Catastrophe theory
Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s.
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.
In the late 1970s, applications of catastrophe theory to areas outside its scope began to be criticized, especially in biology and social sciences.
[1][2] Zahler and Sussmann, in a 1977 article in Nature, referred to such applications as being "characterised by incorrect reasoning, far-fetched assumptions, erroneous consequences, and exaggerated claims".
The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters.
If the potential function depends on two or fewer active variables, and four or fewer active parameters, then there are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by diffeomorphism (a smooth transformation whose inverse is also smooth).
Catastrophe theory studies dynamical systems that describe the evolution[5] of a state variable
Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x.
The cusp geometry is very common when one explores what happens to a fold bifurcation if a second parameter, b, is added to the control space.
Varying the parameters, one finds that there is now a curve (blue) of points in (a,b) space where stability is lost, where the stable solution will suddenly jump to an alternate outcome.
A famous suggestion is that the cusp catastrophe can be used to model the behaviour of a stressed dog, which may respond by becoming cowed or becoming angry.
[6] The suggestion is that at moderate stress (a > 0), the dog will exhibit a smooth transition of response from cowed to angry, depending on how it is provoked.
[7] Catastrophic failure of a complex system with parallel redundancy can be evaluated based on the relationship between local and external stresses.
[10] Fold bifurcations and the cusp geometry are by far the most important practical consequences of catastrophe theory.
They produce the strong gravitational lensing events and provide astronomers with one of the methods used for detecting black holes and the dark matter of the universe, via the phenomenon of gravitational lensing producing multiple images of distant quasars.
Depending on the parameter values, the potential function may have three, two, or one different local minima, separated by the loci of fold bifurcations.
Vladimir Arnold gave the catastrophes the ADE classification, due to a deep connection with simple Lie groups.
The caustics one sees at the bottom of a swimming pool, for example, have a distinctive texture and only has a few types of singular points, even though the surface of the water is ever changing.
Due to the wave nature of light, the catastrophe has fine diffraction details described by the Airy function.
Due to the wave nature of light, the catastrophe has fine diffraction details described by the Pearcey function.
