Catastrophe theory

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamic systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analyzing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide. Catastrophe theory, which was originated with the work of the French mathematician René Thom in the 1960s, and became very popular not least due to the efforts of Christopher Zeeman in the 1970s, considers the special case where the long-run stable solution can be identified with the minimum of a smooth, well-defined potential function. Small changes in parameters can cause a previously stable equilibrium to disappear, leading to a large and sudden transition of the behavior of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.