The Turing--Hopf Bifurcation Problem in Three-Dimensional Reaction-Diffusion Systems

Abstract

The article discusses the issues of studying local bifurcations in the vicinity of spatially homogeneous equilibrium positions of the reaction-diffusion system in a limited area with homogeneous Neumann boundary conditions. The main results concern the interaction between the Turing bifurcation and the Andronov--Hopf bifurcation, which leads to the Turing--Hopf bifurcation. In such a situation, there are two types of diffusive instabilities, which can lead to a variety of effects, accompanied by the emergence of complex space-time regimes. The article discusses in detail the conditions for Turing--Hopf bifurcations and offers new necessary and sufficient criteria bifurcations, leading to effective formulas for studying them. When solving the main problem of the signs of the Turing--Hopf bifurcation, we obtain calculation formulas for studying Turing and Andronov--Hopf bifurcations in two-dimensional and three-dimensional systems; these formulas are of independent theoretical and practical interest.