An Introduction to Catastrophe Theory

A great many of the most interesting phenomena in nature involve discontinuities. These may be in time, like the breaking of a wave, the division of a cell or the collapse of a bridge, or they may be spatial, like the boundary of an object or the frontier between two kinds of tissue. Yet the vast majority of the techniques available to the applied mathematician have been designed for the quantitative study of continuous behaviour. These methods, based primarily on the calculus, though very much refined and extended since the time of Newton and Leibniz, have made possible tremendous advances in our understanding of nature. Their great success has, however, been largely confined to the physical sciences. When we turn to the biological and social sciences, we generally find that we are unable to construct the relatively complete models which would permit us to apply the same methods. Moreover, the observations, which are the raw material from which the theoretician must work and the standard against which he must test his models, are seldom of the same precision as those which are available in physics. In many cases they are only qualitative. There is nothing in biology to compare with the inexorable and accurately predictable motions of the heavenly bodies.

As a part of mathematics, catastrophe theory is a theory about singularities. When applied to scientific problems, therefore, it deals with the properties of discontinuities directly, without reference to any specific underlying mechanism.