A principal characteristic of “flight” is that a significant part of the aerodynamic force is needed to cancel the weight of the organism. Thus, certain features of flying apply to buoyant fish. In forward flight such a force can be obtained by creating horizontal bound vorticity, this being the main purpose of the lifting surfaces of the body. The soaring and gliding of birds provides a familiar example where the classical aerodynamics of fixed-wing aircraft can be applied at once, a problem we consider in Section 11.1. It is worth mentioning that there is evidence that observations of birds led Lanchester to the notions of circulation and induced drag of finite wings [see the engrossing historical summary in Durand (1963)].

However, gliding, insofar as it may be taken to be stationary in time, is a rather special instance of animal flight. The common maneuvers of natural flight – takeoff and landing, flapping flight, and hovering – are fundamentally time-dependent phenomena (see the estimate of the frequency parameter σ given in Chapter 1). In effect, natural fliers, in particular the hummingbirds and certain of the hovering insects, explore the problem of lift generation in a very different parameter range from that conventionally exploited in aeronautics. In later sections we examine some of these time-dependent problems in the context of unsteady airfoil theory.

For general references on this subject, see Gray (1968) and the review articles by Lighthill and Weis-Fogh in Wu et al. (1975, vol. 2). For well-illustrated popular accounts of insect flight, see Nachtigall (1968) and Weis-Fogh (1975b).

Introduction

This is the second basic swimming mechanism in the Stokesian realm. Although the organelles are apparently identical in ultrastructure and physiology, we use the term “flagellum” when there is only one, or a small number of these hairlike appendages on a cell, as in spermatozoans and the flagellates, and “cilia” to denote large numbers of them on the same cell, as in the ciliates. Cilia, then, are just a large number of flagella on the same cell. The ciliates tend to be larger than the cell bodies of spermatozoans by an order of magnitude. Cilia tend to be shorter than ftagella, however, and this fact puts their mechanism firmly in the Stokesian realm.

Confusion about terminology can arise from two sources: (1) in bacteria there are organelles called flagella with very different ultrastructure and physiology (Chapter 4); and (2) the term “cilia” is usually used in connection with various ciliated tissues in metazoans (manycelled animals) as in the lining of our respiratory tracts, but there is usually only one or a small number of cilia per cell in Lhese tissues.

Because of their proximity to the cell wall, we can expect to find that optimal propulsion using cilia will involve movements quite different from those of a single large flagellum. In this regard it would be valuable to understand the possible adaptive advantages that could have led to the evolution of ciliary motion. As both flagellary and ciliary modes are widespread and both are evidently successful strategies, we might suppose that there are trade-offs between them.

We now take up the mechanical principles underlying Eulerian swimming of a thin, flexible creature, with the aim of understanding morphology in terms of the mechanisms of propulsion. For summaries of the related biology see Lighthill (1975, chap. 2) and Wu (1971a).

Small-perturbation theory of slender fish

Many fish change shape rather gradually along the anterior–posterior axis. (There are, however, many exceptions: for example, angelfish.) As a first approximation, we consider the swimming of slender, neutrally bouyant fish, By slender we mean, among other things, that the cross-sectional area of the body changes slowly along its length. Notation and terminology are summarized in Figure 10.1.

Necessary conditions for the validity of slender-body theory are such as to ensure that velocity perturbations caused by the fish are a small fraction of its swimming speed. It is certainly sufficient that the body surface S be smooth and that tangent planes always make a small angle with the x axis. However, it turns out that if these conditions are exactly met and the cross-sectional area is zero at the extremities, the theory predicts zero mean thrust. Fortunately, the model also allows us to treat “slender” fish with sharp downstream edges, at the caudal fin, for example, even though s(x) may be discontinuous there. Also, it is not necessary that surface slope always be small, at the nose of the fish, for example, provided that this occurs over at most a fraction of its length.

This monograph developed from a course given at the Courant Institute in the spring of 1976. At that time a preliminary set of notes was prepared for the class by M. Lewandowsky, who also contributed Chapter 4 and made numerous helpful suggestions. My purpose at the time was to provide a brief complement to the relevant portions of Lighthill's book, Mathematical Biofluiddynamics, as well as a guide to the papers contained in the symposium proceedings Swimming and Flying in Nature, edited by Wu, et al, and to incorporate background and supplementary material as needed. The present book has been changed only by the addition of Chapter 12 on interactions and a number of revisions and corrections.

Although, as a result of these origins, this monograph falls far short of a definitive treatise, I hope that this glimpse of a fascinating and rapidly evolving body of research will be useful to students of fluid mechanics seeking a compact introduction to biological modeling. Those familiar with the aerodynamics of fixed-wing aircraft or the hydrodynamics of ships, to take two examples from many, are well aware that in these applications the theory is concerned with design as well as analysis. The products of our technology are both the subjects and the results of our mathematical modeling. In “natural” swimming and flying the situation is no longer the same. Nature has already provided the answers.

The applications in this chapter represent the opposite end of the spectrum to the physical systems of Chapter 5. When we are trying to analyse the behaviour of an individual, or of a group, we cannot write down a set of equations of motion for the system based on known quantitative laws, and then look to see what catastrophe theory has to say about the solutions of these equations. What we must do is quite different. If we observe in a system some or all of the features which we recognize as characteristic of catastrophes – sudden jumps, hysteresis, bimodality, inaccessibility and divergence – we may suppose, at least as a working hypothesis, that the underlying dynamic is such that catastrophe theory applies. We then choose what appear to be appropriate state and control variables and attempt to fit a catastrophe model to the observations.

Right from the start, we see one of the advantages of catastrophe theory in this sort of problem. The data which are available are often not quantifiable. We can generally order our observations; for example, we can tell whether a person has become more angry or less angry. And we can usually say whether or not a variable is continuous and whether it changes smoothly. On the other hand, algebraic concepts such as addition and multiplication generally have no meaning: it makes little real sense to say that someone has become twice as angry.

In this chapter we derive Thorn's famous list of seven elementary catastrophes. We have already seen that we need only consider Taylor series in one or two variables; we now have to find all the different cases which can arise with codimension no greater than 4.

We begin with some mathematical preliminaries. First, we have to be clear about what we mean by ‘different’ cases. The usual statement is that there are seven qualitatively different catastrophes, and when we come to discuss the applications of the theory it will be clear enough what this means. But we need a more precise definition if we are to know what sorts of mathematical operations we may use in the derivations, so we say that two catastrophes are equivalent if one can be transformed to the other by (i) a diffeomorphism of the control variables, and (ii) at each point in the control space a diffeomorphism of the state variables. The resulting family of state-variable diffeomorphisms must be smooth when considered as a function of the control variables.

A diffeomorphism is a one-to-one continuous differentiable transformation. It is sometimes useful (though not strictly accurate) to think of two geometric objects as being topologically equivalent, or homeomorphic, if one can be continuously deformed into the other without any tearing or pasting together. To the same degree of accuracy we may think of two geometric objects as being diffeomorphic if they are homeomorphic and if, in addition, the deformation involved no creasing or flattening of creases.

In this chapter we bring together the different ways in which we have seen that catastrophe theory can be applied, and we discuss how catastrophe theory compares with other methods as a means of explaining nature.

Applications of catastrophe theory

We began our study of the applications of catastrophe theory by quoting Thorn's observation that they form a spectrum of different types. Now that we have seen a number of applications we can describe the spectrum in more detail.

At the extreme ‘physical’ end, catastrophe theory is used much like any other mathematical method, to help us discover the properties of a known, or at any rate postulated, dynamic. For example, when we are studying the buckling of elastic structures or the stability of ships (Zeeman, 1977a), the concept of a universal unfolding and the techniques for finding one can warn us when our analysis is structurally unstable and can be used to suggest what further effects are to be expected. Catastrophe theory can also be used to establish results which are true for a large class of systems, each with a known dynamic; Berry's (1976) work on caustics, which we discussed in Chapter 5, is a good example.

In the middle of the spectrum are the applications to the study of systems whose mechanisms are not known in sufficient detail for us to write down and solve the equations that describe them, but for which we are reasonably confident that we know the sorts of equations that are involved.

One of the most interesting – and difficult – problems in biology is that of understanding development, the process by which a fertilized egg becomes first an embryo and then a fully formed organism. And an important aspect of development is morphogenesis, the creation of the various forms characteristic of the organism and its constituent parts. Of course the problem of form and the succession of forms is encountered in other branches of science as well, but it is in developmental biology that it is especially important. How is it that out of a single cell there can develop an organism which is to such a large extent the same as all others of the same species? How is it that this can happen even though different individuals within a species may be quite different in size and in certain details of their shape? And how is it that the process is so stable, allowing considerable variation in the environment and resisting many (though by no means all) perturbations?

In seeking an answer to these questions, the first step is to try to establish exactly what it is that the developmental process accomplishes. In the language of the earlier chapters of this book, what do we mean when we say that two individuals are ‘of the same form’? It is clearly impossible to give a completely satisfactory definition, but the mathematical relation that most nearly captures the essential idea is that of topological equivalence.

Like any other major advance, catastrophe theory depends on a number of concepts with which the reader new to the field is unlikely to be familiar. The aim of this chapter is to provide an introduction to these which is as simple as possible. A more sophisticated, but still relatively accessible, account is given by Poston & Stewart (1978a). Mathematicians might prefer Lu (1976) or, for full proofs, Brocker & Lander (1975) or Trotman & Zeeman (1976).

Structural stability

Implicit in science is the belief that there is some sort of order in the universe and that, in particular, experiments are generally repeatable. What is often not recognized is that what we demand of nature in this regard is not mere repeatability, but something rather more. It is never possible to reproduce exactly the conditions under which an experiment was performed. The quantity of one of the reagents may have been altered by 0.001%, the temperature may have increased by 0.0002 K, and the distance from the laboratory to the moon will probably be different as well. So what we really expect is not that if we repeat the experiment under precisely the same conditions we will obtain precisely the same results, but rather that if we repeat the experiment under approximately the same conditions we will obtain approximately the same results. This property is known as structural stability.

Now that we have the list of seven elementary catastrophes, we have to discover their properties. This is a comparatively straightforward task, and we shall carry out almost all the necessary calculations explicitly.

What we have to do is precisely what we did when we analysed the catastrophe machines in the first chapter. Given a potential, V, we define the equilibrium surface, M, by the equation

where the subscript x indicates that the gradient is with respect to the state variables only. This surface is made up of all the critical points of V, i.e. all the equilibria (stable or otherwise) of the system. We denote it by M to indicate that it is a manifold, a well-behaved smooth surface. It is not, by the way, obvious that M must be a manifold, but it can be proved that it is.

Next we find the singularity set, S, which is the subset of M which consists of all the degenerate critical points of V. These are the points at which ∇xV=0 and also

where H(V) is the Hessian of V, the matrix of second order partial derivatives which we defined in Chapter 2. We then project S down into the control space C (by eliminating the state variables from the equations which define it) to obtain the bifurcation set, B, which is the set of all points in C at which changes in the form of Voccur.

A great many of the most interesting phenomena in nature involve discontinuities. These may be in time, like the breaking of a wave, the division of a cell or the collapse of a bridge, or they may be spatial, like the boundary of an object or the frontier between two kinds of tissue. Yet the vast majority of the techniques available to the applied mathematician have been designed for the quantitative study of continuous behaviour. These methods, based primarily on the calculus, though very much refined and extended since the time of Newton and Leibniz, have made possible tremendous advances in our understanding of nature. Their great success has, however, been largely confined to the physical sciences. When we turn to the biological and social sciences, we generally find that we are unable to construct the relatively complete models which would permit us to apply the same methods. Moreover, the observations, which are the raw material from which the theoretician must work and the standard against which he must test his models, are seldom of the same precision as those which are available in physics. In many cases they are only qualitative. There is nothing in biology to compare with the inexorable and accurately predictable motions of the heavenly bodies.

As a part of mathematics, catastrophe theory is a theory about singularities. When applied to scientific problems, therefore, it deals with the properties of discontinuities directly, without reference to any specific underlying mechanism.