, two minimums and one maximum meet together.

When the parameters values are at swallowtail point in control space, the situation is more sophisticated;
At this point in control space, potential curve (germ) has only one minimum which corresponds to the collapse of two minimums and two maximums.

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Author: Lucien Dujardin

Faculté de Pharmacie BP 83 F 59006 Lille Cedex France

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