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Numerical Bifurcation Analysis for Reaction-Diffusion Equations : Springer Computational Mathematics - Zhen Mei

Numerical Bifurcation Analysis for Reaction-Diffusion Equations

Springer Computational Mathematics

By: Zhen Mei

Hardcover Published: 21st June 2000
ISBN: 9783540672968
Number Of Pages: 414

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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.

Reaction-Diffusion Equationsp. 1
Introductionp. 1
Bifurcations and Pattern Formationsp. 2
Boundary Conditionsp. 4
Continuation Methodsp. 7
Parameterization of Solution Curvesp. 8
Natural parameterizationp. 8
Parameterization with arclengthp. 9
Parameterization with pseudo-arclengthp. 11
Local Parameterization of Solution Manifoldsp. 14
Predictor-Corrector Methodsp. 16
Euler-Newton methodp. 19
A continuation-Lanczos algorithmp. 22
A continuation-Arnoldi algorithmp. 25
Computation of Multi-Dimensional Solution Manifoldsp. 27
Detecting and Computing Bifurcation Pointsp. 31
Generic Bifurcation Pointsp. 31
One-parameter problemsp. 32
Two-parameter problemsp. 34
Test Functionsp. 35
Test functions for turning pointsp. 36
Test functions for simple bifurcation pointp. 40
Test functions for Hopf bifurcationsp. 43
Minimally extended systemsp. 46
Computing Simple Bifurcation Pointsp. 47
Simple bifurcation pointsp. 48
Extended systemsp. 49
Newton-like methodsp. 53
Rank-1 corrections for sparse problemsp. 56
A numerical examplep. 59
Computing Hopf Bifurcation Pointsp. 60
Hopf pointsp. 60
Extended systemsp. 62
Newton method for extended systemsp. 67
Branch Switching at Simple Bifurcation Pointsp. 69
Structure of Bifurcating Solution Branchesp. 70
Behavior of the Linearized Operatorp. 73
Euler-Newton Continuationp. 75
Branch Switching via Regularized Systemsp. 80
Other Branch Switching Techniquesp. 84
Bifurcation Problems with Symmetryp. 85
Basic Group Conceptsp. 86
Equivariant Bifurcation Problemsp. 90
Equivariant Branching Lemmap. 92
A Semi-linear Elliptic PDE on the Unite Squarep. 97
Liapunov-Schmidt Methodp. 101
Liapunov-Schmidt Reductionp. 101
Equivariance of the Reduced Bifurcation Equationsp. 104
Derivatives and Taylor Expansionp. 105
Equivalence, Determinacy and Stabilityp. 107
Simple Bifurcation Pointsp. 109
Truncated Liapunov-Schmidt Methodp. 110
Branch Switching at Multiple Bifurcation Pointsp. 112
Branch switching with prescribed tangentsp. 113
Branch switching with scaling techniquesp. 114
Corank-2 Problems with Dm-symmetryp. 118
Semilinear elliptic PDEs on a squarep. 118
A semilinear elliptic PDE on a hexagonp. 123
Center Manifold Theoryp. 129
Center Manifolds and Their Propertiesp. 129
Approximation of Center Manifoldsp. 132
Liapunov-Schmidt Reductionp. 136
Symmetry and Normal Formp. 139
Simple bifurcation pointsp. 140
Hopf bifurcationsp. 143
Waves in Reaction-Diffusion Equationsp. 145
Oscillating wavesp. 148
Long wavesp. 148
Long time and large spatial behaviorp. 150
A Bifurcation Function for Homoclinic Orbitsp. 151
A Bifurcation Functionp. 152
Approximation of Homoclinic Orbitsp. 154
Solving the Adjoint Variational Problemp. 156
Preserving the inner productp. 159
Systems with continuous symmetriesp. 162
The Approximate Bifurcation Functionp. 163
Examplesp. 165
Freire et al.'s circuitp. 165
Kuramoto-Sivashinsky equationp. 167
One-Dimensional Reaction-Diffusion Equationsp. 173
Introductionp. 173
Linear Stability Analysisp. 175
The general systemp. 175
The Brusselator equationsp. 178
Solution Branches at Double Bifurcationsp. 180
The reflection symmetry and its induced actionp. 182
(k,m) = (odd, odd) or (odd, even)p. 182
(k,m) = (even, even)p. 184
The Brusselator equationsp. 186
Central Difference Approximationsp. 187
General systemsp. 187
The Brusselator equationsp. 191
Numerical Results for the Brusselator Equationsp. 193
The length <$>\ell = 1<$>, diffusion rates d1 = 1, d2 = 2p. 193
The length <$>\ell = 10<$>, diffusion rates d1 = 1, d2 = 2p. 197
Reaction-Diffusion Equations on a Squarep. 199
D4-Symmetryp. 200
Eigenpairs of the Laplacianp. 202
Linear Stability Analysisp. 204
Bifurcation Pointsp. 207
Steady state bifurcation pointsp. 208
Hopf bifurcation pointsp. 213
Mode Interactionsp. 213
Steady/steady state mode interactionsp. 213
Hopf/steady state mode interactionsp. 216
Hopf/Hopf mode interactionsp. 217
Kernels of Du G0 and <$>(D_u G_0)^{\ast}<$>p. 217
Liapunov-Schmidt Reductionp. 221
Simple and Double Bifurcationsp. 222
Simple bifurcationsp. 222
Double bifurcations induced by the D4 symmetriesp. 223
Normal Forms for Hopf Bifurcationsp. 231
Introductionp. 231
Domain Symmetries and Their Extensionsp. 233
Actions of D4 on the Center Eigenspacep. 235
The Normal Formp. 237
Analysis of the Normal Formp. 238
Odd parityp. 239
Even parityp. 240
Brusselator Equationsp. 244
Linear stability analysisp. 245
Bifurcation scenariop. 247
Nonlinear degeneracyp. 251
Steady/Steady State Mode Interactionsp. 255
Induced Actionsp. 255
Interaction of Two D4-Modesp. 258
Interaction of two even modesp. 258
Interaction of an even mode with an odd modep. 260
Interaction of two odd modesp. 262
Mode Interactions of Three Modesp. 263
Induced actionsp. 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 Modesp. 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-Symmetryp. 275
Hopf/Steady State Mode Interactionsp. 283
Hopf/Steady State Mode Interactionsp. 283
Induced Actionsp. 286
Normal Formsp. 289
Bifurcation Scenariop. 293
Calculations of the Normal Formp. 299
Homotopy of Boundary Conditionsp. 305
Boundary Conditionsp. 305
Homotopy of boundary conditionsp. 306
Boundary conditions for different componentsp. 307
Mixed boundary conditions along the sidesp. 309
Dynamical boundary conditionsp. 309
A Brief Review of Sturm-Liouville Theoryp. 309
Laplacian with Robin Boundary Conditionsp. 312
Variational Formp. 316
Continuity of Solutions along the Homotopyp. 318
Neumann and Dirichlet Problemsp. 320
Properties of Eigenvaluesp. 322
One-dimensional problemsp. 323
Two-dimensional problemsp. 327
Bifurcations along a Homotopy of BCsp. 331
Introductionp. 332
Stability and Symmetriesp. 333
Normal Formsp. 335
Variations of Bifurcations along the Homotopyp. 337
(¿1, ¿2) = (odd, even) or (even, odd)p. 338
(¿1, ¿2) = (odd, odd)p. 339
(¿1, ¿2) = (even, even)p. 340
A Numerical Examplep. 340
Discretization with finite difference methodsp. 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 BCsp. 349
Bifurcation pointsp. 351
Bifurcation scenariosp. 354
A Mode Interaction on a Homotopy of BCsp. 361
Introductionp. 361
Symmetries and Normal Formsp. 363
Generic Bifurcation Behaviorp. 365
Solutions with the modes ¿1, ¿2p. 366
Pure ¿3-mode solutionsp. 367
Interactions of three modesp. 367
Scales of Solution Branchesp. 368
Secondary Bifurcationsp. 370
Secondary Hopf bifurcationsp. 372
Truncated Bifurcation Equationsp. 373
Derivatives with respect to homotopy parameterp. 376
Reduced Stabilityp. 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 interactionp. 380
A Numerical Examplep. 381
Solution branches along (0; ¿1(¿),¿)p. 381
Solution branches along (0, ¿2(¿),¿)p. 382
Mode interactionp. 383
Switching and continuation of solution branchesp. 385
List of Figuresp. 389
List of Tablesp. 393
Bibliographyp. 395
Indexp. 411
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540672968
ISBN-10: 3540672966
Series: Springer Computational Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 414
Published: 21st June 2000
Country of Publication: DE
Dimensions (cm): 24.13 x 16.23  x 2.72
Weight (kg): 0.69

Earn 456 Qantas Points
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