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# 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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$220.74 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 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 Dm-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<$>, diffusion rates d1 = 1, d2 = 2 p. 197 Reaction-Diffusion Equations on a Square p. 199 D4-Symmetry p. 200 Eigenpairs of the Laplacian p. 202 Linear Stability Analysis p. 204 Bifurcation Points p. 207 Steady state bifurcation points p. 208 Hopf bifurcation points p. 213 Mode Interactions p. 213 Steady/steady state mode interactions p. 213 Hopf/steady state mode interactions p. 216 Hopf/Hopf mode interactions p. 217 Kernels of Du G0 and <$>(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. 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ISBN: 9783540672968
ISBN-10: 3540672966
Series: Springer Computational Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 414
Published: 21st June 2000
Publisher: SPRINGER VERLAG GMBH
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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