Zhen Mei Reihenfolge der Bücher

The book focuses on the numerical analysis of bifurcation problems in reaction-diffusion equations, which are essential in understanding pattern formation and nonlinear dynamics in various scientific fields. It explores how changes in parameters can lead to abrupt shifts in solutions and the emergence of complex patterns, such as convection and waves. The author employs three mathematical approaches: numerical methods for solution continuation, low-dimensional modeling of bifurcation scenarios, and analysis of boundary conditions' effects, aiming for a systematic investigation of generic bifurcations and mode interactions.

Reaction-diffusion equations serve as fundamental mathematical models in biology, chemistry, and physics, influenced by parameters such as temperature, catalyst, and diffusion rate. Typically forming a nonlinear dissipative system, these equations exhibit complex interactions among various substances. The number and stability of solutions can change dramatically with variations in control parameters, leading to pattern formation, such as convection and waves in chemical reactions, a phenomenon known as bifurcation. The inherent nonlinearity of these systems results in constant bifurcation occurrences, introducing uncertainty in reaction outcomes. Therefore, analyzing bifurcations is crucial for understanding pattern formation and the nonlinear dynamics of reaction-diffusion processes. However, analytical bifurcation analysis is limited to exceptional cases. This work focuses on the numerical analysis of bifurcation problems within reaction-diffusion equations, aiming for a systematic exploration of generic bifurcations and mode interactions. This goal is achieved through a combination of three mathematical approaches: numerical methods for the continuation of solution curves and detection of bifurcation points; effective low-dimensional modeling of bifurcation scenarios and long-term dynamics; and analysis of bifurcation scenarios, mode interactions, and the effects of boundary conditions.