Spectral properties of the "reaction-diffusion" system operators and bifurcations signs

The article discusses differential equations that arise when modeling reaction-diffusion systems. Questions about the stability of equilibrium points in critical cases, as well as about bifurcations in the vicinity of such points, are studied. The main attention is paid to the linearized problem operators spectral properties study. The spectrum discreteness was established, the root properties and invariant subspaces were studied, and formulas for eigenfunctions were proposed. As an application, questions about the multiple equilibrium bifurcation signs and Andronov-Hopf bifurcation in the vicinity of equilibrium points are discussed. Examples are given to illustrate the proposed approaches effectiveness in studying stability and bifurcations problems.