Numerical bifurcation analysis for reaction diffusion equations

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.