Reaction-Diffusion Equations / 1: |
Introduction / 1.1: |
Bifurcations and Pattern Formations / 1.2: |
Boundary Conditions / 1.3: |
Continuation Methods / 2: |
Parameterization of Solution Curves / 2.1: |
Natural parameterization / 2.1.1: |
Parameterization with arclength / 2.1.2: |
Parameterization with pseudo-arclength / 2.1.3: |
Local Parameterization of Solution Manifolds / 2.2: |
Predictor-Corrector Methods / 2.3: |
Euler-Newton method / 2.3.1: |
A continuation-Lanczos algorithm / 2.3.2: |
A continuation-Arnoldi algorithm / 2.3.3: |
Computation of Multi-Dimensional Solution Manifolds / 2.4: |
Detecting and Computing Bifurcation Points / 3: |
Generic Bifurcation Points / 3.1: |
One-parameter problems / 3.1.1: |
Two-parameter problems / 3.1.2: |
Test Functions / 3.2: |
Test functions for turning points / 3.2.1: |
Test functions for simple bifurcation point / 3.2.2: |
Test functions for Hopf bifurcations / 3.2.3: |
Minimally extended systems / 3.2.4: |
Computing Simple Bifurcation Points / 3.3: |
Simple bifurcation points / 3.3.1: |
Extended systems / 3.3.2: |
Newton-like methods / 3.3.3: |
Rank-1 corrections for sparse problems / 3.3.4: |
A numerical example / 3.3.5: |
Computing Hopf Bifurcation Points / 3.4: |
Hopf points / 3.4.1: |
Newton method for extended systems / 3.4.2: |
Branch Switching at Simple Bifurcation Points / 4: |
Structure of Bifurcating Solution Branches / 4.1: |
Behavior of the Linearized Operator / 4.2: |
Euler-Newton Continuation / 4.3: |
Branch Switching via Regularized Systems / 4.4: |
Other Branch Switching Techniques / 4.5: |
Bifurcation Problems with Symmetry / 5: |
Basic Group Concepts / 5.1: |
Equivariant Bifurcation Problems / 5.2: |
Equivariant Branching Lemma / 5.3: |
A Semi-linear Elliptic PDE on the Unite Square / 5.4: |
Liapunov-Schmidt Method / 6: |
Liapunov-Schmidt Reduction / 6.1: |
Equivariance of the Reduced Bifurcation Equations / 6.2: |
Derivatives and Taylor Expansion / 6.3: |
Equivalence, Determinacy and Stability / 6.4: |
Simple Bifurcation Points / 6.5: |
Truncated Liapunov-Schmidt Method / 6.6: |
Branch Switching at Multiple Bifurcation Points / 6.7: |
Branch switching with prescribed tangents / 6.7.1: |
Branch switching with scaling techniques / 6.7.2: |
Corank-2 Problems with Dm-symmetry / 6.8: |
Semilinear elliptic PDEs on a square / 6.8.1: |
A semilinear elliptic PDE on a hexagon / 6.8.2: |
Center Manifold Theory / 7: |
Center Manifolds and Their Properties / 7.1: |
Approximation of Center Manifolds / 7.2: |
Symmetry and Normal Form / 7.3: |
Hopf bifurcations / 7.4.1: |
Waves in Reaction-Diffusion Equations / 7.5: |
Oscillating waves / 7.5.1: |
Long waves / 7.5.2: |
Long time and large spatial behavior / 7.5.3: |
A Bifurcation Function for Homoclinic Orbits / 8: |
A Bifurcation Function / 8.1: |
Approximation of Homoclinic Orbits / 8.2: |
Solving the Adjoint Variational Problem / 8.3: |
Preserving the inner product / 8.3.1: |
Systems with continuous symmetries / 8.3.2: |
The Approximate Bifurcation Function / 8.4: |
Examples / 8.5: |
Freire et al.'s circuit / 8.5.1: |
Kuramoto-Sivashinsky equation / 8.5.2: |
One-Dimensional Reaction-Diffusion Equations / 9: |
Linear Stability Analysis / 9.1: |
The general system / 9.2.1: |
The Brusselator equations / 9.2.2: |
Solution Branches at Double Bifurcations / 9.3: |
The reflection symmetry and its induced action / 9.3.1: |
(k,m) = (odd, odd) or (odd, even) / 9.3.2: |
(k,m) = (even, even) / 9.3.3: |
Central Difference Approximations / 9.3.4: |
General systems / 9.4.1: |
Numerical Results for the Brusselator Equations / 9.4.2: |
The length <$>\ell = 1<$>, diffusion rates d1 = 1, d2 = 2 / 9.5.1: |
The length <$>\ell = 10<$>, diffusion rates d1 = 1, d2 = 2 / 9.5.2: |
Reaction-Diffusion Equations on a Square / 10: |
D4-Symmetry / 10.1: |
Eigenpairs of the Laplacian / 10.2: |
Bifurcation Points / 10.3: |
Steady state bifurcation points / 10.4.1: |
Hopf bifurcation points / 10.4.2: |
Mode Interactions / 10.5: |
Steady/steady state mode interactions / 10.5.1: |
Hopf/steady state mode interactions / 10.5.2: |
Hopf/Hopf mode interactions / 10.5.3: |
Kernels of Du G0 and <$>(D_u G_0)^{\ast}<$> / 10.6: |
Simple and Double Bifurcations / 10.7: |
Simple bifurcations / 10.8.1: |
Double bifurcations induced by the D4 symmetries / 10.8.2: |
Normal Forms for Hopf Bifurcations / 11: |
Domain Symmetries and Their Extensions / 11.1: |
Actions of D4 on the Center Eigenspace / 11.3: |
The Normal Form / 11.4: |
Analysis of the Normal Form / 11.5: |
Odd parity / 11.5.1: |
Even parity / 11.5.2: |
Brusselator Equations / 11.6: |
Linear stability analysis / 11.6.1: |
Bifurcation scenario / 11.6.2: |
Nonlinear degeneracy / 11.6.3: |
Steady/Steady State Mode Interactions / 12: |
Induced Actions / 12.1: |
Interaction of Two D4-Modes / 12.2: |
Interaction of two even modes / 12.2.1: |
Interaction of an even mode with an odd mode / 12.2.2: |
Interaction of two odd modes / 12.2.3: |
Mode Interactions of Three Modes / 12.3: |
Induced actions / 12.3.1: |
Interactions of the modes (m,n,k) =(even, odd, odd) / 12.3.2: |
Interactions of the modes (m,n,k) =(even, odd, even) / 12.3.3: |
Interactions of Four Modes / 12.4: |
Interactions of the modes (m, n, k, l) = (even, odd, even, odd) / 12.4.1: |
Interactions of the modes (m, n, k, l) = (even, even, even, odd) / 12.4.2: |
Reactions with Z2-Symmetry / 12.5: |
Hopf/Steady State Mode Interactions / 13: |
Normal Forms / 13.1: |
Bifurcation Scenario / 13.4: |
Calculations of the Normal Form / 13.5: |
Homotopy of Boundary Conditions / 14: |
Homotopy of boundary conditions / 14.1: |
Boundary conditions for different components / 14.1.2: |
Mixed boundary conditions along the sides / 14.1.3: |
Dynamical boundary conditions / 14.1.4: |
A Brief Review of Sturm-Liouville Theory / 14.2: |
Laplacian with Robin Boundary Conditions / 14.3: |
Variational Form / 14.4: |
Continuity of Solutions along the Homotopy / 14.5: |
Neumann and Dirichlet Problems / 14.6: |
Properties of Eigenvalues / 14.7: |
One-dimensional problems / 14.7.1: |
Two-dimensional problems / 14.7.2: |
Bifurcations along a Homotopy of BCs / 15: |
Stability and Symmetries / 15.1: |
Variations of Bifurcations along the Homotopy / 15.3: |
(κ1, κ2) = (odd, even) or (even, odd) / 15.4.1: |
(κ1, κ2) = (odd, odd) / 15.4.2: |
(κ1, κ2) = (even, even) / 15.4.3: |
A Numerical Example / 15.5: |
Discretization with finite difference methods / 15.5.1: |
Homotopy of (κ1(μ), κ2(μ)) from (1,2) to (2,3) / 15.5.2: |
Homotopy of (κl(μ), κ2(μ)) from (1,3) to (2,4) / 15.5.3: |
Homotopy of (κ1(μ), κ2(μ)) from (2,4) to (3,5) / 15.5.4: |
Forced Symmetry-Breaking in BCs / 15.6: |
Bifurcation points / 15.6.1: |
Bifurcation scenarios / 15.6.2: |
A Mode Interaction on a Homotopy of BCs / 16: |
Symmetries and Normal Forms / 16.1: |
Generic Bifurcation Behavior / 16.3: |
Solutions with the modes φ1, φ2 / 16.3.1: |
Pure φ3-mode solutions / 16.3.2: |
Interactions of three modes / 16.3.3: |
Scales of Solution Branches / 16.4: |
Secondary Bifurcations / 16.5: |
Secondary Hopf bifurcations / 16.5.1: |
Truncated Bifurcation Equations / 16.6: |
Derivatives with respect to homotopy parameter / 16.6.1: |
Reduced Stability / 16.7: |
Stability of solution branches at (0, λ1(μ),μ) / 16.7.1: |
Stability of solution branches at (0, λ2(μ), μ) / 16.7.2: |
Stability of solution branches at mode interaction / 16.7.3: |
Solution branches along (0; λ1(μ),μ) / 16.8: |
Solution branches along (0, λ2(μ),μ) / 16.8.2: |
Mode interaction / 16.8.3: |
Switching and continuation of solution branches / 16.8.4: |
List of Figures |
List of Tables |
Bibliography |
Index |