Operator Methods for Investigating Stability and BifurcationProblems in a Reaction–Diffusion System and Applications

The article discusses the issues of studying stability and bifurcations in the reaction–diffusion system in a bounded domain with homogeneous Neumann boundary conditions. The main results of the article concern the study of problems of local bifurcations in the vicinity of spatially homogeneous equilibrium positions. A general scheme is proposed that allows us to obtain new formulas for studying the main characteristics of multiple equilibrium bifurcation and bifurcation Andronov–Hopf: sufficient signs of bifurcations, their type, approximate construction of solutions, stability analysis. The proposed approaches do not require complex and cumbersome transformations, the results obtained are brought to calculation formulas and algorithms. Some applications in problems of diffusion instability and corresponding bifurcations in reaction–diffusion systems are also discussed. The distributed model of the “brusselator” is considered as the main illustrative example.