Catastrophe theory

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical 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, analysing 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 (Lyapunov function).

Small changes in parameters can cause previously stable equilibria to disappear, leading to a large and sudden transition of the behaviour 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.

Elementary catastrophes
Catastrophe theory analyses degenerate critical points of the potential function -- points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters.

When the degenerate points are not merely accidental, but are structurally stable, the degenerate points exist as organising centres for particular geometric structures of lower degeneracy critical features in parameter space around them. If the potential function depends on three or fewer active variables, and five 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 change of variables. These are now presented, with the names that Thom gave them.

Fold catastrophe

 * $$V = x^3 + ax\,$$

At negative values of a, the potential has two extrema - one stable, and one unstable. If the parameter a is slowly increased, the system can follow the stable minimum point. But at a=0 the stable and unstable extrema meet, and annihilate. This is the bifurcation point. At a>0 there is no longer a stable solution. If a physical system is followed through a fold bifurcation, one therefore finds that as a reaches 0, the stability of the a<0 solution is suddenly lost, and the system will make a sudden transition to a new, very different behaviour. This bifurcation value of the parameter a is sometimes called the tipping point. 

Cusp catastrophe

 * $$V = x^4 + ax^2 + bx \,$$

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.

But in a cusp geometry the bifurcation curve loops back on itself, giving a second branch where this alternate solution itself loses stability, and will make a jump back to the original solution set. By repeatedly increasing b and then decreasing it, one can therefore observe hysteresis loops, as the system alternately follows one solution, jumps to the other, follows the other back, then jumps back to the first.

However, this is only possible in the region of parameter space a<0. As a is increased, the hysteresis loops become smaller and smaller, until above a=0 they disappear altogether (the cusp catastrophe), and there is only one stable solution.

One can also consider what happens if one holds b constant and varies a. In the symmetrical case b=0, one observes a pitchfork bifurcation as a is reduced, with one stable solution suddenly splitting into two stable solutions and one unstable solution as the physical system passes to a<0 through the cusp point a=0, b=0 (an example of spontaneous symmetry breaking). Away from the cusp point, there is no sudden change in a physical solution being followed: when passing through the curve of fold bifurcations, all that happens is an alternate second solution becomes available.

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. 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. But higher stress levels correspond to moving to the region (a<0). Then, if the dog starts cowed, it will remain cowed as it is irritated more and more, until it reaches the 'fold' point, when it will suddenly, discontinuously snap through to angry mode. Once in 'angry' mode, it will remain angry, even if the direct irritation parameter is considerably reduced.

Fold bifurcations and the cusp geometry are by far the most important practical consequences of catastrophe theory. They are patterns which reoccur again and again in physics, engineering and mathematical modelling.

The remaining simple catastrophe geometries are very specialised in comparison, and presented here only for curiosity value.

Swallowtail catastrophe

 * $$V = x^5 + ax^3 + bx^2 + cx \, $$

The control parameter space is three dimensional. The bifurcation set in parameter space is made up of three surfaces of fold bifurcations, which meet in two lines of cusp bifurcations, which in turn meet at a single swallowtail bifurcation point.

As the parameters go through the surface of fold bifurcations, one minimum and one maximum of the potential function disappear. At the cusp bifurcations, two minima and one maximum are replaced by one minimum; beyond them the fold bifurcations disappear. At the swallowtail point, two minima and two maxima all meet at a single value of x. For values of a>0, beyond the swallowtail, there is either one maximum-minimum pair, or none at all, depending on the values of b and c. Two of the surfaces of fold bifurcations, and the two lines of cusp bifurcations where they meet for a<0, therefore disappear at the swallowtail point, to be replaced with only a single surface of fold bifurcations remaining.

Butterfly catastrophe

 * $$V = x^6 + ax^4 + bx^3 + cx^2 + dx \, $$

Depending on the parameter values, the potential function may have three, two, or one different local minima, separated by the loci of fold bifurcations. At the butterfly point, the different 3-surfaces of fold bifurcations, the 2-surfaces of cusp bifurcations, and the lines of swallowtail bifurcations all meet up and disappear, leaving a single cusp structure remaining when a>0.

Potential functions of two active variables
Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in optics in the focal surfaces created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces. Thom proposed that the Hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair like structures.

Hyperbolic umbilic catastrophe

 * $$V = x^3 + y^3 + axy + bx + cy \, $$

Elliptic umbilic catastrophe

 * $$V = x^3/3 - xy^2 + a(x^2+y^2) + bx + cy \, $$

Parabolic umbilic catastrophe

 * $$V = x^2y + y^4 + ax^2 + by^2 + cx + dy \, $$

Arnold's notation
Vladimir Arnol'd gave the catastrophes the ADE classification, due to a deep connection with simple Lie groups. There are objects in singularity theory which correspond to most of the other simple Lie groups.
 * A0 - a non singular point: $$V = x$$.
 * A1 - a local extrema, either a stable minimum or unstable maximum $$V = \pm x^2 + a x$$.
 * A2 - the fold
 * A3 - the cusp
 * A4 - the swallowtail
 * A5 - the butterfly
 * Ak - an infinite sequence of one variable forms $$V=x^{k+1}+\cdots$$
 * D4- - the elliptical umbilic
 * D4+ - the hyperbolic umbilic
 * D5 - the parabolic umbilic
 * Dk - an infinite sequence of further umbilic forms
 * E6 - the symbolic umbilic $$V = x^3+y^4+a x y^2 +bxy+cx+dy$$
 * E7
 * E8