Springer Computational Mathematics

By: Zhen Mei

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Hardcover | 21 June 2000

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432 Pages

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Reaction-diffusion equations are typical mathematical models in biology, chemistry and physics. These equations often depend on various parame- ters, e. g. temperature, catalyst and diffusion rate, etc. Moreover, they form normally a nonlinear dissipative system, coupled by reaction among differ- ent substances. The number and stability of solutions of a reaction-diffusion system may change abruptly with variation of the control parameters. Cor- respondingly we see formation of patterns in the system, for example, an onset of convection and waves in the chemical reactions. This kind of phe- nomena is called bifurcation. Nonlinearity in the system makes bifurcation take place constantly in reaction-diffusion processes. Bifurcation in turn in- duces uncertainty in outcome of reactions. Thus analyzing bifurcations is essential for understanding mechanism of pattern formation and nonlinear dynamics of a reaction-diffusion process. However, an analytical bifurcation analysis is possible only for exceptional cases. This book is devoted to nu- merical analysis of bifurcation problems in reaction-diffusion equations. The aim is to pursue a systematic investigation of generic bifurcations and mode interactions of a dass of reaction-diffusion equations. This is realized with a combination of three mathematical approaches: numerical methods for con- tinuation of solution curves and for detection and computation of bifurcation points; effective low dimensional modeling of bifurcation scenario and long time dynamics of reaction-diffusion equations; analysis of bifurcation sce- nario, mode-interactions and impact of boundary conditions.

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Non-Fiction Science Physics Mathematical Physics Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations

Reaction-Diffusion Equations	p. 1
Introduction	p. 1
Bifurcations and Pattern Formations	p. 2
Boundary Conditions	p. 4
Continuation Methods	p. 7
Parameterization of Solution Curves	p. 8
Natural parameterization	p. 8
Parameterization with arclength	p. 9
Parameterization with pseudo-arclength	p. 11
Local Parameterization of Solution Manifolds	p. 14
Predictor-Corrector Methods	p. 16
Euler-Newton method	p. 19
A continuation-Lanczos algorithm	p. 22
A continuation-Arnoldi algorithm	p. 25
Computation of Multi-Dimensional Solution Manifolds	p. 27
Detecting and Computing Bifurcation Points	p. 31
Generic Bifurcation Points	p. 31
One-parameter problems	p. 32
Two-parameter problems	p. 34
Test Functions	p. 35
Test functions for turning points	p. 36
Test functions for simple bifurcation point	p. 40
Test functions for Hopf bifurcations	p. 43
Minimally extended systems	p. 46
Computing Simple Bifurcation Points	p. 47
Simple bifurcation points	p. 48
Extended systems	p. 49
Newton-like methods	p. 53
Rank-1 corrections for sparse problems	p. 56
A numerical example	p. 59
Computing Hopf Bifurcation Points	p. 60
Hopf points	p. 60
Extended systems	p. 62
Newton method for extended systems	p. 67
Branch Switching at Simple Bifurcation Points	p. 69
Structure of Bifurcating Solution Branches	p. 70
Behavior of the Linearized Operator	p. 73
Euler-Newton Continuation	p. 75
Branch Switching via Regularized Systems	p. 80
Other Branch Switching Techniques	p. 84
Bifurcation Problems with Symmetry	p. 85
Basic Group Concepts	p. 86
Equivariant Bifurcation Problems	p. 90
Equivariant Branching Lemma	p. 92
A Semi-linear Elliptic PDE on the Unite Square	p. 97
Liapunov-Schmidt Method	p. 101
Liapunov-Schmidt Reduction	p. 101
Equivariance of the Reduced Bifurcation Equations	p. 104
Derivatives and Taylor Expansion	p. 105
Equivalence, Determinacy and Stability	p. 107
Simple Bifurcation Points	p. 109
Truncated Liapunov-Schmidt Method	p. 110
Branch Switching at Multiple Bifurcation Points	p. 112
Branch switching with prescribed tangents	p. 113
Branch switching with scaling techniques	p. 114
Corank-2 Problems with D_m-symmetry	p. 118
Semilinear elliptic PDEs on a square	p. 118
A semilinear elliptic PDE on a hexagon	p. 123
Center Manifold Theory	p. 129
Center Manifolds and Their Properties	p. 129
Approximation of Center Manifolds	p. 132
Liapunov-Schmidt Reduction	p. 136
Symmetry and Normal Form	p. 139
Simple bifurcation points	p. 140
Hopf bifurcations	p. 143
Waves in Reaction-Diffusion Equations	p. 145
Oscillating waves	p. 148
Long waves	p. 148
Long time and large spatial behavior	p. 150
A Bifurcation Function for Homoclinic Orbits	p. 151
A Bifurcation Function	p. 152
Approximation of Homoclinic Orbits	p. 154
Solving the Adjoint Variational Problem	p. 156
Preserving the inner product	p. 159
Systems with continuous symmetries	p. 162
The Approximate Bifurcation Function	p. 163
Examples	p. 165
Freire et al.'s circuit	p. 165
Kuramoto-Sivashinsky equation	p. 167
One-Dimensional Reaction-Diffusion Equations	p. 173
Introduction	p. 173
Linear Stability Analysis	p. 175
The general system	p. 175
The Brusselator equations	p. 178
Solution Branches at Double Bifurcations	p. 180
The reflection symmetry and its induced action	p. 182
(k,m) = (odd, odd) or (odd, even)	p. 182
(k,m) = (even, even)	p. 184
The Brusselator equations	p. 186
Central Difference Approximations	p. 187
General systems	p. 187
The Brusselator equations	p. 191
Numerical Results for the Brusselator Equations	p. 193
The length <$>\ell = 1<$>, diffusion rates d₁ = 1, d₂ = 2	p. 193
The length <$>\ell = 10<$>, diffusion rates d₁ = 1, d₂ = 2	p. 197
Reaction-Diffusion Equations on a Square	p. 199
D₄-Symmetry	p. 200
Eigenpairs of the Laplacian	p. 202
Linear Stability Analysis	p. 204
Bifurcation Points	p. 207
Steady state bifurcation points	p. 208
Hopf bifurcation points	p. 213
Mode Interactions	p. 213
Steady/steady state mode interactions	p. 213
Hopf/steady state mode interactions	p. 216
Hopf/Hopf mode interactions	p. 217
Kernels of D_u G₀ and <$>(D_u G_0)^{\ast}<$>	p. 217
Liapunov-Schmidt Reduction	p. 221
Simple and Double Bifurcations	p. 222
Simple bifurcations	p. 222
Double bifurcations induced by the D₄ symmetries	p. 223
Normal Forms for Hopf Bifurcations	p. 231
Introduction	p. 231
Domain Symmetries and Their Extensions	p. 233
Actions of D₄ on the Center Eigenspace	p. 235
The Normal Form	p. 237
Analysis of the Normal Form	p. 238
Odd parity	p. 239
Even parity	p. 240
Brusselator Equations	p. 244
Linear stability analysis	p. 245
Bifurcation scenario	p. 247
Nonlinear degeneracy	p. 251
Steady/Steady State Mode Interactions	p. 255
Induced Actions	p. 255
Interaction of Two D₄-Modes	p. 258
Interaction of two even modes	p. 258
Interaction of an even mode with an odd mode	p. 260
Interaction of two odd modes	p. 262
Mode Interactions of Three Modes	p. 263
Induced actions	p. 264
Interactions of the modes (m,n,k) =(even, odd, odd)	p. 265
Interactions of the modes (m,n,k) =(even, odd, even)	p. 268
Interactions of Four Modes	p. 269
Interactions of the modes (m, n, k, l) = (even, odd, even, odd)	p. 271
Interactions of the modes (m, n, k, l) = (even, even, even, odd)	p. 272
Reactions with Z₂-Symmetry	p. 275
Hopf/Steady State Mode Interactions	p. 283
Hopf/Steady State Mode Interactions	p. 283
Induced Actions	p. 286
Normal Forms	p. 289
Bifurcation Scenario	p. 293
Calculations of the Normal Form	p. 299
Homotopy of Boundary Conditions	p. 305
Boundary Conditions	p. 305
Homotopy of boundary conditions	p. 306
Boundary conditions for different components	p. 307
Mixed boundary conditions along the sides	p. 309
Dynamical boundary conditions	p. 309
A Brief Review of Sturm-Liouville Theory	p. 309
Laplacian with Robin Boundary Conditions	p. 312
Variational Form	p. 316
Continuity of Solutions along the Homotopy	p. 318
Neumann and Dirichlet Problems	p. 320
Properties of Eigenvalues	p. 322
One-dimensional problems	p. 323
Two-dimensional problems	p. 327
Bifurcations along a Homotopy of BCs	p. 331
Introduction	p. 332
Stability and Symmetries	p. 333
Normal Forms	p. 335
Variations of Bifurcations along the Homotopy	p. 337
(¿₁, ¿₂) = (odd, even) or (even, odd)	p. 338
(¿₁, ¿₂) = (odd, odd)	p. 339
(¿₁, ¿₂) = (even, even)	p. 340
A Numerical Example	p. 340
Discretization with finite difference methods	p. 341
Homotopy of (¿₁(¿), ¿²(¿)) from (1,2) to (2,3)	p. 345
Homotopy of (¿_l(¿), ¿₂(¿)) from (1,3) to (2,4)	p. 345
Homotopy of (¿₁(¿), ¿₂(¿)) from (2,4) to (3,5)	p. 347
Forced Symmetry-Breaking in BCs	p. 349
Bifurcation points	p. 351
Bifurcation scenarios	p. 354
A Mode Interaction on a Homotopy of BCs	p. 361
Introduction	p. 361
Symmetries and Normal Forms	p. 363
Generic Bifurcation Behavior	p. 365
Solutions with the modes ¿₁, ¿₂	p. 366
Pure ¿₃-mode solutions	p. 367
Interactions of three modes	p. 367
Scales of Solution Branches	p. 368
Secondary Bifurcations	p. 370
Secondary Hopf bifurcations	p. 372
Truncated Bifurcation Equations	p. 373
Derivatives with respect to homotopy parameter	p. 376
Reduced Stability	p. 378
Stability of solution branches at (0, ¿₁(¿),¿)	p. 379
Stability of solution branches at (0, ¿₂(¿), ¿)	p. 380
Stability of solution branches at mode interaction	p. 380
A Numerical Example	p. 381
Solution branches along (0; ¿₁(¿),¿)	p. 381
Solution branches along (0, ¿₂(¿),¿)	p. 382
Mode interaction	p. 383
Switching and continuation of solution branches	p. 385
List of Figures	p. 389
List of Tables	p. 393
Bibliography	p. 395
Index	p. 411
Table of Contents provided by Publisher. All Rights Reserved.

Springer Computational Mathematics

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Hardcover

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