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.
For citation: Vasenina, N. A. and Yumagulov, M. G. The Turing-Hopf Bifurcation Problem in Three-Dimensional Reaction-Diffusion Systems, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp. 60-73. DOI 10.46698/f6031-1077-3134-o
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