As one might expect, the applications of catastrophe theory in biology tend to occupy a position on the spectrum somewhere between those in physics and those in the social sciences. We do not usually know the dynamic, but we do generally have at least some idea of the processes involved. As a result, we are often in a position to judge whether or not the conditions necessary for catastrophe theory to be applicable are likely to be satisfied, and this puts us on much firmer ground than in the social sciences. Indeed, one of the aims of applying catastrophe theory in biology is to help us in the task of deducing the mechanism.

The two examples that we discuss in this chapter differ from those in Chapters 5 and 6 in an important respect. So far we have seen catastrophe theory applied to problems which had previously been studied by other methods. Here, in contrast, we have two case studies of catastrophe theory in action. In both cases, new results were obtained and (which should satisfy those who see it as the sole criterion for the usefulness of theory in science) further experiments were suggested.

The movement of a frontier

This is one of the first real applications of catastrophe theory. We begin with the statement and proof in more or less the same form as that originally given by Zeeman (1974). We then discuss part of the rest of that paper in order to see precisely what it is that the analysis accomplishes.

Almost every scientist has heard of catastrophe theory and knows that there has been a considerable amount of controversy surrounding it. Yet comparatively few know anything more about it than they may have read in some article written for the general public. The aim of this book is to make it possible for anyone with only a modest background in mathematics – no more than is usually included in a first year university course for students not specializing in the subject – to understand the theory well enough to follow the arguments in papers in which it is used and, if the occasion arises, to use it himself.

Most readers will find a number of concepts which are new to them; it would have been impossible to avoid this altogether and still give an adequate account of the theory. But wherever possible I have tried to keep to familiar ground. My object is to explain the theory, rather than to provide formal proofs, and it is almost always harder to understand anything if it is explained in terms of ideas which have themselves only just been introduced. For the same reason, I have sometimes carried out calculations by brute strength and awkwardness when a more elegant derivation was available. I have, however, tried to keep to the spirit, if not always the letter, of the mathematics, and the reader who uses this book as an introduction and then goes on to study the theorems in their full rigour should find that there is nothing he has to unlearn.

An important feature of catastrophe theory is that it can be used not only in many different problems, but also in many different ways. In an essay entitled ‘The two-fold way of catastrophe theory’, Thorn (1976) has characterized the two ends of the spectrum of the theory's applications as the ‘physical’ and the ‘metaphysical’:

It seems natural in a textbook to begin with the more secure examples, and so in this chapter we shall be discussing three applications of catastrophe theory in physics. Because the dynamics are known, almost all the calculations we shall perform will be standard, yet in each case catastrophe theory throws some new light on the problem. In return, these examples contribute to our understanding of catastrophe theory by serving as relatively straightforward illustrations, and also by showing how the range of applicability of the theory extends far beyond systems with gradient dynamics. Caustics For the study of many optical phenomena we may ignore the wave nature of light and consider the energy to be transported along curves known as light rays.

Introduction

A direct method for finding characteristic values is one in which values, which are exact apart from the inevitable rounding off errors, are found after a certain finite number of steps. Direct methods may thus be contrasted with iterative methods, which proceed by means of a sequence of approximate values. In this chapter we present one direct method by which the characteristic values of a symmetric matrix may be found, and another method for finding the characteristic values of an unsymmetric matrix. Thus the chapter is not, and is not intended to be, a compendium of methods; for this the reader must look elsewhere.

Both of the methods presented depend on the reduction of the matrix to ‘triple-diagonal form’ The technique applicable to symmetric matrices is described in § 9.3, and that applicable to unsymmetric matrices in § 9.8. When a symmetric matrix has been reduced to a symmetric triple diagonal form its characteristic values may be found by using certain convenient properties of what are called ‘Sturm's sequences’. These are described in §§ 9.4 and 9.5. A method for finding the characteristic vectors of the reduced, triple-diagonal matrix, and the corresponding vectors of the original matrix, is described in §§ 9.6 and 9.7.

In Chapter 1 we gave an account of elementary matrix theory and this was sufficient for the discussion of simple dynamical systems presented in Chapter 2. The reader who wishes to follow up this introductory theory by a systematic study of matrix methods has now three possibilities from which to choose.

The reader who is new to the subject, or whose interests are essentially practical, may wish to pass on immediately to Chapter 5. By this means he will escape what is necessarily a more mathematical portion of the book without missing any of the physically more important matters of dynamics.

Secondly, the reader who wants a fairly complete treatment of dynamical problems, but whose mathematical background is limited, should study §§3.1, 3.2, the first part of §3.7 and the whole of Chapter 4. In this way he will become familiar with the matrix analysis of systems having repeated natural frequencies. These systems do not occur exactly in reality, but they are important because they may be approximated very closely by real systems. Their analysis requires a little more care, but this is rewarded by added insight into the nature of vibration in general.

In this chapter we shall commence our discussion of the small oscillations of dynamical systems having a finite number of degrees of freedom. It will be assumed that there are no forces proportional to velocity acting on the systems; this will exclude damping and gyroscopic effects. It will mean also that the systems are ‘conservative’, since forces depending on displacements may always be obtained from a potential function. Our purpose in this chapter is to cast the theory of these systems into matrix form.

The reader who is already familiar with the basic theory, and who wishes immediately to pass on to the application of matrices in it should turn directly to § 2.1. For the benefit of readers who require either an introduction to the theory or a brief review of it, however, an introductory ‘§2.0’ is given which has no direct connection with matrix theory.

Introduction

The analytical determination of the vibration behaviour of a physical system may be divided into a number of stages:

1. The choice of a mathematical model which is to be used to represent the physical system. If the system were a wing of an aircraft, for instance, it would have to be decided whether to treat it as a continuous beam in flexure and/or torsion, as a plate or a box-like structure, as some lumped-mass system, or as some other type of system.

2. […]

The methods described in this chapter for the solution of linear equations spring from that taught to school children, elimination of the variables until a single equation in a single unknown is reached. In their hands, the method often leads to the conclusion that 0 = 0. To avoid this conclusion being drawn when it does not reflect any special property of the equations, the process will be systematised. The number of arithmetic operations (essentially multiplications) required to solve a set of equations of order n varies as n3. For quite moderate values of n, say 20, this provides plenty of opportunity for mistakes. A running check technique will be described to detect faults before much work based on wrong values has been carried out.

The large number of operations involved has a further consequence which is still of importance when automatic digital computers are used to carry out the arithmetic and checking techniques. This is that the number of round-off errors that are made can affect the result of a calculation to a very great extent. When work is carried out by hand, a skilled computer will see what is happening, and take avoiding action if it becomes necessary. The machine is not endowed with any native intelligence, and if the programmer does not provide for any necessary avoiding action, the machine will continue to run.

The analysis of systems having finite freedom is found to require manipulation of linear algebraic equations. These equations are clumsy in form, particularly if the system to which they relate has more than three or four degrees of freedom, so that some form of algebraic shorthand becomes desirable. Matrix notation serves this purpose. The essence of matrix methods is orderliness, and this feature makes the techniques particularly helpful in the setting out of problems for numerical solution by digital computers or by semi-skilled human computers.

The theme of this presentation of vibration theory is that sets of linear equations may be written concisely in matrix form, by the use of certain conventions. The matrices can then be manipulated (for instance to give other matrices) according to certain rules, re-interpretation at any stage being possible through the conventions. The purpose of this first chapter is merely to provide a background in matrix algebra before the treatment of the dynamical problems is presented.

Matrix notation and preliminary definitions

A matrix is an array of numbers (or ‘elements’), the positions as well as the magnitudes of which have significance. It is thus like a determinant, but with one essential difference.

The theory of vibration which is presented in the previous chapters relates to an idealised vibrating system having a finite number of degrees of freedom. This chapter is concerned mainly with continuous systems, that is, systems with an infinity of degrees of freedom. First it will be shown how receptances may be defined for continuous systems, then it will be shown how composite systems, whether continuous or discrete, may be analysed by the use of matrices. Finally, it will be shown how the theory developed for systems with a finite number of degrees of freedom may be applied to the vibration of continuous systems.

Throughout this chapter only undamped vibration will be discussed. But the reader will observe that much of the analysis may be generalised to cover damped vibrating systems. Throughout this chapter, therefore, all displacements and forces will be harmonic, and will have the same frequency and phase.

The receptances of continuous systems

The receptances of a discrete system—that is, a system with a finite number of degrees of freedom—were defined in § 2.2. Although they were first introduced as elements of the receptance matrix, their fundamental meaning is such that the receptance αxy gives the displacement at the coordinate x due to a harmonic force at the coordinate y.