Catastrophe Theory and Its Applications

Catastrophe Theory and Its Applications

Catastrophe Theory and Its Applications

Catastrophe Theory and Its Applications

eBook

$22.49  $29.95 Save 25% Current price is $22.49, Original price is $29.95. You Save 25%.

Available on Compatible NOOK Devices and the free NOOK Apps.
WANT A NOOK?  Explore Now

Related collections and offers

LEND ME® See Details

Overview

First integrated treatment of main ideas behind René Thom's theory of catastrophes stresses detailed applications in the physical sciences. Mathematics of theory explained with a minimum of technicalities. Over 200 illustrations clarify text designed for researchers and postgraduate students in engineering, mathematics, physics and biology. 1978 edition. Bibliography.

Product Details

ISBN-13: 9780486143781
Publisher: Dover Publications
Publication date: 05/05/2014
Sold by: Barnes & Noble
Format: eBook
Pages: 512
Sales rank: 994,428
File size: 28 MB
Note: This product may take a few minutes to download.

Read an Excerpt

Catastrophe Theory and its Applications


By Tim Poston, Ian Stewart

Dover Publications, Inc.

Copyright © 1978 T Poston and I N Stewart
All rights reserved.
ISBN: 978-0-486-14378-1



CHAPTER 1

Smooth and Sudden Changes

Classical physics (from Newton to General Relativity) is essentially the theory of various kinds of smooth behaviour; above all the awe-inspiring fall of the planets around the sun: unresting, unhasting and utterly regular. Even the wobbles that have dethroned Earth's rotation as the standard clock happen smoothly. No coherent and mathematical description of celestial mechanics can allow, say, a huge comet falling into the solar system, parting the Red Sea as it passes Earth, and then losing most of its kinetic energy and settling down into an almost perfectly circular orbit as the planet Venus (a widely held pseudoscientific theory). Planets interact much too evenly for that.


1 Catastrophes

Other things, however, jump. Water suddenly boils. Ice melts. Earths and moons quake. Buildings fall. The back of a camel is stable, we are told, under a load of N straws, but breaks suddenly under a load of N+1. Stock markets collapse.

These are sudden changes caused by smooth alterations in the situation: an analogous astronomical event would be the Sun's steady motion around the galaxy causing the Earth to switch (instantly or in a matter of days) to an orbit ten million miles wider, when some critical position was reached. Such changes are far more awkward for prediction and analysis than the stars in their courses, and the sciences (from physics to economics) are still gathering together the analytical techniques to handle such jumping behaviour.

Now there are many kinds of jump phenomena. There are forces that build up until friction can no longer hold them: the roar of an earthquake, and the rustle of rhubarb growing, are made by the movements when friction gives way. There is a critical population density below which certain creatures grow up as grasshoppers, above which as locusts: this is why locusts, when they do occur, do so in a huge swarm. A cell suddenly changes its reproductive rhythm and doubles and redoubles, cancerously. A man has a vision on the road to Tarsus.

Many of these still defy analysis: many have been analysed, with a tremendous variety of mathematical methods. We shall be concerned in this book with one particular mathematical context which covers a broad range of such phenomena in a coherent manner. The techniques involved were developed by the French mathematician René Thom and became widely known through his book Stabilité Structurelle et Morphogénèse in which he proposed them as a foundation for biology. The sudden changes involved were christened by Thom catastrophes, to convey the feeling of abrupt or dramatic change: the word's overtones of disaster are, for most applications, misleading. The subject has since become known as catastrophe theory, a phrase which is open to a variety of interpretations depending on the scope accorded it.

These techniques apply most directly (but far from exclusively) to systems that through varying situations seek at each moment to minimize some function (e.g. energy) or maximize one (e.g. entropy). We shall clarify in Chapter 3 what this means mathematically. For the present a good picture is that of a ball rolling around a landscape and 'seeking' through the agency of gravitation to settle in some position which, if not the lowest possible, is at least lower than any other nearby. (But meanwhile the landscape itself is changing.) The particular geometrical forms that arise in this setting have become known, following Thom, as elementary catastrophes, in the sense of fundamental entities (like chemical elements) and their use as expounded in this book is thus 'elementary catastrophe theory' (a phrase misinterpreted by Sussman and Zahler [la] to resemble' 'elementary arithmetic'), though it is deep both mathematically and scientifically. For some systems more complicated phenomena can occur (we give an easily explained example in Chapter 17 Section 7), whose onset Thom classes collectively as generalized catastrophes. Their theory is by no means so complete.

Physical intuition is important for the understanding of catastrophe theory. In this chapter we shall describe three simple physical systems exhibiting typical catastrophic behaviour, having the advantage that (unlike earthquakes or stock markets) they are simple enough to build, and small enough to carry around. In addition they may be used for elementary experiments. They are well adapted to analysis, although we shall not analyse them at this stage, and will be used repeatedly as examples. The reader will find his intuition very much assisted if he actually makes them (for which reason we give some practical indications as to their construction) and plays with them. No description can compete with direct experience. But it must be emphasized that these machines bear a similar relation to catastrophe theory as do the toys known as 'Newton's cradle' and 'the simple pendulum' to Newtonian mechanics.


2 The Zeeman Catastrophe Machine

We begin with the first machine invented. E. C. Zeeman, of the University of Warwick, devised it in 1969: after three weeks of experimentation with rubber bands and paperclips he refined it to the version we describe. The first appearance in print was Zeeman: other references include Poston and Woodcock and Dubois and Dufour.

It consists of a wheel (Fig. 1.1) mounted flat against a board, able to turn freely, and not too heavy: too much friction resisting the movement or inertia prolonging it obscure the behaviour we wish to study. To one point (B) on its edge are attached two lengths of elastic. One of these has its other end fixed to the board at point (A), far enough from the hub (O) of the wheel to keep the elastic BA always tight. The second has its other end (C) attached to a pointer, to be held in the hand. (The position of C can thus be controlled from a little distance without obscuring it.) Dimensions which work well in practice are a wheel of radius 3 cm, OA of length 12 cm, and each piece of elastic of unstretched length 6 cm.

Regardless of the radius r of the wheel, the unstretched lengths a and b of the elastic BA and BC, and the distance OA (as long as this is more than a + r) the machine will show qualitatively the behaviour to be described below. This is a part of the property of 'structural stability' which we discuss later: changes in the parameters make no essential qualitative difference, in a sense to be made precise at the relevant time. However, we shall be analysing the machine for one particular set of numbers, with the aid of computer graphics; so we now give detailed instructions for a machine whose behaviour has exactly the geometry drawn by the computer.

Photocopy Fig. 1.2. An enlarged or reduced photograph is perfectly acceptable, since scale does not affect the behaviour, or its subsequent analysis. Mount the result on board or heavy card, and attach a wheel at point O. Attach a stiff wire to the wheel, perpendicular to the plane of the board, at radius r from O. (It may be convenient to make the wheel itself a little larger, since only the position of the wire matters, and combine the mounting of the wire with that of the wheel as in Fig. 1.3.) Fix another stiff wire perpendicular to the board at A. File a groove round each wire, both at the same distance above the board, and higher than any central raised point of the wheel. Take a piece of good quality rubber cord (not a cut rubber band or sewing elastic: better the square section cord sold for catapults or model aeroplanes), some-what longer than four times the diameter 2r shown for the wheel in your copy of Fig. 1.2. Attach the middle to the wire at A, binding it with cotton to form a loop round the groove (Fig. 1.4(a)). Mark the point whose distance along the unstretched elastic is 2r, holding the elastic doubled and straight: bind on each side of it to form a «tight loop around the groove in wire B (Fig. 1.4(b)). Bind the point whose distance is 2r further along to a pointer (Fig. 1.4(c)). Care will be needed in making AB exactly 2r long: it may help to bind the doubled end to a loop, attach point B, and only then slip the loop over wire A.

Now experiment with the machine. You will find that if C is held equidistant with A from the board over a point on the board outside the four-pointed region [??] only one position can be occupied by the wheel under the influence of the elastic alone. If you push it to some other position and release it, it jumps back again. This one position will depend on that of C, but a smooth change in C will lead to a smooth change in the position of the wheel.

When C is inside [??], however, two such positions are possible. By entering [??] smoothly from one side, the wheel moves smoothly to one of them; entering on the other side and taking C to the same point carries the wheel to the other.

Only if you can now decide by pure thought what happens if you enter from the left by the upper/lower edge and leave on the right by the upper/lower edge of [??] (four possibilities altogether) do you not need to make this machine to understand it properly.

If time is of the essence, a few minutes will make a qualitatively accurate version of the machine, using stiff card for the board, drawing pins as wheel axle and wires, and elastic bands (all found in any office). Link the bands over pin B as shown in Fig. 1.5: if you fasten them down to the wheel they get tangled when it rotates. The position and exact shape of [??] will change a little, but can be found experimentally. (How?)


3 Gravitational catastrophe machines

Photocopy Fig. 1.6 (again scale does not matter) and back it with light card, about postcard thickness. Cut round the figure accurately (a knife or razor blade is best) and cut another piece of card into a ring a few centimetres wide whose outer edge is identical to that of the first. Make six triangular beams of equal length, about one quarter that of the axis of the parabola, as in Fig. 1.7. Glue them to points near the edge of the parabola, evenly spaced, with one at each corner; and to the corresponding positions on the ring, so that when laid on its face, the solid card has its boundary directly below that of the ring. A small heavy magnet behind the solid card will grip a light piece of metal in front (Fig. 1.8) and can be slid to any desired position while retaining a good grip.

Since most of the mass of the assembled device is in the magnet, we may take the centre of gravity of the whole to be the position of the magnet. When the machine balances steadily on edge, the centre of gravity must be vertically above the point of contact. If the machine rests on a level plane, the plane must be a tangent to the edge, so the centre of gravity lies on the corresponding normal (the line through the point of contact perpendicular to the tangent). The straight lines in Fig. 1.6 are some of these normals.

Experiments with the machine, or geometric thought along the above lines, will answer the following questions.

(a)What, if any, positions of the magnet give the machine N possible angles at which it can balance (where N = 0,1, 2, 3, ... and there is a new question for each choice of N)?

(b)Putting the magnet anywhere on the normal at a point P places the centre of gravity vertically above P, so the machine can balance at P. However, for some positions of the magnet on this normal the machine will return to a point of balance after a small wobble (that is, the equilibrium is stable), for others, it will topple over like an egg that has been stood on end (the equilibrium is unstable). What distinguishes the two?

(c)When does a small change in the position of the magnet leave the machine sitting in a position which is also slightly different, and when does it make the machine roll right over (a dramatic 'catastrophic change' which is unmistakable in a practical experiment)?


Now repeat the construction, with the parabola replaced by an ellipse. Fig. 1.6 is replaced by Fig. 1.9. Answer the same three questions in this case.

These machines are not as artificial as they may seem. Both of them turn out to correspond closely (Chapter 10) to larger scale phenomena in the behaviour of ships.


4 Catastrophe Theory

The complicated behaviour of the above machines shows that even simple problems in classical statics conceal many subtleties. A deeper analysis reveals that there are some underlying regularities in the mathematical structure which permit routine calculations of how such systems behave, based on the traditional applied mathematician's use of Taylor series approximations. But these techniques also conceal many subtleties. The main mathematical thrust of this book is to develop a proper understanding of the geometric and algebraic methods used to handle Taylor series properly. Once developed, the methods provide powerful tools for tackling a wide range of problems, going far beyond simple statics of artificial machines, and opening up perspectives to which the traditional use of Taylor expansions as a source of approximations, to be justified post hoc by experiment, is blind.

Catastrophe theory is not a single thread of ideas; it resembles more closely a web, with innumerable interconnected strands; these include physical intuition and experiment, geometry, algebra, calculus, topology, singularity theory and many others. This web is itself connected to and embedded in a larger web: the theory of dynamical systems. A proper perspective on the theory involves some appreciation of all of these strands and the way they combine. The elementary catastrophes of René Thom are but one strand, though an important one. That they only come in seven basically different shapes is an intriguing fact, but it is not the only significant feature to be dealt with. It is not Thom's theorem, but Thom's theory, that is the important thing: the assemblage of mathematical and physical ideas that lie behind the list of elementary catastrophes and make it work.

CHAPTER 2

Multidimensional Geometry

A proper understanding of catastrophe theory involves a feeling for the geometry of space of many dimensions, backed up by suitable algebraic and analytic techniques. This permits a geometric approach to the calculus of several variables: an important viewpoint which can motivate and simplify calculations by relating them to geometric insights.

The first few sections of this chapter review essential linear algebra, presenting the geometric view that is sometimes missed in treatments of 'matrix theory'. (For a more detailed geometric account, with proofs and many more pictures, see Dodson and Poston, which also develops the rigorous geometry of calculus in several variables.) We then take our first, classical steps in catastrophe theory. The most widely publicized feature of the theory has been the classification theorem mentioned above and discussed in Chapter 7: up to suitable changes of coordinates, a small number of standard forms are 'typical' for many phenomena. Coordinate changes thus play a key role in the theory. Here we show linear coordinate changes in action, reducing polynomial functions to a few standard expressions. This is both a key example of the kind of 'classification' the theory achieves in a far more glorious context, and a vital ingredient in what we do later. Much of the subsequent material aims to reduce other problems to those we solve in this chapter.


(Continues...)

Excerpted from Catastrophe Theory and its Applications by Tim Poston, Ian Stewart. Copyright © 1978 T Poston and I N Stewart. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

Preface
1 Smooth and sudden changes
1. Catastrophes
2. The Zeeman catastrophe machine
3. Gravitational catastrophe machines
4. Catastrophe theory
2 Multidimensional geometry
1. Set-theoretic notation
2. Euclidean space
3. Linear transformations
4. Matrices
5. Quadratic forms
6. Two-variable cubic forms
7. Polynomial geometry
3 Multidimensional calculus
1. Distance in Euclidean space
2. The derivative as tangent
3. Contours
4. Partial derivatives
5. Higher derivatives
6. Taylor series
7. Truncated algebra
8. The Inverse Function Theorem
9. The Implicit Function Theorem
4 Critical points and transversality
1. Critical points
2. The Morse Lemma
3. Functions of a single variable
4. Functions of several variables
5. The Splitting Lemma
6. Structural stability
7. Manifolds
8. Transversality
9. Transversality and stability
10. Transversality for mappings
11. Codimension
5 Machines revisited
1. The Zeeman machine
2. The canonical cusp catastrophe
3. Dynamics of the Zeeman machine
4. The gravitational machines
5. Formulation of a general problem
6 Structural stability
1. Equivalence of families
2. Structural stabillty of families
3. Physical interpretations of structural stability
4. The Morse and Splitting Lemmas for families
5. Catastrophe geometry
7 Thom's classification theorem
1. Functions and families of functions
2. One-parameter families
3. Non-transversaliity and symmetry
4. Two-parameter families
5. "Three-, four- and five-parameter families"
6. Higher catastrophes
7. Thom's theorem
8 Determinacy and unfoldings
1. Determine and strong determinacy
2. One-variable jet spaces
3. Infinitesimal changes of variable
4. Weaker determinacy conditions
5. Transformations that move the origin
6. Tangency and transversality
7. Codimension and unfoldings
8. Transversality and universality
9. Strong equivalence of unfoldings
10. Numbers associated with singularities
11. Inequalities
12. Summary of results and calculation methods
13. Examples and calculations
14. Compulsory remarks on terminology
9 The first seven catastrophe geometries
1. The objects of study
2. The fold catastrophe
3. The cusp catastrophe
4. The swallowtail catastrophe
5. The butterfly catastrophe
6. The elliptic umbilic
7. The hyperbolic umbilic
8. The parabolic umbilic
9. Ruled surfaces
10 Stability of ships
Static equilibrium
1. Buoyancy
2. Equilibrium
3. Stability
4. The vertical-sided ship
5. Geometry of the buoyancy locus
6. Metacentres
Ship shapes
7. The elliptical ship
8. The rectangular ship
9. Three dimensions
10. Oil-rigs
11. Comparison with current methods
11. The geometry of fluids
Background on fluid mechanics
1. What we are describing
2. Stream functions
3. Examples of flows
4. Rotation
5. Complex variable methods
Stability and experiment
6. Changes of variable
7. Heuristic programme
8. Experimental realization
Combining polymer molecules
9. Non-Newtonian behaviour
10. Extensional flows
Degenerate flows
11. The six-roll mill
12. The non-local bifurcation set of the elliptic umbilic
13. The six-roll mill with polymer solution
14. The 2n-roll mill
12 Optics and scattering theory
Ray optics
1. Caustics
2. The rainbow
3. Variational principles
4. Scattering
Wave optics
5. Asymptotic solutions of wave equations
6. Oscillatory integrals
7. Universal unfoldings
8. Orders of caustics
Applications
9. Scattering from a crystal lattice
10. Other caustics
11. Mirages
12. Sonic booms
13. Giant ocean waves
13 Elastic structures
General theory
1. Objects under stress
2. Elastic equilibria
3. Infinite-dimensional peculiarities
Euler struts
4. Finite element vision
5. Classical (1744) variational version
6. Perturbation analysis
7. Modern functional analysis
8. The buckling of a spring
9. The pinned strut
The geometry of collapse
10. Imperfection sensitivity
11. "(r, s)-Stability"
12. Optimization
13. Symmetry: rods and shells
Buckling plates
14. The von Kármán equations
15. Unfolding a double eigenvalue
Dynamics
16. Soft modes
17. Stiffness
14 Thermodynamics and phase transitions
Equations of state
1. van der Waals' equation
2. Ferromagnetism
Thermodynamic potentials
3. Entropy
4. Transforming the maximum entropy principle
5. Legendre transformations
6. Explicit potentials
7. The Landau theory
Fluctuations and critical exponents
8. Classical exponents
9. Topological tinkering
10. The rôle of fluctuations
11. Spatial variation
12. Partition functions
13. Renormalization group
14. Structural stability of renormalization
The rôle of symmetry
15. Even functions
16. The shapes of rotating stars
17. Symmetry breaking
18. Tricritical points
19. Crystal symmetries
20. Spectrum singularities
15 Laser physics
Preliminaries
1. Atoms
2. Field
3. Interaction
4. Measurement
The laser catastrophe
5. Unfolded Hamiltonian
6. Equations of motion
7. Mean field approximation
8. Boundary conditions
9. Non-equilibrium stationary manifold
Experiments
10. Laser transition
11. Optical bistability
12. Photocount distributions
Analytic correspondence
13. Equilibrium boundary conditions
14. Equilibrium manifold
15. Thermodynamic phase transition
16. Critical behaviour
17. Analytic correspondence of experiments
&
From the B&N Reads Blog

Customer Reviews