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.