Explanations

Explanations mainly from Poston & Stewart (1978). With 35 fragment identifiers referred to in the text "Elementary catastrophe theory: an introduction", 24.11.01. The list is in working order from this date, from here on there will only be minor additions and adjustments for better stringency.

Harold Marston Morse American mathematician, who a.o. things generalized the notion of nondegenerate critical point (see this word).
Frederik Stjernfelt, b. 1957, mag. art. and D. Ph. in Nordic literature, lecturer at the Institute for Literature Science, University of Copenhagen. "Formens betydning" (Kbh. 1992), "Rationalitetens himmel" (1997) and articles in Nordic and international papers. Editor of KRITIK. For the time being occupied with a larger work on semiotic similarity (da.: lighed) and generality based on Peirce and Husserl.
René Thom, French mathematician and philosopher, b. 1923. In the fifties Thom was instrumental in the development of differential geometry and topology; in 1958 he received the Field award for the cobordism theory in topology. His interest has been focused on the theory of critical points (in plain language: points where "something happens"). From here he developed in the sixties a general philosophy of science, that has gone public under the name of catastrophe theory. The leading idea of this theory is, that we do not have a continuous causal description at the micro level of many complicated systems, so we have to confine ourselves to a "superficial" description, that points out when and where a system changes its behaviour. These transitions can be modelized by means of topological singularities. In general the focus has been on the concept of form, in biology as well as in linguistics. The theory was first put forward in "Stabilité structurelle et morphogenèse" (1973), by the mid-seventies it appeared for a few years in the limelight of media and public discussion. Later on Thom has developed the general theory in "Modèles mathématiques de la morphogenèse" (1980) and in "Apologie du logos" (1990). /courtesy by Frederik Stjernfelt/
Erik Cristopher Zeeman, b. 1925, head of Mathematical Research Centre, University of Warwick, 1964 - 88. His role and position in the advent of catastrophe theory is clear from the dedication in Poston & Stewart (two of his pupils; reference see below): "To Cristopher Zeeman, at whose feet we sit, on whose shoulders we stand". He is present at every event, in every context where catastrophe theory is discussed and forwarded in the 70's and early 80's; without his efforts, both scientific and introductory, much of the theoretical body would not exist. He is one of many examples in British history, where the nurse shoulders the role of parent, for good and bad. In this case, it seems to me the child would have been at least rachitic without him being there - but it may have contracted some other English disease instead. /CP/
affine An affine subspace is a vector space translated away from the origin: X = V + a = {v + a such that v is contained in V}.
analytic A function f is analytic if its Taylor expansion about a point x_o converges in some neighbourhood U of x_o and its sum is equal to f(x + x_o) in U.
(Thom's) classification theorem. Typically an r-parameter family of smooth functions Rⁿ-->R, for any n and for all r less than or equal to 5, is structurally stable, and is equivalent around any point to one of the following forms:
Non-critical: x (Df non-zero)

Nondegenerate critical (Morse): x₁² + ... + x_k² - x_k+1² - ... - x_n²

These two types are not catastrophe forms, the rest are catastrophes.

First come the cuspoid catastrophes (with 5 parameters there are five of them), three of which are listed below:

the fold:x³ + ax + (M)

the cusp: ą(x⁴ + ay² + bx) + (M)

the swallowtail: x⁵ + ax³ + by² + cx + (M)
Here x, y (or x₁, x₂ etc.) denote regular function (state) variables, a, b, c... auxiliary parameters and (M) is a Morse function of the nondegenerate critical type, listed above, minus one or two terms: x₂² + ... + x_k² - x_k+1² - ... - x_n² (Morse (n - k)-saddle)
codimension A subspace W in Rⁿ must have dimension in the range 0 - n. The difference n - dim W is called the codimension of W in Rⁿ. It is also the number of equations in (x₁, ..., x_n) locally needed to describe W as a subset of Rⁿ.
corank In a non-Morse (degenerate) critical point: the number of independent directions in which it is degenerate. If a quadratic form has rank r less than the number of variables n, it is degenerate and the difference n - r is its corank.
critical point. In the function f: Rⁿ-->R a point where all partial derivatives of first order vanish. Cf. singularity, nondegenerate and degenerate critical points.
cubic form The general form with two variables is c(x, y) = ax³ + 2bx²y + cxy² + dy³, all such equations can by a non-singular linear transformation be transferred into either of the four types x²y - y³, x²y + y³, x²y and x³.
degenerate critical point. In the function f: Rⁿ-->R a point where the partial derivatives of first order vanish, and the derivatives of second order are degenerate quadratic forms (i.e.: the Hessian matrix is singular, its determinant = 0). Cf. nondegenerate critical point.
degree The highest power of the (any) variable appearing in a function.
diffeomorphism A change of coordinates, that doesn't alter or destroy information given by the old system. f is a diffeomorphism if a/ it is smooth, b/ it has an inverse and c/ the inverse is smooth.
dimension of manifold: see manifold.
Euclidean space Let R be the set of real numbers. Then, for any integer n > 0, n-dimensional Euclidean space Rⁿ is defined as the set of all x_i (x₁,..., x_n) such that x_i belongs to R for i = 1, ..., n.
equivalence Two smooth functions f, g: Rⁿ are said to be equivalent around 0 if there is a local diffeomorphism y: Rⁿ-->Rⁿ around 0 and a constant c (the shear term) such that, around 0
g(x) = f(y(x)) + c
Then the Morse lemma says, that near a nondegenerate critical point a function is equivalent to one of the Morse standard forms (see Morse lemma) and another theorem (mentioned before the Splitting lemma in the word-list) that a function f: R-->R with non-zero Taylor series is equivalent to ąx^k for some k. By introducing one diffeomorphism e taking care of the parameter(s), one family of diffeomorphisms, y_s: Rⁿ-->Rⁿ, taking care of the state variable(s), and a shear term c(s), we can in a similar way define equivalence for families: f, g: Rⁿ-->R are equivalent if there exist e, y, c defined in a neighbourhood of 0 such that g(x, s) = f(y_s(x), e(s)) + c(s)) for all (x, s) contained in Rⁿ x R^r in that neighbourhood.
germ Function defined locally, near a point.
Hessian matrix The matrix of all second partial derivatives in conventional arrangement; if x = (x₁, ..., x_n), d²f/dx₁² is placed in the upper left corner, d²f/dx₁dx₂ next to it in the same row etc.
Inverse Function Theorem See Jacobian determinant.
Jacobian determinant The Inverse Function Theorem states: Let f: U-->R^m be smooth and let x be contained in U. If the linear map Df |_x (derivative of f at x) is non-singular, then f is a local diffeomorphism at x. Df |_x is non-singular if and only if its Jacobian determinant Jf |_x = det Df |_x is non-zero.
jet The k-jet, j^kf(x) = SUM from 0 to k of 1/k! D^kf(0) x^k is the Taylor series of a function f at the origin, truncated at degree k. This is a perfect polynomial function, but it need not converge. The tail [f - j^kf] has its first k derivatives 0 at the origin. (Poston & Stewart (1978) call it the "Tayl"...).
linear transformation (map) A function f: Rⁿ-->R^m with the properties:
f(x+y) = f(x) + f(y)

f(kx) = kf(x) for all (x, y) contained in Rⁿ, and all k contained in R.
A linear map preserves straight lines through the origin.
manifold A higher-dimensional analogue of a smooth curve or surface. A smooth sub-manifold of Rⁿ is a subset with the following properties:

a. locally it looks like a piece of R^m
b. it is embedded in Rⁿ smoothly, that is, has a unique tangent hyperplane at each point.
The number m is called the dimension of the manifold. A one-dimensional manifold is a curve, a two-dimensional a surface, a three-dimensional e.g. a doughnut.
map or mapping is another name for function: a rule which associates for each x contained in X a unique element y contained in Y. (f : X-->Y)
Morse critical see: nondegenerate critical point.
Morse lemma see: Splitting lemma.
nondegenerate critical point. In the function f: Rⁿ-->R a point where the partial derivatives of first order vanish, and the derivatives of second order are nondegenerate quadratic forms (i.e., its rank is equal to the number of variables, or equivalently: the Hessian matrix is non-singular, its determinant not equal to zero).
order The lowest power of the variable appearing in a function. A function g: R-->R has order k at the origin if g(0) = Dg(0) = ... = D^k-1g(0) = 0
organizing centre Changes between states, change from one behaviour to another occur at bifurcation points or along bifurcation sets. Bifurcation doesn't occur as long as the critical points are Morse (see this word). There must be a degenerate critical point (or better: a function f as a degenerate c. p., near it) to have bifurcation, and it could be said that this point "organizes" the transition from one kind of nondegenerate function to another, hence it is the organizing centre.
quadratic form. The general form is q(x) = SUM c_ijx_i x_j, but each quadratic form in n variables can by a non-singular linear transformation (see this word) of the variables be reduced to the shape d₁y₁² + ... + d_ny_n². The two-variable form is of general type ax² + 2bxy + y², all such equations can be transformed into either of the possibilities u² + v², u² - v², -u² - v², u², -u² or 0. A quadratic form is degenerate if its rank (see this word) is less than the number of variables n. The difference n - r is the corank (see this word) and measures the number of independent directions in which it is degenerate. Degeneracy happens if and only if the determinant of the matrix, ac - b², is zero. This determinant is called the discriminant of the quadratic form.
rank The rank of a linear transformation f is the dimension of its image f(Rⁿ) = {f(x) such that x is contained in Rⁿ}. Any quadratic form can, by a non-singular real linear transformation (see this word) be put in the form z₁² + ... + z_r² - z_r+1² - ... - z_s², where s is equal to or less than n. The number s is called the rank of the quadratic form, and the difference n-r the corank. The corank measures the number of directions in which the quadratic form is degenerate.
singularity is sometimes used to denote a critical point (see this word), but it is better reserved for really "singular" points where the derivative is undefined, e.g. the point of the cusp y³ = x².
smooth A smooth function possesses derivatives of arbitrary order, this means for example that its Taylor series has an infinite number of terms.
Splitting lemma.
1. First there is a Morse lemma (or theorem) that goes: Let u be a nondegenerate critical point of the smooth function f : Rⁿ-->R. Then there is a local coordinate system (y₁, ..., y_n) in a neighbourhood U of u, with y_i(u) = 0 for all i, such that f = f(u) - y₁² - ... - y_k² + y_k+1² + ... + y_n² for all y contained in U.
  This lemma allows us to recognize a nondegenerate critical point for any number of variables. A function of the form z₁² + ... + z_n-k² - z_n-k+1² - ... - z_n² is called a Morse k-saddle. Hence the Morse lemma implies that every nondegenerate critical point may be transformed by a diffeomorphism to a Morse k-saddle. With k = n we have a maximum, with k = 0 a minimum. From the Morse lemma follows, that near to a critical point a function of one variable may be replaced by the first non-zero term of its Taylor series without any qualitative changes. A simple case is a function f: R-->R, with f(0) = 0 and a nondegenerate critical point at the origin. The Morse lemma then states, that there is a smooth local change of coordinates under which f becomes
  ą x²
  near the critical point (which always means: to the next critical point) and where the sign is that of the second derivative.
2. From here we come to another important lemma, dealing with a coordinate transformation that makes functions "handier", amenable to classification. Let f: R-->R be a smooth function, such that f(0) = df(0) = ... = d^k-1f(0) = 0, but d^kf(0) is not zero. Then there exists a smooth local change of coordinates under which f takes the form
  x^k (k odd) or
  
  ą x^k (k even)
  and in the latter case the sign is that of d^kf(0).
3. Finally the time is ripe for the Splitting lemma. What it says, is that a function may be split into a Morse piece on one set of variables and a degenerate piece on a different set of variables, whose number is equal to the corank (see this word) of the function. The lemma goes: Let f: Rⁿ-->R be a smooth function, with df(0) = 0, whose Hessian matrix at 0 has rank r (and corank n - r). Then f is equivalent, around 0, to a function of the form ą x₁² ą ... ą x_r² + g (x_r+1, ..., x_n) where g: R^n-r-->R is smooth. The corank of the Hessian matrix is also the corank of the function at the critical point.
structural stability We say that a function p is small if all of its derivatives are small for x in a fixed neighbourhood of 0. We say that f is structurally stable if, for all sufficiently small functions p, the critical points of f and f + p have the same type. Near a Morse critical point any function is structurally stable. It can be proved, that a critical point is structurally stable if and only if it is nondegenerate; every degenerate critical point is structurally unstable.
Taylor series, Taylor expansion Around a point x_o a function f: R-->R can be expandend into a Taylor series, such that f(x + x_o) = a_o + a₁x + a₂x² + ... If this series is analytic, it may be differentiated term by term in a neighbourhood of x_o, and the coefficients a_r are given by
a_r = 1/r! D^rf(x_o)
If f(x) = f(-x) the series has no odd powers, if f(x) = -f(-x) it has only odd powers (cf. expansion of sin and cos functions).
transversality Two subspaces U, V of Rⁿ are transverse if they meet in a subspace whose dimension is as small as possible. If dim U = s and dim V = t then this minimal dimension is max (0, s + t - n). (So, transversality depends on the dimension of the surrounding space). Two submanifolds of Rⁿ meet transversely at a given point if they do not meet or their tangent affine hyperplanes meet transversely.
For example, two planes in R³ are transverse if they meet in a line (or do not coincide). Thom is connected with a transversality theorem stating that transversality is typical in the following sense: two manifolds taken at random are infinitely unlikely to intersect non-traversely. Transversality is a stable property, maintained under small perturbations. With one variable the condition for the first derivative Df to cross the zero function (zero line) transversely is that the Jacobian determinant is non-zero, i.e. that d²f/dx² is non-zero. This is the Morse condition, identifying a nondegenerate critical point. E.g. x³ and x⁴ do not meet the zero line y = 0 transversely in R².
typical, typicality, preservation of type under small perturbations A feature - e.g. transversality or the character of critical points - that is retained under perturbations or is valid throughout a family of functions is said to be typical.
unfolding, universal unfolding In loose language it could be said: "inside" a degenerate singularity f there are [codimension(f) + 1] critical points waiting to be "shaken loose" by a small perturbation, or a family of small perturbations. Such a family Thom called an unfolding of f. An r-unfolding of a function f: Rⁿ-->R is a function F: R^n+r-->R such that F(x₁,..., x_n, 0,..., 0) = f(x₁,..., x_n).
An r-unfolding of f is versal if all other unfoldings of f can be induced from it. It is universal if r is the smallest dimension for which a versal r-unfolding of f exists.

Till innehållsförteckningen för EKT/To ECT contents

Till artiklar på engelska/To articles in English

Till startsidan/Back to start page