Catastrophe Theory for Scientists and Engineers

Front Cover
Courier Corporation, Jan 1, 1993 - Technology & Engineering - 666 pages
Grounded in the work of Henri Poincare, R. Thom and others, catastrophe theory attempts to study how the qualitative nature of the solutions of equations depends on the parameters that appear in the equations. This advanced-level treatment describes the mathematics of catastrophe theory and its applications to problems in mathematics, physics, chemistry and the engineering disciplines. 1981 edition. References. Includes 28 tables and 397 black-and-white illustrations.
 

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Contents

PART
1
Change of Variables 1 Canonical Forms
15
Change of Variables 2 Perturbations
33
The Crowbar Principle
51
Summary
92
Catastrophe Organization
107
1 1
111
1 2
118
Extended MGL Models
384
Extended Dicke Models
387
Algorithm for Thermodynamic Phase Transitions
388
The MGL and Dicke Models
391
Extended MGL and Dicke Models
394
Crossover Theorem
401
Structural Stability and Canonical Kernels
404
Dynamical Equations of Motion
408

Contour Representations
132
Abutment
135
Summary
139
Catastrophe Conventions
141
The Conventions
142
Which Convention to Use
146
The Inadequacy of Conventions
154
Summary
155
Catastrophe Flags
157
Modality
158
Sudden Jumps
159
Divergence
160
Hysteresis
161
Divergence of Linear Response
162
Critical Slowing DownMode Softening
166
Anomalous Variance
170
Summary
182
PART 2
185
Thermodynamics
187
General Description of Phase Transitions
188
GinzburgLandau SecondOrder Phase Transitions
189
Topological Remarks
194
Critical Point of a Fluid
199
van der Waals Equation
203
Predictions of the van der Waals Equation
206
GinzburgLandau FirstOrder Phase Transitions
210
Tricritical Points
214
Maxwell Set for A5
220
Triple Point
222
Catastrophe Topology and Thermodynamics
224
Metric Geometry and Thermodynamics
229
Thermodynamic Partial Derivatives
235
Alternative Variational Representations
240
Fluctuations
246
Additional Questions
247
Summary
250
Structural Mechanics
254
Systems Governed by a Potential
255
Euler Buckling of a Beam
258
Collapse of a Shallow Arch
265
Exchange of Stability
271
Compound Systems
277
Engineering Optimization
281
Dangers of Engineering Optimization
282
Propped Cantilever C3v
286
Mode Softening
292
Summary
293
Aerodynamics
296
Truncated Dynamical System Equations
298
Linear Stability Analysis
300
Description of Aircraft
301
Aircraft Equations of Motion
302
SteadyState Manifolds
304
Application to a Particular Aircraft
305
The Bifurcation Set
306
Catastrophe Phenomena
308
Integration of the Equations of Motion
312
Energy as Lyapunov Function
315
Summary
317
Caustics and Diffraction Patterns
319
Stationary Phase Method
324
Degree of a Singularity
326
Caustics and Catastrophe Germs
331
Diffraction Patterns and Catastrophe Functions
337
Summary
342
JordanArnold Canonical Form
345
Program of Catastrophe Theory for Linear Systems
346
Perturbation
349
A2
355
A3
356
Ak
358
Bifurcation Set for JordanArnold Canonical Forms
360
Summary
365
Quantum Mechanics
367
The Operators
369
The Classical Limit
371
Collective Model Hamiltonians
374
Algorithm for Ground State Energy Phase Transitions
378
The MGL Model
379
The Dicke Model
382
Steady States Far from Equilibrium
412
Multiple Stability
417
Summary
423
Climate
428
Pacemaker of the Ice Ages
429
The Milankovitch Theory
433
Test of the Milankovitch Theory
439
Connection with Catastrophe Theory
441
Reduction to Cusp Form
444
Toward a Useful Program
446
Epilogue
447
PART 3
449
Beyond the Elementary Catastrophes
451
1 457 4 SymmetryRestricted Catastrophes
458
Constraint Catastrophes
462
Summary
466
Gradient Dynamical Systems
468
Noncanonical Form for Gradient Systems
469
Equilibria
470
Degeneracies
472
The Branching Tree
480
Marching
488
Relation Between Catastrophe Theory and Bifurcation Theory
494
Summary
499
Autonomous Dynamical Systems
501
Reduction to Gradient Form
502
F0
503
F 0
505
n 2
508
Perturbations Around Equal Nonzero Eigenvalues
510
n 2
513
Perturbations of a Saddle Node
518
The Hopf Bifurcation
524
Summary
529
Equations Exhibiting Catastrophes
531
Aufbau Principle 1
532
Van der Pol Oscillator 1 Hopf Bifurcation
535
Van der Pol Oscillator 2 Relaxation Oscillations
539
Aufbau Principle 2
542
Strange Building Blocks 1 Spiral Chaos
551
Strange Building Blocks 2 Lorenz Attractor
553
Hydrodynamics The Bénard Instability
562
Electrodynamics The LaserSpiking Instability
565
Aufbau Principle 3
568
Center Manifold Theorem
573
RuelleTakens Picture of Turbulence
575
Summary
581
PART 4
583
Thoms Theorem
585
Topology
586
Stable Functions
588
Generic Properties
591
Singularities of Mappings
593
Thoms Theorem
595
Summary
596
Transversality
597
Transversality
598
Linear Terms
606
Quadratic Terms
607
Germs with 1 1
611
Germs with 1 2
612
Germs with 1 3 k 2 6
613
Determinacy and Unfolding
615
Change of Variables
616
Determinacy Truncating the Taylor Series
617
Unfolding Universal Perturbations
620
Rules for Determinacy
623
Rules for Unfolding
626
Rules for Germs
629
The Implicit Function Theorem
634
The Morse Lemma
635
E6 E7 Eg
636
Multiple Cusps
637
Some Simple Germs
639
Summary
641
Epilogue
643
Author Index
647
Subject Index
651
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