Catastrophe Theory

Catastrophe Theory Things that change suddenly, by fits and starts, have long resisted mathematical analysis. A method derived fi-om topology describes these phenomena as examples of seven "elementary catastrophes" by E. C. Zeeman cientists often describe events by con Etudes Scientifique at Bures-sur-Yvette in probable and the dog will most likely flee. structing a mathematical model. In France. He presented his ideas in a book Prediction is also straightforward if neither S deed, when such a model is particular published in 1972, Stabilite Structurelle et stimulus is present; then the dog is likely to ly successful, it is said not only to describe Morphogenese; an English translation by express some neutral kind of behavior unre the events but also to "explain" them; if the David H. Fowler of the University of War lated to either aggression or submission. model can be reduced to a simple equation, wick has recently been published. The the What if the dog is made to feel both rage it may even be called a law of nature. For ory is derived from topology, the branch of and fear simultaneously? The two control 300 years the preeminent method in build mathematics concerned with the properties ling factors are then in direct conflict. Sim ing such models has been the differential of surfaces in many dimensions. Topology ple models that cannot accommodate dis calculus invented by Newton and Leibniz. is involved because the underlying forces in continuity might predict that the two stim Newton himself expressed his laws of mo nature can be described by smooth surfaces uli would cancel each other, leading again tion and gravitation in terms of differential of equilibrium; it is when the equilibrium to neutral behavior. That prediction merely equations, and James Clerk Maxwell em breaks down that catastrophes occur. The reveals the shortcomings of such simplistic ployed them in his theory of electromagnet problem for catastrophe theory is therefore models, since neutrality is in fact the least ism. Einstein's general theory of relativity to describe the shapes of all possible equilib likely behavior. When a dog is both angry also culminates in a set of differential equa rium surfaces. Thorn has solved this prob and frightened, the probabilities of both ex tions, and to these examples could be added lem in terms of a few archetypal forms, treme modes of behavior are high; the dog many less celebrated ones. Nevertheless, as which he calls the elementary catastrophes. may attack or it may flee, but it will not a descriptive language differential equations For processes controlled by no more than remain indifferent. It is the strength of the have an inherent limitation: they can de four factors Thorn has shown that there model derived from catastrophe theory that scribe only those phenomena where change are just seven elementary catastrophes. The it can account for this bimodal distribution is smooth and continuous. In mathematical proof of Thorn's theorem is a difficult one, of probabilities. Moreover, the model pro terms, the solutions to a differential equa but the results of the proof are relatively vides a basis for predicting, under particular tion must be functions that are differenti easy to comprehend. The elementary catas circumstances, which behavior the dog will able. Relatively few phenomena are that or trophes themselves can be understood and choose. derly and well behaved; on the contrary, the applied to problems in the sciences without To construct the model we first plot the world is full of sudden transformations and reference to the proof. two control parameters, rage and fear, as unpredictable divergences, which call for axes on a horizontal plane, called the con functions that are not differentiable. A Model of Aggression trol surface. The behavior of the dog is then A mathematical method for dealing with measured on a third axis, the behavior axis, discontinuous and divergent phenomena The nature of the models derived from which is perpendicular to the first two. We has only recently been developed. The catastrophe theory can best be illustrated by might assume that there is a smooth contin method has the potential for describing the example, and I shall begin by considering a uum of possible modes of behavior, ranging, evolution of forms in all aspects of nature, model of aggression in the dog. Konrad Z. for example, from outright retreat through and hence it embodies a theory of great Lorenz has pointed out that aggressive cowering, avoidance, neutrality, growling generality; it can be applied with particular behavior is influenced by two conflicting and snarling to attacking. The most aggres effectiveness in those situations where grad drives, rage and fear, and he has proposed sive modes of behavior are assigned the ually changing forces or motivations lead to that in the dog these factors can be mea largest values on the behavior axis, the least abrupt changes in behavior. For this reason sured with some reliability. A dog's rage is aggressive the smallest values. For each the method has been named catastrophe correlated with the degree to which its point on the control surface (that is, for theory. Many events in physics can now be mouth is open or its teeth are bared; its fear each combination of rage and fear) there is recognized as examples of mathematical ca is reflected by how much its ears are flat at least one most probable behavior, which tastrophes. Ultimately, however, the most tened back. By employing facial expression we represent as a point directly above the important applications of the theory may be as an indicator of the dog's emotional state point on the control surface and at a height in biology and the social sciences, where we can attempt to learn how the dog's be appropriate to the behavior. For many discontinuous and divergent phenomena havior varies as a function of its mood. points on the control surface, where either are ubiquitous and where other mathemati If only one of the conflicting emotional rage or fear is predominant, there will be cal techniques have so far proved ineffec factors is present, the response of the dog is just one behavior point. Near the center of tive. Catastrophe theory could thus provide relatively easy to predict. If the dog is en the graph, however, where rage and fear are a mathematical language for the hitherto raged but not afraid, then some aggressive roughly equal, each point on the control "inexact" sciences. action, such as attacking, can be expected. surface has two behavior points, one at a Catastrophe theory is the invention When the dog is frightened but is not large value on the behavior axis represent of Rene Thorn of the Institut des Hautes provoked to anger, aggression becomes im- ing aggressive action, the other at a small 65 © 1976 SCIENTIFIC AMERICAN, INC value representing submissive action. In ad surface. The surface has an overall slope creating a pleat without creases, which dition we can note a third point that will from high values where rage predominates grows narrower from the front of the sur always fall between these two, representing to low values in the region where fear is the face to the back and eventually disappears the least likely neutral behavior. prevailing state of mind, but the slope is not in a singular point where the three sheets of If the behavior points for the entire con its most distinctive feature. Catastrophe the pleat come together [see illustration be trol surface are plotted and then connected, theory reveals that in the middle of the sur low]. It is the pleat that gives the model they form a smooth surface: the behavior face there must be a smooth double fold, its most interesting characteristics. All the GROWLING C ATIACKING BEHAVIOR SURFACE AVOIDING FLIGHT CATASTROPHE CONTROL SURFACE - -----.... A AGGRESSION IN DOGS can be described by a model based on becomes narrower, and eventually it vanishes. The line defining the one of the elementary catastrophes. The model assumes that aggres edges of the pleat is called the fold curve, and its projection onto the sive behavior is controlled by two conflicting factors, rage and fear, control surface is a cusp-shaped curve. Because the cusp marks the which are plotted as axes on a horizontal plane: the control surface. boundary where the behavior becomes bimodal it is called the bifur The behavior of the dog, which ranges from attacking to retreating, is cation set and the model is called a cusp catastrophe. If an angry dog represented on a vertical axis. For any combination of rage and fear, is made more fearful, its mood follows the trajectory on the control A and thus for any point on the control surface, there is at least one like surface. The corresponding path on the behavior surface moves to the ly form of behavior, indicated as a point above the corresponding left on the top sheet until it reaches the fold curve; the top sheet then point on the control surface and at the appropriate height on the be vanishes, and the path must jump abruptly to the bottom sheet. Thus havior axis. The set of all such points makes up the behavior surface. the dog abandons its attack and suddenly flees. Similarly, a fright In most cases there is only one probable mode of behavior, but where ened dog that is angered follows the trajectory The dog remains B. rage and fear are roughly equal there are two modes: a dog both angry on the bottom sheet until that sheet disappears, then as it jumps to and fearful may either attack or retreat.

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