Ebook: Numerical Bifurcation Analysis for Reaction-Diffusion Equations

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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This book provides the readers numerical tools for a systematic analysis of bifurcation problems in reaction- diffusion equations. Emphasis is put on combination of numerical analysis with bifurcation theory and application to reaction-diffusion equations. Many examples and figures are used to illustrate analysis of bifurcation scenario and implementation of numerical schemes. The reader will have a thorough understanding of numerical bifurcation analysis and the necessary tools for investigating nonlinear phenomena in reaction-diffusion equations.

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This book provides the readers numerical tools for a systematic analysis of bifurcation problems in reaction- diffusion equations. Emphasis is put on combination of numerical analysis with bifurcation theory and application to reaction-diffusion equations. Many examples and figures are used to illustrate analysis of bifurcation scenario and implementation of numerical schemes. The reader will have a thorough understanding of numerical bifurcation analysis and the necessary tools for investigating nonlinear phenomena in reaction-diffusion equations.
Content:
Front Matter....Pages i-xiv
Reaction-Diffusion Equations....Pages 1-6
Continuation of Nonsingular Solutions....Pages 7-29
Detecting and Computing Bifurcation Points....Pages 31-68
Branch Switching at Simple Bifurcation Points....Pages 69-84
Bifurcation Problems with Symmetry....Pages 85-100
Liapunov-Schmidt Method....Pages 101-127
Center Manifold Theory....Pages 129-150
A Numerical Bifurcation Function for Homoclinic Orbits....Pages 151-172
One-Dimensional Reaction-Diffusion Equations....Pages 173-198
Reaction-Diffusion Equations on a Square....Pages 199-229
Normal Forms for Hopf Bifurcations....Pages 231-254
Steady/Steady State Mode Interactions....Pages 255-281
Hopf/Steady State Mode Interactions....Pages 283-303
Homotopy of Boundary Conditions....Pages 305-329
Bifurcations along a Homotopy of Boundary Conditions....Pages 331-359
A Mode Interaction on a Homotopy of Boundary Conditions....Pages 361-388
Back Matter....Pages 389-414

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This book provides the readers numerical tools for a systematic analysis of bifurcation problems in reaction- diffusion equations. Emphasis is put on combination of numerical analysis with bifurcation theory and application to reaction-diffusion equations. Many examples and figures are used to illustrate analysis of bifurcation scenario and implementation of numerical schemes. The reader will have a thorough understanding of numerical bifurcation analysis and the necessary tools for investigating nonlinear phenomena in reaction-diffusion equations.
Content:
Front Matter....Pages i-xiv
Reaction-Diffusion Equations....Pages 1-6
Continuation of Nonsingular Solutions....Pages 7-29
Detecting and Computing Bifurcation Points....Pages 31-68
Branch Switching at Simple Bifurcation Points....Pages 69-84
Bifurcation Problems with Symmetry....Pages 85-100
Liapunov-Schmidt Method....Pages 101-127
Center Manifold Theory....Pages 129-150
A Numerical Bifurcation Function for Homoclinic Orbits....Pages 151-172
One-Dimensional Reaction-Diffusion Equations....Pages 173-198
Reaction-Diffusion Equations on a Square....Pages 199-229
Normal Forms for Hopf Bifurcations....Pages 231-254
Steady/Steady State Mode Interactions....Pages 255-281
Hopf/Steady State Mode Interactions....Pages 283-303
Homotopy of Boundary Conditions....Pages 305-329
Bifurcations along a Homotopy of Boundary Conditions....Pages 331-359
A Mode Interaction on a Homotopy of Boundary Conditions....Pages 361-388
Back Matter....Pages 389-414
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