Catastrophe Theory and Its Applications

by Poston, Tim; Stewart, Ian

• ISBN13: 9780486692715
• ISBN10: 048669271X
• Format: Paperback
• Copyright: 2012-02-29
• Publisher: Dover Publications

Summary

First integrated treatment of main ideas behind Rene 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.

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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