# CATASTROPHE TEACHER

## an introduction for experimentalists

# The seven elementary catastrophes

## Catastrophes in systems with only one state variable:

### The butterfly catastrophe

#### Germ

#### Unfolding

V_{(a, b, c, d)} = x^{6} - ax^{4} - bx^{3} - cx^{2} - dx

#### An applet

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

Bottom : Control panel. The butterfly catastrophe has four control parameters;
the a parameter is named "butterfly factor", the b one "bias factor", the c one "splitting factor, and the d one "normal factor".
Left : space control representation, control parameters values and bifurcation set on a plane (a = butterfly value, b = bias value) through the four 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 four-dimensional; bifurcation set is made of fold hypersurfaces (lines on a two-dimensional slice of control space)
which meet on surface of cusp points which themselves meet on lines of swallowtail points.
Finally, this lines of swalowtail points meet together at a butterfly point.
At this point in control space, potential curve (germ) has only one minimum which corresponds to the collapse of three 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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