Underlined terms, person names etc. are connected to explanations by means of fragment identifiers

Now, let us look at a dynamical process in a hypothetical environment, a process producing n variants of some outcome. It could for example be an ecosystem, producing a sequence of delicate balances between its constituent species. Let it be given by an equation system

dxwhere x measures the population level of each species, t is time and a, b... are auxiliary parameters, describing factors affecting the overall process. A_{j}/dt = X_{j}(x_{j}, t, a, b,...)

Local

This implies: we can leave the complete, incalculable systems of differential equations describing the "interior" qualities of a dynamical system aside and concentrate on e.g. a more manageable gradient function (some quadratic or cubic form, more about this later on), producing the catastrophes. As early as 1971 Thom had outlined this shift away from a purely quantitative (and in most cases: vain) approach to a

How does a small perturbation influence a critical point? We say thatEven if the right hand side of(the system of differential equations: dx_{j}/dt = X(x_{j}, t, etc.))is given explicitly, it is nevertheless impossible to integrate formally. To get the solution one has to use approximating procedures.

For these two reasons, one has to know to what extent a slight perturbation on the right hand side of the equation system may effect the global behaviour of the solutions. To overcome - at least partially - these difficulties, the mathematician Henri Poincaré introduced in 1881 a radically new approach, the theory of 'Qualitative Dynamics'. Instead of trying to get explicit solutions of the equation system, one aims for a global geometrical picture of the system of trajectories, defined by the fieldX.If this can be done, one is able to describe qualitatively the asymptotic behaviour of any solution. This is in fact what really matters: in most practical situations, one is interested not in a quantitative result, but in the qualitative outcome of the evolution. Thus, qualitative dynamics, despite the considerable weakening of its program, remains a very useful - although very difficult - theory. (2)

Transversality and structural stability are the topics of Thom's important transversality and isotopy theorems; the first one says that transversality is a stable property, the second one that

So, in a given state function, catastrophe theory separates between two kinds of functions: one "Morse" piece, containing the nondegenerate critical points, and one piece, where the (parametrized)

This catastrophe, says Thom, can be interpreted as "the start of something" or "the end of something", in other words as a "limit", temporal or spatial. In this particular case (and only in this case) the complete graph in internal (x) and external space (y) with the control parameter

One point should be stressed already at this stage, it will be repeated again later on. In "Topological models..." (2) Thom remarks on the "informational content" of the degenerate critical point:

This notion of universal unfolding plays a central role in our biological models. To some extent, it replaces the vague and misused term of 'information', so frequently found in the writings of geneticists and molecular biologists.(My emphasis; CP)The 'information' symbolized by the degenerate singularity V(x) is 'transcribed', 'decoded' or 'unfolded' into the morphology appearing in the space of external variables which span the universal unfolding family of the singularity V(x).

Finally we look for the position of the degenerate critical points projected on (a,b)-space, this projection has given the catastrophe its name: the "cusp" (Fig. 4). (An arrowhead or a spearhead is a cusp). The edges of the cusp, the

TO BE CONTINUED

- Stjernfelt, F. (1992): Formens betydning. Katastrofeteori og semiotik. Akademisk forlag.
*In Danish, with semiotics as main subject field, written by the foremost introducer of the catastrophe theory in Scandinavia; this book was my first really helpful introduction to catastrophe theory after reading Thom (2). The proofreading of the text is catastrophic; it could be said in its favour that it tests the attention of the reader.* - Thom, R. (1971): Topological models in biology. In: Towards a Theoretical Biology, ed. C. H. Waddington.
*Four volumes: 1969 (Prolegomena), 1970 (Drafts), 1971 (Sketches), 1972 (?). This is a wonderful project, with utopian overtones; a couple of texts are among the most valuable biological papers (or: "meta"-papers) written in the 20th century.* - Zeeman, E. C. (1977): Catastrophe Theory. Selected Papers 1972 - 1977. Addison-Wesley Publ. Co.
*This book is a patchwork: a collection of lecture scripts and papers, but still very illuminating. Zeeman has a typically British inclination towards empiricism, 'positive' thought, always grabbing for any possible application; throughout the book he tries to reconcile this attitude with the more esoteric, lofty approach of Thom, the result being a highly creative culture clash between "English" and "French" basic attitudes to thinking. (On the other hand Thom's "Songeries ferroviaires" from 1979 seems to be British-inspired; this fruitful exchange has never been unilateral). In this way the book also comes to depict the practical difficulties of the catastrophe theory ("Where are the damn morphogens?") as much as its possibilities.* - Poston, T. & I. Stewart (1978): Catastrophe Theory and its Applications. Pitman Publishers. A more strict textbook with all the mathematics one can need and use, dedicated to E. C. Zeeman, but the authors also acknowledge: "In the beginning was Thom." This book is extremely rewarding reading, function theory should be taught this way! I particularly noted the following lines in the preface:
*It has been said more than once that it is possible to apply Thom's theorem without understanding the mathematics behind it: we disagree. In fact we disagree with the implication that it is Thom's*(My bold type; CP). I am inclined to agree - but in the long run most applications are dead without the metaphysical "spark" from Thom! The optimal approach in these matters is bimodal, at least in this respect.**theorem**that should be applied: analysis of the most solid and successful applications shows that**the methods and concepts that lie behind the theorem are often of greater importance than the result itself.** - Rosen, R. (1970): Dynamical System Theory in Biology. Vol. 1: Stability Theory and its Applications. John Wiley and Sons.
*The name Rosen is still among the first to come up when the question is put: Who understands e. g. structural stability - more than the "inner circle" of catastrophe theory? The early year of publication indicates, that Robert Rosen should be included among the pioneers of at least stability theory. I haven't read much yet, but to my delight I understand what I didn't understand when I opened the book twenty years ago. There is still hope for us: the human mind can be improved.* - An "engineering", practical approach - in the tradition of Thom and Zeeman, who did similar things - to catastrophe theory can be found on http://perso.wanadoo.fr/l.d.v.dujardin/ct, "An introduction to Catastrophe Theory for experimentalists" by Lucien Dujardin, Laboratory of Parasitology, University of Lille. I love labyrinthic web structures of this kind, with small surprises behind every new corner. Anyone who wants to feel his or her way to catastrophe theory by means of the computer keyboard should visit this page. One French, one English version, I will return with more comment when I have read it all. [CP]

(This excurse has been spotted by readers from the first second - the effectivity of search engines always amazes me! It is my first attempt with catastrophe theory, so there will be initial errors, in language, in thinking. I will detect them in due time - but any suggestion saving me from a blunder will be welcome! /