Catastrophe

by Manfred Füllsack

WHAT IS IT?

This model simulates the Cusp Catastrophe, one of the seven elementary catastrophes of mathematical Catastrophe Theory. See below how it works.

Imagine yourself as having a more or less clear conception 1.) of what kind of activities have to be undertaken to procure means for a living – let them be called laborious activities or labor – and 2.) of  what kind of activities would be appropriate for a good living – let them be called creative activities or work. Labor and work are excluding eachother to some degree, but depend on eachother. You have to minimize labor in order to do as much as possible work, but you will be able to do work only when doing at least some labor as well.

Now imagine labor conditions being normal and copious for some time. You don't have to think too much about it. Labor is just a necessary nuisance, but due to conditions can be easily adjusted to the amount you need in order to optimize time for your work. As a consequence you might establish some sort of balance between labor and work, a balance you treasure and feel comfortably with because it allows you to do work and not too much of labor.

Suddenly labor conditions start to get worse. You are trying to keep to your habits by maintaining your balance. And you succeed for some time. But nevertheless, labor conditions gradually but constantly get worse. Eventually your balance might reach a tipping point, mathematically a singularity, at which your habits cave in and the relation of labor and work abruptly changes its form. A catastrophe has afflicted your work activities.

HOW IT WORKS

In the model above, the blue ball that is displayed when pushing <setup> symbolizes labor, an activity that you would like to minimize in order to do work. This minimization here is expressed as gravity that would cause the ball to fall down wouldn't there be the red curve holding it. The red curve stands for the necessity to make a living. Pushing the button <go> and then moving the slider <a> to the right or to the left lets you find a balance between labor and work. The ball falls to the deepest point of the curve. The plotter to the right of the display marks a certain level, that is, a straight line is displayed. When moving the slider backwards again (which here might be interpreted as worsening of labor conditions) the line in the plotter stays constant at first. However, when reaching the tipping point, the ball in the display falls to the other side of the curve and the line in the plotter avalanches to an other extreme.  A catastrophe has occured causing a new balance.

Note, that the tipping point is different when moving the slider from left to the right and from the right to the left.

The slider <b> changes the distiction of the local minima of the curve. In the above example, this parameter could be interpreted as expressing individual experiences with past labor-work-relations. A low b produces deep local minima and might stand for well established and thus quite stable labor-work-relation habits, whereas a b closer to zero produces rather indistinct local minima. The labor-work-relation here is more susceptible to changes in labor conditions.

In general, Catastrophe Theory investigates and explains processes in which gradual changes of one parameter can cause sudden, and therewith unexpected and unforeseen changes in others. A catastrophe in this regard is a point in this process at which a vanishingly small change in the input can produce a large change in the output. There is reason to assume that catastrophic, i.e. non-linear processes are more widespread in social interactions and developments than gradual and linear ones.

Mathematically, the curve in the model above is drawn by the function y = x⁴ + ax + bx². The parameters a and b are the ones changed with the sliders.

Good introductions to mathematical Catastrophe Theory can be found at the site of the American Mathematical Society. Lucien Dujardin maintains an interesting website called Catastrophe Teacher.

Literature:

Arnol’d, Vladimir I. (1992): Catastrophe Theory. Third, Revised and Expanded Edition. Berlin Heidelberg (1.Edition 1984, 2.Edition 1986).

Castrigiano, Domenico P. L. / Hayes, Sandra A.  (1993): Catastrophe Theory. Reading Mass.

Gilmore, Robert (1981): Catastrophe Theory for Scientists and Engineers. New York.

Majthay, Antal (1985): Foundations of Catastrophe Theory. Marshfield Mass.

Poston, Tim / Stewart, Ian (1978): Catastrophe Theory and its Applications. London.

Saunders, Peter Timothy (1980): An Introduction to Catastrophe Theory, Cambridge.

Thom, René (1975): Structural Stability and Morphogenesis – An Outline of a General Theory of Models. Reading Mass.

Woodcock, Alexander E. R. / Davis, Monte (1978): Catastrophe Theory. New York.

Zeeman, Eric Christopher (1976a): Catastrophe Theory. Scientific American 234 (April 1976) S. 65-83.

Zeeman, Eric Christopher (1977): Catastrophe Theory – Selected Papers 1972-1977. Reading Mass.