It can be said that catastrophe theory is a special case of singularity theory, which is in turn the key to bifurcation theory, part of the study of nonlinear dynamical systems. Bifurcation theory is widely argued to have been invented/discovered by the great French mathematician, Henri Poincaré, as part of his qualitative analysis of systems of nonlinear differential equations (18801890). This arose from his study of celestial mechanics and the famous three body problem in particular. Would the orbits of the planets in the solar system escape to infinity, remain within certain bounds, or would the planets crash into each other or the sun? Beyond this question he investigated the structural stability of the system, studying if small perturbations to the system would leave it relatively unchanged in its behavior or cause it to move in a very different manner. It was this particular investigation that led to bifurcation theory.
Although Poincaré was the first to formally analyze bifurcation theory, there was already a fairly well established body of knowledge in mathematics about it. The Russian mathematician, Vladimir Arnol’d (1992, Appendix) has provided a list of precursors to the work of Poincaré. According to him, although an eager enough observer can find hints of it in some of the work of Leonardo da Vinci, the first clear presentation of the structural stability of a cusp point came in the study of light caustics and wave fronts was due to Huygens in 1654. Critical points in geometrical optics were studied by Hamilton in 183738. By the late nineteenth century many algebraic geometers were examining the singularities of curves and smooth surfaces with some of these discussions even ending up in textbooks on algebraic geometry. Among those engaged in such studies included Cayley, Kronecker, and Bertini. Nevertheless, it was Poincaré who brought structure to the discussion of these topics.
Consider a general family of differential equations whose behavior is determined by a kdimensional control parameter, μ:
dx/dt = f_{μ}(x); x in R^{n}, μ in R^{k}. (1)
Equilibrium solutions are given by f_{μ}(x) = 0. This set of equilibria will bifurcate into separate branches at a singularity, or a degenerate critical point. More precisely, a singularity occurs where the Jacobian matrix Dxf_{μ}(x) has a zero real part for one of its eigenvalues. Intuitively a single stable curve of equilibrium points may split into several curves at such a point, with some stable and others unstable locally. At such points the first derivative may be zero but the function may not be at an extremum. There are many different kinds of bifurcations, with Guckenheimer and Holmes (1983) providing a good summary of the various types.
The distinction between critical points of functions that are nondegenerate (associated with extrema) and degenerate ones (singular, nonextremal) was further studied by George Birkhoff (1927), a follower of Poincaré, and also by Birkhoff’s follower, Marston Morse (1931). In particular, Morse showed how a function with a degenerate singularity could be slightly perturbed to a new function that would now exhibit two distinct nondegenerate critical points instead of the singularity. This was a bifurcation of the degenerate equilibrium and indicates the close connection between the singularity of a mapping and its structural stability (see Figure 1).
[insert Figure 1]
Hassler Whitney (1955) followed Morse by studying different kinds of singularities and their stabilities. This was the sort of approach that was essentially the origin of catastrophe theory, although nobody was using that term yet. In fact, Whitney discovered/invented the two singularities associated with the two most commonly studied kinds of elementary catastrophes, and the only ones that are stable in all their forms, the fold and the cusp (see Figure 2). He showed that these were the only two kinds of structurally stable singularities for differentiable mappings between two planar surfaces. Thus Whitney can be viewed as the real inventor/discoverer of catastrophe theory.
[insert Figure 2]
Following his discovery of transversality (1956), René Thom developed further the classification of singularities, or of elementary catastrophes, although a more complete categorization would eventually be carried out by Arnol’d, GuseinZade, and Varchenko (1985), who showed that for systems beyond a dimensionality of eleven, the categories of catastrophes become infinite and thus difficult to categorize. Thom (1972, pp, 10308) would label such catastrophes as “generalized” or “nonelementary.” More particularly, Thom (1972) studied the seven elementary catastrophes going up through six dimensions in control and state variables. This became standard catastrophe theory.
Consider a dynamical system given by n functions on r control variables, ci. The n equations determine n state variables, x_{j}:
x_{j} = f_{j}(c_{1}…c_{r}). (2)
Let V be a potential function on the set of control and state variables:
V = V(c_{i},x_{j}) (3)
such that for all x_{j}
V/x_{j}= 0. (4)
This set of points constitutes the equilibrium manifold, M, and an example is seen in the cusp catastrophe seen in Figure 2, which is characterized by two control variables and one state variable. In much discussion the control variables are characterized as being “slow,” whereas the state variables are characterized as being “fast.” The usual presumption has been that the state variables adjust quickly to be on the equilibrium manifold while the control variables move the system around on the manifold. In turn, the catastrophe function is the projection of the equilibrium manifold into the rdimensional control variable space, with its singularities the main focus of catastrophe theory (not to be confused with projection functions found in game theory)..
Thom’s Theorem, which was rigorously demonstrated by Malgrange (1966) and Mather (1968), states that if the underlying functions fj are generic (qualitatively stable under slight perturbations), if r < 6, and if n is finite with all but two state variables being represented by linear and nondegenerate quadratic terms, then any singularity of a catastrophe function will be structurally stable (generic) under slight perturbations and can be classified into eleven different types. There are seven such types for r < 5, and Thom (1972) provided colorful names for each of these, along with detailed discussions of their various characteristics, with further discussion carried out by Trotman and Zeeman (1976).^{4} For r > 5 and more than two control parameters, the set of possible catastrophes is infinite.
Although there have been some applications of catastrophes of somewhat higher dimensionality in economics, especially in urban and regional economics, most of the applications have involved the two simplest forms already known to Hassler Whitney in 1955, the fold and the cusp depicted in Figure 2. In order to analyze a particular model using one of these one must make assumptions regarding how the system moves between equilibria in situations of multiple equilibria. For the simplest case of the fold catastrophe four kinds of behavior can occur: hysteresis, bimodality, inaccessibility, and sudden jumps. An additional phenomenon that can occur in the case of the cusp catastrophe is divergence, which involves the increasing separation of the two planes of the equilibrium manifold as the value of the socalled splitting factor control variable increases in value. When its value is sufficiently low, there are no discontinuities and the system is controlled by variations of the socalled normal factor, the other control variable.
