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.

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.

First monograph on numerical aspects of bifurcation theory Includes supplementary material: sn.pub/extras

Klappentext

This monograph is the first to provide readers with numerical tools for a systematic analysis of bifurcation problems in reaction-diffusion equations. Many examples and figures illustrate analysis of bifurcation scenario and implementation of numerical schemes. Readers will gain a thorough understanding of numerical bifurcation analysis and the necessary tools for investigating nonlinear phenomena in reaction-diffusion equations.

Inhalt

1. Reaction-Diffusion Equations.- 2. Continuation Methods.- 3. Detecting and Computing Bifurcation Points.- 4. Branch Switching at Simple Bifurcation Points.- 5. Bifurcation Problems with Symmetry.- 6. Liapunov-Schmidt Method.- 7. Center Manifold Theory.- 8. A Bifurcation Function for Homoclinic Orbits.- 9. One-Dimensional Reaction-Diffusion Equations.- 10. Reaction-Diffusion Equations on a Square.- 11. Normal Forms for Hopf Bifurcations.- 12. Steady/Steady State Mode Interactions.- 13. Hopf/Steady State Mode Interactions.- 14. Homotopy of Boundary Conditions.- 15. Bifurcations along a Homotopy of BCs.- 16. A Mode Interaction on a Homotopy of BCs.- List of Figures.- List of Tables.