An introduction to catastrophes

Even social and biological scientists familiar with the uses of mathematical models in their fields will find conceptually novel and sometimes mathematically forbidding elements in catastrophe models. One major difficulty is that these models use concepts that are more familiar to mathematicians and physicists than to behavioral scientists: potential functions and gradient systems, for instance. Starting from the basic elements of linear systems and their usual analysis (trajectories, equilibrium, stability of equilibria, comparative statics) it is possible to construct a conceptual and mathematical path from these familiar notions to more unfamiliar ideas and techniques dealing with structural stability and catastrophes. This is the aim: to present such a path from simple to complex, from old to new, from familiar to unfamiliar. In a later section of the paper, the approaches of Thom and Zeeman to the scientific use of the catastrophe concept are presented. The conceptual framework employed in the paper is that of general systems theory and the basic notion within which all the remaining techniques and ideas find their place is that of a state-determined dynamic system. Apart from such an expository function, the paper also shows that the use of catastrophe concepts need not imply a special catastrophe model if analysis of any given dynamic model is extended to the problem of the implications for the events in state space of trajectories in parameter space. In this way, an effort is made to make this notion more accessible and useful in science.