Catastrophe Theory and Its Applications

by Poston, Tim; Stewart, Ian

• ISBN13: 9780486789330
• ISBN10: 0486789330
• Format: Hardcover
• Copyright: 2013-12-23
• Publisher: Dover Publications

Summary

This first integrated treatment of the main ideas behind René Thom's theory of catastrophes aims to make them accessible to scientists wishing to apply the theory in their own fields of research. The mathematical basis of the theory is therefore explained with a minimum of technicalities, although some knowledge of the calculus of variables is assumed.
Thom's now-famous list of seven elementary catastrophes, broadly classifying various types of discontinuous change, is elucidated, as are the reasons for its appearance. Nearly half the book concentrates on detailed applications of the theory, emphasizing its uses in the physical sciences where applications can be made quantitative and can be experimentally verified. The more controversial and speculative applications to areas in the social sciences are also mentioned, but not discussed in detail.
Over 200 illustrations help clarify the ideas and applications in this volume, which will be of interest to researchers and postgraduate students in such diverse disciplines as engineering, mathematics, physics, and biology. 1978 edition. Bibliography.

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
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