Reaction-diffusion equations are typical mathematical models in biology, chemistry and physics. These equations often depend on various parameยญ ters, e. g. temperature, catalyst and diffusion rate, etc. Moreover, they form normally a nonlinear dissipative system, coupled by reaction among differยญ ent substances. The number and stability of solutions of a reaction-diffusion system may change abruptly with variation of the control parameters. Corยญ respondingly we see formation of patterns in the system, for example, an onset of convection and waves in the chemical reactions. This kind of pheยญ nomena is called bifurcation. Nonlinearity in the system makes bifurcation take place constantly in reaction-diffusion processes. Bifurcation in turn inยญ 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 nuยญ 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 conยญ 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 sceยญ nario, mode-interactions and impact of boundary conditions
CONTENT
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
