*Thanks to Tony Phillips who helped with the text of the English version.*

The question is: How can we describe (and interpret), in a synthetic way, the observations we made with the applet ?

We now consider
this applet as a "black
box" : it has an input (control parameters *i.e.* mouse
coordinates on the square) and an output (the blue-to-green colour of the
square). How can we interpret its behaviour ? Here is a new design for
the applet (version 2) : The control space is the pink square (move
the mouse in this pink square) and the output appears in the bottom right
rectangle. This new design shows a black box between input and output. Our
goal is to imagine what the black box may contain
in order to understand the relationships
between inputs and outputs.

- For many systems, what we need to put in the "black box" is a function : output = f(input); the same input always gives the same output.
- In this case, two different output values, a blue color and a green colour, may be observed for the same input (the middle bottom point of the pink square, for example). The output is controlled by the mouse coordinates but also by the way the mouse has arrived at those coordinates. The system has a memory ; internal state and mouse coordinates together determine the output. Another difficulty is that continuous variation of input does not produce continuous variation of output ; at some ``critical'' values of the input, the output value jumps suddenly. A system such as that of this applet is a non-linear system ; the observed behaviour is a critical phenomenon.

Let me try to present the experimental data as a three dimensional graphic.

- The input (control) space is two dimensional ; In Fig. 1,
it is coloured in pink. The two control parameters are called
*a*and*b.* - The output space is one-dimensional. On a computer screen, the pixel
colour is encoded by three numbers which determine the amount of red, green
and blue. Here, the red component is null and the sum of the blue and green
components is constant; so we may use one number,
*x*, to locate an output colour on a scale, the code for the blue component for example. In fact, we do not have to know how the colours are encoded. We need only observe that we are able to locate the observed colour on a one-dimensional scale from blue to green. - In a three-dimensional space (input space x output space) data are put together on a surface which seems torn or split. Above some parts of the control space, there are two sheets of the data surface. When the representative point of the system crosses the rip, it jumps from one sheet to the other one.
- Fig. 1 describes the observed data in a synthetic way. There are jumps but there are also continuous pathways from green to blue which bypass the rip.
- Now, we have to interpret this split surface.

Fig 1 : The data as a torn surface

What mathematical surface may be adjusted to the data ? Mathematicians don't like discontinuities, they don't like split surfaces. They like smoothness. How can we find smoothness behind this split surface ? Is the surface really torn along the rip ? We may imagine a surface which is only folded by adding a third sheet between the two overlying sheets. Then we obtain what a dressmaker would call a pleat. Mathematicians know it as a Riemann-Hugoniot surface or as a cusp. Now the surface is smooth, but the added piece does not belong to the data: it is inacessible to the system.

An equation for the Riemann-Hugoniot surface (in x, y, z coordinates)
is : z^{3} - xz - y = 0 .
In our input-output coordinates this becomes: x^{3} - bx - a = 0 .
The b parameter is called the
splitting factor, while the a parameter is called the normal
factor.

Let me now talk about a potential function. Some systems are known
to be controlled by energy dissipation or more generally by potential minimization.

(**here are some examples
from the physical and chemical sciences**)

Imagine that the system we are studying (the applet or protein synthesis by a cell) is one which minimizes a potential. What is the system doing ? It chooses its internal state to make the potential value is as small as possible,compatibly with the input.

Notice that the experimentalist doesn't know the values of all the internal variables. Most of these are hidden. The experimentalist knows only the input and the output. The main contribution of Catastrophe theory is to show that a local model of a potential-minimizing system (a dissipative system) will ignore the very high number of internal variables. The complexity of the local observed behaviour is in fact constrained by the number of output variables and the number of control parameters.

The potential V is a function of x which is controlled by a and b;
we write this as
V_{a,b}(x). The system may only choose x. We know that the
system has two possible behaviours for some
inputs ; so we are searching
for a potential V_{a,b}(x) which may have two minima. The simplest
function which has these properties is a fourth degree polynomial function.
A canonical form, from which all two minima functions may be obtained,
is :
V_{a,b}(x) = x^{4}/4 - bx^{2}/2 - ax .
Notice that the Riemann-Hugoniot surface is the set of points such that V'_{a,b}(x) = 0,
in other words, the set of minima and maxima of V_{a,b}(x). Fig.
2 summarizes all this material :
the data surface in three-dimensional
space is the set of minima of the potential function. The graph shows
this function for the specific point of the control space picked
out by the vertical, grey line. The three intersection
points of the grey line with the surface correspond to the x-coordinates
of the
two minima and the one maximum on the curve V_{a,b}(x)
shown in the graph.
The surface sheet corresponding to maxima (The red part on Fig. 2)
is inaccessible by the system. This part acts as a repeller, while the
other parts act as attractors.

Fig 2

Now, look at Fig. 3. In the middle, you see a pink square which represents the control space. Around the pink square, you see five graphs which show the shape of the potential curve for five points in the control space. The control space is divided in two areas by a black line which turns around and goes back. This line is called the bifurcation set. It is the set of critical points in the control space. Points in the two remaining areas are called regular points. Near a regular point, the shape of the potential function doesn't change; that is, the number of its minima and maxima is constant. Critical points are points at which one maximum and one minimum meet together to disappear. They are the points of control space where the system state jumps.

One point of the bifurcation set is different from
the others : that is
the point where the bifurcation set line turns around and goes back. At
this point, the two mimima and the maximum are merged. The corresponding
potential is
a "flat" minimum whose equation is V = x^{4}/4 .

Fig 3

Work in progress: Here are some terms which have to be explained

fast and slow dynamics, Maxwell or delay law

equilibrium manifold

position stability (minimization of potential, fast dynamic, local situation)

structural stability ( slow dynamic, global behaviour of the system)

attractor, repeller, basin

divergence

hysteresis

position stability (behaviour variable perturbation = equilibrium perturbation): not possible with the applet, but sometimes possible for other systems (see gravitational catastrophe machines for example).

note on experimental control: cf nervous exitability, slow dynamic control in H & H experiments

Now we can make a third version of the applet : the model
replaces the black box. The model is made of a potential V_{a,b}
(x), the red curve, which is minimized by the system. There is
a delay law which
picks out one minimum, in case there are two minima. See
this new design.

Now, you may read the applet source code, if you want. In fact, our mystery system is the same as the model.

Playing with the applet brought out two strange behaviours nbsp;; they caused us to search for a model. The proposed model is the cusp and the two strange behaviours are two properties of the cusp : bimodality and sudden jumps. Does the cusp have other properties ? What new experiments may we devise to test expected properties of the model ?

When you move the mouse on the bottom of the control space from left to right the output colour is green and jumps suddenly to blue. If you go back to the left, then the output colour doesn't turn immediately to green. You have to go far to the left to obtain the jump to green. This phenomenon is called hysteresis by analogy with a similar effect in magnetism. Investigate this property with the applet.

The horizontal axis on the control space corresponds to the "normal factor", while the vertical axis is the ``splitting factor''. It is along this direction that the behaviour surface is split. For some systems, the control factors do not correspond to the normal and splitting directions of the behaviour surface and they are then called conflicting factors.

Try to move the mouse along two paths in the control space : these two paths start in the same place,at the top center (there, the output is a colour halfway between blue and green), and they both stop in the same place, at the bottom center. The two paths go alongside each other but one starts out slightly to the left and the other starts out slightly to the right. Although the control parameter values are the same at the ends of these two paths, the output has continuously (without any jump) changed colour to blue for the left-hand path and to green for the right-hand one. This property of the cusp is called divergence. Investigate this property with the applet.

This property corresponds to the fact that the data surface is torn. A sheet is added to the data surface in the cusp model. This sheet corresponds to an unstable equilibrium of the system. If we try to place the system at a local maximum of potential, it will return to a minimum. To control this property we have to be able to push or to pull the state (behaviour) variable. We have no possibility of controlling the applet state variable, so this property may not be tested with this first applet. Experimentalists often need to imagine some device to control some variable of the studied system. One of these devices is that of Hodgkin-Huxley which allow control of the nerve membrane potential and measurement of membrane permeability to sodium and potassium ions.

The Zeeman catastrophe machine allows to experiment with this property

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This small toy was invented by E. C. Zeeman . References are: Zeeman E. C. A catastrophe machine. in Towards a Theoretical Biology (C. H. Waddington, ed.). Edinburgh University Press, Edinburgh, 1972, 4: 276-282.

The same article may be found in Zeeman E. C. Selected papers. Addison-Wesley Publishing Company, Reading, Massachussetts, 1977.

This machine consists of a disk able to turn freely around its center (O). On one point (B) near its edge are attached two elastic bands. One of these has its other end fixed. The end (C) of the other elastic may be moved on the plane, so the control space is two dimensional. Although this small toy has high pedagogic value, few people have experimented with it. I am happy to present here an applet which simulates Zeeman's machine.

To use the Zeeman machine applet, put the cursor on the point (C), click the mouse button, use the mouse to drag the point slowly and observe the motion of the disk. You will see either continuous or sudden rotation of the disk. You can locate the bifurcation set. In this case, you can understand the behaviour of the machine: this system is a dissipative one which minimizes the energy of the two stretched elastic bands.

Here is the applet code.

Another way to use Zeeman's machine is to directly perturb the position of the disk. Put the cursor on the point B where the two elastic bands are linked together, click and drag to rotate the disk, release the button and watch what happens.

A reference for this machine is: Poston T. & Stewart I. - Catastrophe theory and its applications. Pitman London, 1978.

The machine is made of two parabola-shaped pieces of light cardboard held together along their edges by short rods. "A small heavy magnet behind the solid card will grip a light piece of metal in front and can be slid to any desired position while retaining a good grip."

"Since most of the mass of the assembled device is in the magnet, we may take the centre of gravity of the whole to be the position of the magnet. When machine balance steadily on edge, the centre of gravity must be vertically above the point of contact. If the machine rests on a level plane, the plane must be tangent to the edge, so the center of gravity lies on the corresponding normal."

The applet shows two copies of the gravitational machine. On the left-hand one you choose a position for the magnet by clicking and dragging (this copy may be considered as the control space). When you release the mouse button, the right-hand copy moves to the resulting equilibrium position of the machine.

Here is the applet code.