Catastrophe theory is an attempt to go beyond classical physics -- the physics of structurally stable systems -- and to provide a mathematical framework for discontinuous processes. It was primarily developed by René Thom. Thom developed catastrophe theory as a mathematical way of addressing the work on morphogenesis done by C.H. Waddington in the 1950's.
If the attractor points of a state, or phase space of a system are thought of as a surface, catastrophe points are moments of discontinuity, where the system can move to more than one possible point for the same input values. According to catastrophe theory, the system jumps to one or another of these points in only a small number of inequivalent ways. (see also bifurcation)
The generic geometry of a surface with catastrophes consists of S-shaped folded areas, or conflict regions, where more than one stable local maximum is separated by local unstable minima. A fold which comes to a point has a cusp singularity, where the two fold lines meet. Divergence in cusp catastrophes is similar to sensitivity to initial conditions see also non-linear.
The classification theorem of catastrophe theory gives the standard geometry for all functions having at most six input parameters and one or two output variables. It states that when a system is characterized by one input variable and one output, the universal or canonical picture is the fold, and with two input parameters and a single behavioural output, the universal picture of all stable descriptions will display cusp geometry. (three inputs and one output yields the swallowtail, four inputs and one output the butterfly(which contains a pocket of compromise-- note that the surface is a 4D one --, three inputs and two outputs the hyperbolic umbilic or elliptic umbilic, four inputs and two outputs the parabolic umbilic) These catastrophes can be given spatial or temporal interpretations. (see Casti, Complexification, fig. 2.2) Some examples analysed by Casti include failure of bending beams, developmental biology, collapse of civilization...
Since the "noble death" of catastrophe theory, Thom has tried to reduce the scope of expectation that it ellicited. He points out that the theory deals with qualities not quantitites and is not meant to be predictive. see science / philosophy regarding some of the controversy sourrounding catastrophe theory. Russian catastrophe theory describes perestroika (metamorphosis) in mathematical terms while acknowledging that their successful study is undoubtedly a result of political perestroika . (see political systems. )
Stuart Kauffman speaks of complexity catastrophes, referring to moments when the complexity of an organic system makes it resistant to selection pressures.The central problem for any theory of the origin of replication is that the replicative apparatus has to function almost perfectly if it is to function at all. Otherwise, it will give rise to errors that will result in "error catastrophes," the progressive deterioration of the system until it is totally disorganized.