ISBN: 3-540-67296-6
TITLE: Numerical Bifurcation Analysis for Reaction-Diffusion Equations
AUTHOR: Mei, Zhen
TOC:

1. Reaction-Diffusion Equations 1
1.1 Introduction 1
1.2 Bifurcations and Pattern Formations 2
1.3 Boundary Conditions 4
2. Continuation Methods 7
2.1 Parameterization of Solution Curves 8
2.1.1 Natural parameterization 8
2.1.2 Parameterization with arclength 9
2.1.3 Parameterization with pseudo-arclength 11
2.2 Local Parameterization of Solution Manifolds 14
2.3 Predictor-Corrector Methods 16
2.3.1 Euler-Newton method 19
2.3.2 A continuation-Lanczos algorithm 22
2.3.3 A continuation-Arnoldi algorithm 25
2.4 Computation of Multi-Dimensional Solution Manifolds 27
3. Detecting and Computing Bifurcation Points 31
3.1 Generic Bifurcation Points 31
3.1.1 One-parameter problems 32
3.1.2 Two-parameter problems 34
3.2 Test Functions 35
3.2.1 Test functions for turning points 36
3.2.2 Test functions for simple bifurcation point 40
3.2.3 Test functions for Hopf bifurcations 43
3.2.4 Minimally extended systems 46
3.3 Computing Simple Bifurcation Points 47
3.3.1 Simple bifurcation points 48
3.3.2 Extended systems 49
3.3.3 Newton-like methods 53
3.3.4 Rank-l corrections for sparse problems 56
3.3.5 A numerical example 59
3.4 Computing Hopf Bifurcation Points 60
3.4.1 Hopf points 60
3.4.2 Extended systems 62
3.4.3 Newton method for extended systems 67
4. Branch Switching at Simple Bifurcation Points 69
4.1 Structure of Bifurcating Solution Branches 70
4.2 Behavior of the Linearized Operator 73
4.3 Euler-Newton Continuation 75
4.4 Branch Switching via Regularized Systems 80
4.5 Other Branch Switching Techniques 84
5. Bifurcation Problems with Symmetry 85
5.1 Basic Group Concepts 86
5.2 Equivariant Bifurcation Problems 90
5.3 Equivariant Branching Lemma 92
5.4 A Semi-linear Elliptic PDE on the Unite Square 97
6. Liapunov-Schmidt Method 101
6.1 Liapunov-Schmidt Reduction 101
6.2 Equivariance of the Reduced Bifurcation Equations 104
6.3 Derivatives and Taylor Expansion 105
6.4 Equivalence, Determinacy and Stability 107
6.5 Simple Bifurcation Points 109
6.6 Truncated Liapunov-Schmidt Method 110
6.7 Branch Switching at Multiple Bifurcation Points 112
6.7.1 Branch switching with prescribed tangents 113
6.7.2 Branch switching with scaling techniques 114
6.8 Corank-2 Problems with D_m-symmetry 118
6.8.1 Semilinear elliptic PDEs on a square 118
6.8.2 A semilinear elliptic PDE on a hexagon 123
7. Center Manifold Theory 129
7.1 Center Manifolds and Their Properties 129
7.2 Approximation of Center Manifolds 132
7.3 Liapunov-Schmidt Reduction 136
7.4 Symmetry and Normal Form 139
7.4.1 Simple bifurcation points 140
7.4.2 Hopf bifurcations 143
7.5 Waves in Reaction-Diffusion Equations 145
7.5.1 Oscillating waves 148
7.5.2 Long waves 148
7.5.3 Long time and large spatial behavior 150
8. A Bifurcation Function for Homoclinic Orbits 151
8.1 A Bifurcation Function 152
8.2 Approximation of Homoclinic Orbits 154
8.3 Solving the Adjoint Variational Problem 156
8.3.1 Preserving the inner product 159
8.3.2 Systems with continuous symmetries 162
8.4 The Approximate Bifurcation Function 163
8.5 Examples .165
8.5.1 F'reire et al.'s circuit 165
8.5.2 Kuramoto-Sivashinsky equation 167
9. One-Dimensional Reaction-Diffusion Equations 173
9.1 Introduction 173
9.2 Linear Stability Analysis 175
9.2.1 The general system 175
9.2.2 The Brusselator equations 178
9.3 Solution Branches at Double Bifurcations 180
9.3.1 The reflection symmetry and its induced action 182
9.3.2 (k, m) = (odd, odd) or (odd, even) 182
9.3.3 (k, m) = (even,even) 184
9.3.4 The Brusselator equations 186
9.4 Central Difference Approximations 187
9.4.1 General systems 187
9.4.2 The Brusselator equations 191
9.5 Numerical Results for the Brusselator Equations 193
9.5.1 The length l = 1, diffusion rates d_1 = 1, d_2 = 2 193
9.5.2 The length l = 10, diffusion rates d_1 = 1, d_2 = 2 197
10. Reaction-Diffusion Equations on a Square 199
10.1 D_4-Symmetry 200
10.2 Eigenpairs of the Laplacian 202
10.3 Linear Stability Analysis 204
10.4 Bifurcation Points 207
10.4.1 Steady state bifurcation points 208
10.4.2 Hopf bifurcation points 213
10.5 Mode Interactions 213
10.5.1 Steady/steady state mode interactions 213
10.5.2 Hopf/steady state mode interactions 216
10.5.3 Hopf/Hopf mode interactions 217
10.6 Kernels of D_uG_0 and (D_uG_0)* 217
10.7 Liapunov-Schmidt Reduction 221
10.8 Simple and Double Bifurcations 222
10.8.1 Simple bifurcations 222
10.8.2 Double bifurcations induced by the D_4-symmetries 223
11. Normal Forms for Hopf Bifurcations 231
11.1 Introduction 231
11.2 Domain Symmetries and Their Extensions 233
11.3 Actions of D_4 on the Center Eigenspace 235
11.4 The Normal Form 237
11.5 Analysis of the Normal Form 238
11.5.1 Odd parity 239
11.5.2 Even parity 240
11.6 Brusselator Equations 244
11.6.1 Linear stability analysis 245
11.6.2 Bifurcation scenario 247
11.6.3 Nonlinear degeneracy 251
12. Steady/Steady State Mode Interactions 255
12.1 Induced Actions 255
12.2 Interaction of Two D_4-Modes 258
12.2.1 Interaction of two even modes 258
12.2.2 Interaction of an even mode with an odd mode 260
12.2.3 Interaction of two odd modes 262
12.3 Mode Interactions of Three Modes 263
12.3.1 Induced actions 264
12.3.2 Interactions of the modes (m, n, k) =(even, odd, odd) 265
12.3.3 Interactions of the modes (m, n, k) =(even, odd, even) 268
12.4 Interactions of Four Modes 269
12.4.1 Interactions of the modes (m, n, k, I) = (even, odd, even,odd) 271
12.4.2 Interactions of the modes (m, n, k, I) = (even, even, even,odd) 272
12.5 Reactions with Z_2-Symmetry 275
13. Hopf/Steady State Mode Interactions 283
13.1 Hopf/Steady State Mode Interactions 283
13.2 Induced Actions 286
13.3 Normal Forms 289
13.4 Bifurcation Scenario 293
13.5 Calculations of the Normal Form 299
14. Homotopy of Boundary Conditions 305
14.1 Boundary Conditions 305
14.1.1 Homotopy of boundary conditions 306
14.1.2 Boundary conditions for different components 307
14.1.3 Mixed boundary conditions along the sides 309
14.1.4 Dynamical boundary conditions 309
14.2 A Brief Review of Sturm-Liouville Theory 309
14.3 Laplacian with Robin Boundary Conditions 312
14.4 Variational Form 316
14.5 Continuity of Solutions along the Homotopy 318
14.6 Neumann and Dirichlet Problems 320
14.7 Properties of Eigenvalues 322
14.7.1 One-dimensional problems 323
14.7.2 Two-dimensional problems 327
15. Bifurcations along a Homotopy of BCs 331
15.1 Introduction .332
15.2 Stability and Symmetries 333
15.3 Normal Forms 335
15.4 Variations of Bifurcations along the Homotopy 337
15.4.1 (k_1, k_2 = (odd, even) or (even, odd) 338
15.4.2 (k_1, k_2) = (odd, odd) 339
15.4.3 (k_1, k_2)= (even, even) 340
15.5 A Numerical Example 340
15.5.1 Discretization with finite difference methods 341
15.5.2 Homotopy of (k_1(mu), k_2(mu)) from (1,2) to (2,3) 345
15.5.3 Homotopy of (k_1(mu), k_2(mu)) from (1,3) to (2,4) 345
15.5.4 Homotopy of (k_1(mu), k_2(mu)) from (2,4) to (3,5) 347
15.6 Forced Symmetry-Breaking in BCs 349
15.6.1 Bifurcation points 351
15.6.2 Bifurcation scenarios 354
16. A Mode Interaction on a Homotopy of BCs 361
16.1 Introduction 361
16.2 Symmetries and Normal Forms 363
16.3 Generic Bifurcation Behavior 365
16.3.1 Solutions with the modes phi_1, phi_2 366
16.3.2 Pure phi_3-mode solutions 367
16.3.3 Interactions of three modes 367
16.4 Scales of Solution Branches 368
16.5 Secondary Bifurcations 370
16.5.1 Secondary Hopf bifurcations 372
16.6 Truncated Bifurcation Equations 373
16.6.1 Derivatives with respect to homotopy parameter 376
16.7 Reduced Stability 378
16.7.1 Stability of solution branches at (0,lambda_1(mu),nu) 379
16.7.2 Stability of solution branches at (0,lambda_2(mu),nu) 380
16.7.3 Stability of solution branches at mode interaction 380
16.8 A Numerical Example 381
16.8.1 Solution branches along (0,lambda_1(mu),mu) 381
16.8.2 Solution branches along (0,lambda_2(mu),mu) 382
16.8.3 Mode interaction 383
16.8.4 Switching and continuation of solution branches 385
List of Figures 389
List of Tables 393
Bibliography 395
Index 411

