French version

CATASTROPHE TEACHER

an introduction for experimentalists

 

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


The cusp and its properties

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.

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

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 : z3 - xz - y = 0 . In our input-output coordinates this becomes: x3 - 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 Va,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 Va,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 : Va,b(x) = x4/4 - bx2/2 - 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 Va,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 Va,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 = x4/4 .

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 Va,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.

Activities: use the applet to explore other properties of the cusp catastrophe.

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 ?

Hysteresis

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.

divergence

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.

Inaccessibility

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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Some other catastrophe machines for more precise exploration of the notion of stability.

Zeeman's catastrophe machine

The original Zeeman machine

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.

Zeeman's catastrophe machine as an applet

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.

Inaccessibility

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.

The gravitational catastrophe machine (JMT Thompson)

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

The original gravitational catastrophe machine

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 gravitational catastrophe machine as an applet

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.

Stability and perturbation

Perturbation of control parameters and structural stability

Perturbation of state variable and position stability

Inaccessibility (unstable equilibrium on a maximum of potential)


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Auteur: Lucien Dujardin

Faculté de Pharmacie BP 83 F 59006 Lille Cedex France

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