Introduction ................................................. xvii
1 Introduction to Dynamic Transitions ........................ 1
1.1 First Principles and Dynamic Models ........................ 1
1.1.1 Physical Laws and Mathematical Models ............... 1
1.1.2 Rayleigh-Benard Convection .......................... 2
1.1.3 Mathematical Formulation of Physical Problems ....... 5
1.2 Introduction to Dynamic Transition Theory .................. 7
1.2.1 Motivation and Key Philosophy ....................... 7
1.2.2 Principle of Exchange of Stability .................. 7
1.2.3 Equation of Critical Parameters ..................... 8
1.2.4 Classifications of Dynamic Transitions .............. 9
1.2.5 Structure and Characterization of Dynamic
Transitions ........................................ 11
1.2.6 General Features of Dynamic Transitions ............ 12
1.3 Examples of Typical Phase Transition Problems ............. 14
1.3.1 Rayleigh-Benard Convection ......................... 14
1.3.2 El Nino Southern Oscillation ....................... 15
1.3.3 Dynamic Transition Versus Transition in Physical
Space .............................................. 17
1.3.4 Andrews Critical Point and Third-Order Gas-Liquid
Transition ......................................... 20
1.3.5 Binary Systems ..................................... 21
2 Dynamic Transition Theory ................................. 25
2.1 General Dynamic Transition Theory ......................... 25
2.1.1 Classification of Dynamic Transitions .............. 25
2.1.2 Characterization of Transition Types ............... 28
2.1.3 Local Topological Structure of Transitions ......... 30
2.2 Continuous Transition ..................................... 34
2.2.1 Finite-Dimensional Systems ......................... 34
2.2.2 S'-Attractor Bifurcation ........................... 35
2.2.3 5'"-Attractor Bifurcation .......................... 42
2.2.4 Structural Stability of Dynamic Transitions ........ 42
2.2.5 Infinite-Dimensional Systems ....................... 44
2.3 Transition from Simple Eigenvalues ........................ 46
2.3.1 Real Simple Eigenvalues ............................ 46
2.3.2 Transitions from Complex Simple Eigenvalues ........ 49
2.3.3 Computation of b ................................... 55
2.4 Transition from Eigenvalues with Multiplicity Two ......... 57
2.4.1 Index Formula for Second-Order Nondegenerate
Singularities ...................................... 57
2.4.2 Bifurcation at Second-Order Singular Points ........ 60
2.4.3 The Case ind (F,0) = -2 ............................ 65
2.4.4 TheCaseind (F,0) = 2 ............................... 70
2.4.5 The Case ind (F,0) = 0 ............................. 78
2.4.6 Indices of kth-order Nondegenerate Singularities ... 80
2.4.7 Structure of kth-Order Nondegenerate
Singularities ...................................... 91
2.4.8 Transition from kth-Order Nondegenerate
Singularities ...................................... 93
2.4.9 Bifurcation to Periodic Orbits ..................... 97
2.4.10 Application to Parabolic Systems ................... 99
2.5 Singular Separation ...................................... 101
2.5.1 General Principle ................................. 101
2.5.2 Saddle-Node Bifurcation ........................... 103
2.5.3 Singular Separation of Periodic Orbits ............ 106
2.6 Perturbed Systems ........................................ 107
2.6.1 General Eigenvalues ............................... 107
2.6.2 Simple Eigenvalues ................................ 109
2.6.3 Complex Eigenvalues ............................... 113
2.7 Notes .................................................... 121
3 Equilibrium Phase Transition in Statistical Physics ...... 123
3.1 Dynamic Models for Equilibrium Phase Transitions ......... 124
3.1.1 Thermodynamic Potentials .......................... 125
3.1.2 Time-Dependent Equations .......................... 127
3.2 Classification of Equilibrium Phase Transitions .......... 131
3.3 Third-Order Gas-Liquid Phase Transition .................. 133
3.3.1 Introduction ...................................... 133
3.3.2 Time-Dependent Models for PVT Systems ............. 134
3.3.3 Phase Transition Dynamics for PVT Systems ......... 136
3.3.4 Physical Conclusions .............................. 139
3.4 Ferromagnetism ........................................... 142
3.4.1 Classical Theory of Ferromagnetism ................ 142
3.4.2 Dynamic Transitions in Ferromagnetism ............. 146
3.4.3 Physical Implications ............................. 150
3.4.4 Asymmetry of Fluctuations ......................... 152
3.5 Phase Separation in Binary Systems ....................... 153
3.5.1 Modeling .......................................... 154
3.5.2 Phase Transition in General Domains ............... 155
3.5.3 Phase Transition in Rectangular Domains ........... 160
3.5.4 Spatial Geometry, Transitions, and Pattern
Formation ......................................... 171
3.5.5 Phase Diagrams and Physical Conclusions ........... 174
3.6 Superconductivity ........................................ 181
3.6.1 Ginzburg-Landau Model ............................. 182
3.6.2 TGDL as a Gradient-Type System .................... 186
3.6.3 Phase Transition Theorems ......................... 189
3.6.4 Model Coupled with Entropy ........................ 197
3.6.5 Physical Conclusions .............................. 198
3.7 Liquid Helium-4 .......................................... 208
3.7.1 Dynamic Model for Liquid Helium-4 ................. 209
3.7.2 Dynamic Phase Transition for Liquid 4He ........... 212
3.8 Superfluidity of Helium-3 ................................ 218
3.8.1 Dynamic Model for Liquid 3He with Zero Applied
Field ............................................. 221
3.8.2 Critical Parameter Curves and РГ-Phase Diagram .... 223
3.8.3 Classification of Superfluid Transitions .......... 229
3.8.4 Liquid 3He with Nonzero Applied Field ............. 235
3.8.5 Physical Remarks .................................. 239
3.9 Mixture of He-3 and He-4 ................................. 239
3.9.1 Model for Liquid Mixture of 3He and 4He ........... 240
3.9.2 Critical Parameter Curves ......................... 242
3.9.3 Transition Theorems ............................... 243
3.9.4 Physical Conclusions .............................. 248
4 Fluid Dynamics ........................................... 249
4.1 Rayleigh-Benard Convection ............................... 250
4.1.1 Benard Problem .................................... 250
4.1.2 Boussinesq Equations .............................. 251
4.1.3 Dynamic Transition Theorems ....................... 253
4.1.4 Topological Structure and Pattern Formation ....... 258
4.1.5 Asymptotic Structure of Solutions for the Benard
Problem ........................................... 267
4.1.6 Structure of Bifurcated Attractors ................ 269
4.1.7 Physical Remarks .................................. 277
4.2 Taylor-Couette Flow ...................................... 277
4.2.1 Taylor Problem .................................... 278
4.2.2 Governing Equations ............................... 279
4.2.3 Narrow-Gap Case with Axisymmetric Perturbations ... 282
4.2.4 Asymptotic Structure of Solutions and Taylor
Vortices .......................................... 286
4.2.5 Taylor Problem with z-Periodic Boundary
Condition ......................................... 293
4.2.6 Other Boundary Conditions ......................... 309
4.2.7 Three-Dimensional Perturbation for the Narrow-
Gap Case .......................................... 313
4.2.8 Physical Remarks .................................. 315
4.3 Boundary-Layer and Interior Separations in the Taylor-
Couette-Poiseuille Flow .................................. 317
4.3.1 Model for the Taylor-Couette-Poiseuille Problem ... 318
4.3.2 Phase Transition of the TCP Problem ............... 321
4.3.3 Boundary-Layer Separation from the Couette-
Poiseuille Flow ................................... 325
4.3.4 Interior Separation from the Couette-Poiseuille
Flow .............................................. 332
4.3.5 Nature of Boundary-Layer and Interior
Separations ....................................... 336
4.4 Rotating Convection Problem .............................. 340
4.4.1 Rotating Boussinesq Equations ..................... 341
4.4.2 Eigenvalue Problem ................................ 342
4.4.3 Principle of Exchange of Stabilities .............. 347
4.4.4 Transition from First Real Eigenvalues ............ 354
4.4.5 Transition from First Complex Eigenvalues ......... 360
4.4.6 Physical Remarks .................................. 365
4.5 Convection Scale Theory .................................. 368
5 Geophysical Fluid Dynamics and Climate Dynamics .......... 373
5.1 Modeling and General Characteristics of Geophysical
Flows .................................................... 374
5.2 El Nino-Southern Oscillation ............................. 376
5.2.1 Walker Circulation and ENSO ....................... 377
5.2.2 Equatorial Circulation Equations .................. 378
5.2.3 Walker Circulation Under Idealized Conditions ..... 380
5.2.4 Walker Circulation Under Natural Conditions ....... 387
5.2.5 ENSO: Metastable Oscillation Theory ............... 393
5.3 Thermohaline Ocean Circulation ........................... 396
5.3.1 Boussinesq Equations .............................. 398
5.3.2 Linear Analysis ................................... 400
5.3.3 Nonlinear Dynamic Transitions ..................... 409
5.3.4 Convection Scales and Dynamic Transition .......... 416
5.4 Arctic Ocean Circulations ................................ 425
5.4.1 Model ............................................. 426
5.4.2 Linear Theory ..................................... 427
5.4.3 Transition Theorems ............................... 432
5.4.4 Revised Transition Theory ......................... 435
5.4.5 Physical Conclusions .............................. 437
5.5 Large-Scale Meridional Atmospheric Circulation ........... 438
5.5.1 Polar, Ferrel, and Hadley Cells ................... 438
5.5.2 β-Plane Assumption ............................... 439
5.5.3 Meridional Circulation Under Idealized
Conditions ........................................ 441
5.5.4 Physical Implications ............................. 445
6 Dynamical Transitions in Chemistry and Biology ........... 447
6.1 Modeling ................................................. 448
6.1.1 Dynamical Equations of Chemical Reactions ......... 448
6.1.2 Population Models of Biological Species ........... 450
6.2 Belousov-Zhabotinsky Chemical Reactions: Oregonator ...... 452
6.2.1 The Field-Koros-Noyes Equations ................... 452
6.2.2 Transition Under the Dirichlet Boundary
Condition ......................................... 455
6.2.3 Transitions Under the Neumann Boundary Condition .. 459
6.2.4 Phase Transition in the Realistic Oregonator ...... 470
6.3 Belousov-Zhabotinsky Reactions: Brusselator .............. 475
6.3.1 Prigogine-Lefever Model ........................... 476
6.3.2 Linearized Problem ................................ 477
6.3.3 Transition from Real Eigenvalues .................. 479
6.3.4 Transition from Complex Eigenvalues ............... 480
6.4 Bacterial Chemotaxis ..................................... 482
6.4.1 Keller-Segel Models ............................... 483
6.4.2 Dynamic Transitions for a Rich Stimulant System ... 489
6.4.3 Transition of Three-Component Systems ............. 499
6.4.4 Biological Conclusions ............................ 506
6.5 Biological Species ....................................... 508
6.5.1 Modeling .......................................... 508
6.5.2 Predator-Prey Systems ............................. 510
6.5.3 Three-Species Systems ............................. 517
Appendix A .................................................... 527
A.1 Formulas for Center Manifold Functions ................... 527
A.2 Dynamics of Gradient-Type Systems ........................ 533
References .................................................... 541
Index ......................................................... 553
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