1683-34141814-0807Владикавказский математический журналVladikavkaz Mathematical JournalЮжный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН)The Turing-Hopf Bifurcation Problem in Three-Dimensional Reaction-Diffusion SystemsЗадача о бифуркации Тьюринга - Хопфа в трехмерных системах "реакция-диффузия"10.46698/f6031-1077-3134-o198630920262836073https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=19899&SECTION_ID=673ВасенинаН. А.VaseninaN. A.zhiber.na@gmail.comЮмагуловМ. Г.YumagulovM. G.yum_mg@mail.ruУфимский университет науки и технологий, Россия, 450076, Уфа, ул. Заки Валиди, 32Ufa University of Science and Technology, 32 Zaki Validi St., Ufa 450076, RussiaThe 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.В статье обсуждаются вопросы исследования локальных бифуркаций в окрестностях пространственно однородных положений равновесия системы "реакция-диффузия" в ограниченной области с однородными краевыми условиями Неймана. Основные результаты касаются изучения задачи о взаимодействии бифуркаций Тьюринга и Андронова - Хопфа, приводящего к бифуркации Тьюринга - Хопфа. В такой бифуркации проявляются два типа диффузионных неустойчивостей, что может приводить к разнообразным эффектам, сопровождающимся возникновением сложных пространственно-временных режимов. С целью изучения основной задачи предлагаются новые расчетные формулы для исследования задач о бифуркациях Тьюринга и Андронова - Хопфа в одномерных, двумерных и трехмерных системах "реакция-диффузия". Полученные формулы представляют самостоятельный интерес. Проведено исследование условий, приводящих к бифуркации Тьюринга - Хопфа. Основным результатом является установление ее реализуемости в трехмерных системах "реакция-диффузия". Приводится соответствующий пример. Обсуждаются также возможные направления развития предложенных результатов в задачах исследования диффузионной неустойчивости и соответствующих бифуркаций.реакция-диффузияположение равновесияустойчивостьбифуркацияТьюрингАндронов - ХопфТьюринг - Хопфreaction-diffusionequilibrium positionstabilitybifurcationTuringAndronov-HopfTuring-HopfVanag, V. K. Dissipative Structures in Reaction-Diffusion Systems [Dissipativnye struktury v reaktsionno-diffuzionnykh sistemakh], Izhevsk, Institute of Computer Research, Research Center “Regular and Chaotic Dynamics”, 2008, 300 p. (in Russian).Vanag, V. K. Dissipative Structures in Reaction-Diffusion Systems [Dissipativnye struktury v reaktsionno-diffuzionnykh sistemakh], Izhevsk, Institute of Computer Research, Research Center “Regular and Chaotic Dynamics”, 2008, 300 p. (in Russian).Svirezhev, Yu. M. Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology [Nelineynye volny, dissipativnye struktury i katastrofy v ekologii], Moscow, Nauka, 1987, 368 p. (in Russian).Svirezhev, Yu. M. Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology [Nelineynye volny, dissipativnye struktury i katastrofy v ekologii], Moscow, Nauka, 1987, 368 p. (in Russian).Murray, J. Mathematical Biology. Volume 1 [Matematicheskaya biologiya. Tom 1], Izhevsk, Scientific Research Center “Regular and Chaotic Dynamics”, Institute of Space Research, 2009, 776 p. (in Russian).Murray, J. Mathematical Biology. Volume 1 [Matematicheskaya biologiya. Tom 1], Izhevsk, Scientific Research Center “Regular and Chaotic Dynamics”, Institute of Space Research, 2009, 776 p. (in Russian).Bratus, A. S., Novozhilov, A. S., and Platonov, A. P. Dynamical Systems and Models of Biology [Dinamicheskie sistemi i modeli biologii], Moscow, Fizmatlit Publishing House, 2010, 436 p. (in Russian).Bratus, A. S., Novozhilov, A. S., and Platonov, A. P. Dynamical Systems and Models of Biology [Dinamicheskie sistemi i modeli biologii], Moscow, Fizmatlit Publishing House, 2010, 436 p. (in Russian).Magnitsky, N. A. Theory of Dynamic Chaos [Teoriya dinamicheskogo haosa], Moscow, URSS, 2011, 320 p. (in Russian).Magnitsky, N. A. Theory of Dynamic Chaos [Teoriya dinamicheskogo haosa], Moscow, URSS, 2011, 320 p. (in Russian).Hassard, B., Kazarinov, N., and Ven, I. Theory and Applications of Cycle Birth Bifurcation [Teoriya i prilozheniya bifurkacii rozhdeniya cikla], Moscow, Mir, 1985, 280 p. (in Russian).Hassard, B., Kazarinov, N., and Ven, I. Theory and Applications of Cycle Birth Bifurcation [Teoriya i prilozheniya bifurkacii rozhdeniya cikla], Moscow, Mir, 1985, 280 p. (in Russian).Borina, M. Yu., and Polezhaev, A. A. Diffusion Instability in a Three-Variable Reaction–Diffusion Model, Computer Research and Modeling, 2011, vol. 3, no. 2, pp. 135–146 (in Russian). DOI: 10.20537/2076-7633-2011-3-2-135-146.Borina, M. Yu., and Polezhaev, A. A. Diffusion Instability in a Three-Variable Reaction–Diffusion Model, Computer Research and Modeling, 2011, vol. 3, no. 2, pp. 135–146 (in Russian). DOI: 10.20537/2076-7633-2011-3-2-135-146.Kuznetsov, M. B. Investigation of the Formation of Turing Structures under the Influence of Wave Instability, Computer Research and Modeling, 2019, vol. 11, no. 3, pp. 397–412 (in Russian). DOI: 10.20537/2076-7633-2019-11-3-397-412.Kuznetsov, M. B. Investigation of the Formation of Turing Structures under the Influence of Wave Instability, Computer Research and Modeling, 2019, vol. 11, no. 3, pp. 397–412 (in Russian). DOI: 10.20537/2076-7633-2019-11-3-397-412.Meixner, M., De Wit, A., Bose, S. and Schöll, E. Generic Spatiotemporal Dynamics Near Codimension-Two Turing–Hopf Bifurcations, Physical Review E, 1997, vol. 55, no. 6, Article no. 6690. DOI: 10.1103/PhysRevE.55.6690.Meixner, M., De Wit, A., Bose, S. and Schöll, E. Generic Spatiotemporal Dynamics Near Codimension-Two Turing–Hopf Bifurcations, Physical Review E, 1997, vol. 55, no. 6, Article no. 6690. DOI: 10.1103/PhysRevE.55.6690.Vanag, V. K. Subcritical Wave Instability in Reaction-Diffusion Systems, The Journal of Chemical Physics, 2004, vol. 121, no. 2, pp. 890–894. DOI: 10.1063/1.1760072.Vanag, V. K. Subcritical Wave Instability in Reaction-Diffusion Systems, The Journal of Chemical Physics, 2004, vol. 121, no. 2, pp. 890–894. DOI: 10.1063/1.1760072.Yang, L. A. Jumping Solitary Waves in an Autonomous Reaction-Diffusion System with Subcritical Wave Instability, Physical Chemistry Chemical Physics, 2004, vol. 6, no. 40, pp. 4647–4651.Yang, L. A. Jumping Solitary Waves in an Autonomous Reaction-Diffusion System with Subcritical Wave Instability, Physical Chemistry Chemical Physics, 2004, vol. 6, no. 40, pp. 4647–4651.Yang, L. A. and Epstein, I. R. Oscillatory Turing Patterns in Reaction-Diffusion Systems with Two Coupled Layers, Physical Review Letters, 2003, vol. 90, Article no. 178303. DOI: 10.1103/PhysRevLett.90.178303.Yang, L. A. and Epstein, I. R. Oscillatory Turing Patterns in Reaction-Diffusion Systems with Two Coupled Layers, Physical Review Letters, 2003, vol. 90, Article no. 178303. DOI: 10.1103/PhysRevLett.90.178303.Jiang, W., An, Q. and Shi, J. Formulation of the Normal Form of Turing–Hopf Bifurcation in Partial Functional Differential Equations, Journal of Differential Equations, 2020, vol. 268, no. 10, pp. 6067–6102. DOI: 10.1016/j.jde.2019.11.039.Jiang, W., An, Q. and Shi, J. Formulation of the Normal Form of Turing–Hopf Bifurcation in Partial Functional Differential Equations, Journal of Differential Equations, 2020, vol. 268, no. 10, pp. 6067–6102. DOI: 10.1016/j.jde.2019.11.039.Song, Y., Zhang, T. and Peng, Y. Turing–Hopf Bifurcation in the Reaction–Diffusion Equations and Its Applications, Communications in Nonlinear Science and Numerical Simulation, 2016, vol. 33, pp. 229–258. DOI: 10.1016/j.cnsns.2015.10.002.Song, Y., Zhang, T. and Peng, Y. Turing–Hopf Bifurcation in the Reaction–Diffusion Equations and Its Applications, Communications in Nonlinear Science and Numerical Simulation, 2016, vol. 33, pp. 229–258. DOI: 10.1016/j.cnsns.2015.10.002.Ducrot, A., Fu, X. and Magal, P. Turing and Turing–Hopf Bifurcations for a Reaction–Diffusion Equation with Nonlocal Advection, Journal of Nonlinear Science, 2018, vol. 28, pp. 1959–1997. DOI: 10.1007/s00332-018-9472-z.Ducrot, A., Fu, X. and Magal, P. Turing and Turing–Hopf Bifurcations for a Reaction–Diffusion Equation with Nonlocal Advection, Journal of Nonlinear Science, 2018, vol. 28, pp. 1959–1997. DOI: 10.1007/s00332-018-9472-z.Yumagulov, M. G., Vasenina, N. A. and Gabdrahmanov, R. I. Operator Methods for Investigating Stability and Bifurcation Problems in a Reaction–Diffusion System and Applications, Differential Equations, 2025, vol. 61, no. 4, pp. 525–541. DOI: 10.1134/S0012266125040081.Yumagulov, M. G., Vasenina, N. A. and Gabdrahmanov, R. I. Operator Methods for Investigating Stability and Bifurcation Problems in a Reaction–Diffusion System and Applications, Differential Equations, 2025, vol. 61, no. 4, pp. 525–541. DOI: 10.1134/S0012266125040081.Abushahmina, G. R., Gusarova, N. I. and Yumagulov, M. G. Lyapunov Quantities for Andronov–Hopf Bifurcation Problem in Reaction–Diffusion Systems, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 15, pp. 3567–3573. DOI: 10.1134/S1995080222030027.Abushahmina, G. R., Gusarova, N. I. and Yumagulov, M. G. Lyapunov Quantities for Andronov–Hopf Bifurcation Problem in Reaction–Diffusion Systems, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 15, pp. 3567–3573. DOI: 10.1134/S1995080222030027.Yumagulov, M. G. and Sidelnikova, N. A. Reaction–Diffusion Type Systems: Signs of Stability and Bifurcation, Bulletin of the Bashkir University, 2023, vol. 28, no. 4, pp. 303–309 (in Russian). DOI: 10.33184/bulletin-bsu-2023.4.1.Yumagulov, M. G. and Sidelnikova, N. A. Reaction–Diffusion Type Systems: Signs of Stability and Bifurcation, Bulletin of the Bashkir University, 2023, vol. 28, no. 4, pp. 303–309 (in Russian). DOI: 10.33184/bulletin-bsu-2023.4.1.Yumagulov, M. G. and Vasenina, N. A. Spectral Properties of the Operators of the Reaction–Diffusion System and Signs of Bifurcation, Bulletin of Perm University, Mathematics, Mechanics, Computer Science, 2024, vol. 65, no. 2, pp. 17–25 (in Russian). DOI: 10.17072/1993-0550-2024-2-17-25.Yumagulov, M. G. and Vasenina, N. A. Spectral Properties of the Operators of the Reaction–Diffusion System and Signs of Bifurcation, Bulletin of Perm University, Mathematics, Mechanics, Computer Science, 2024, vol. 65, no. 2, pp. 17–25 (in Russian). DOI: 10.17072/1993-0550-2024-2-17-25.Kulikov, A. N. Some Remarks on the Phenomenon of Diffusion Instability in Homogeneous Nonlinear Systems of the Third Order, Research on Stability and Oscillation Theory, Yaroslavl, 1981, pp. 113–123 (in Russian).Kulikov, A. N. Some Remarks on the Phenomenon of Diffusion Instability in Homogeneous Nonlinear Systems of the Third Order, Research on Stability and Oscillation Theory, Yaroslavl, 1981, pp. 113–123 (in Russian).Henry, D. Geometric Theory of Semilinear Parabolic Equations [Geometricheskaya teoriya polulineynykh parabolicheskikh uravneniy], Moscow, Mir, 1985, 376 p. (in Russian).Henry, D. Geometric Theory of Semilinear Parabolic Equations [Geometricheskaya teoriya polulineynykh parabolicheskikh uravneniy], Moscow, Mir, 1985, 376 p. (in Russian).Kulikov, A. N., Kulikov, D. A. and Frolov, D. G. The Keynes Model of the Business Cycle and the Problem of Diffusion Instability, Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, 2022, vol. 207, pp. 77–90 (in Russian). DOI: 10.36535/0233-6723-2022-207-77-90.Kulikov, A. N., Kulikov, D. A. and Frolov, D. G. The Keynes Model of the Business Cycle and the Problem of Diffusion Instability, Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, 2022, vol. 207, pp. 77–90 (in Russian). DOI: 10.36535/0233-6723-2022-207-77-90.Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. and Chua, L. Methods of Qualitative Theory in Nonlinear Dynamics. Part 2 [Metody kachestvennoy teorii v nelineynoy dinamike. Chast’ 2], Izhevsk, Scientific Research Center “Regular and Chaotic Dynamics”, Institute of Space Research, 2009, 549 p. (in Russian).Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. and Chua, L. Methods of Qualitative Theory in Nonlinear Dynamics. Part 2 [Metody kachestvennoy teorii v nelineynoy dinamike. Chast’ 2], Izhevsk, Scientific Research Center “Regular and Chaotic Dynamics”, Institute of Space Research, 2009, 549 p. (in Russian).Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields [Nelineynye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh poley], Moscow–Izhevsk, Institute of Computer Research, 2002, 560 p. (in Russian).Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields [Nelineynye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh poley], Moscow–Izhevsk, Institute of Computer Research, 2002, 560 p. (in Russian).Yumagulov, M. G. An Operator Method for Investigating Correct Bifurcation in Multiparametric Systems, Dokladi akademii nauk [Reports of the Academy of Sciences], 2008, vol. 423, no. 5, pp. 1–4 (in Russian).Yumagulov, M. G. An Operator Method for Investigating Correct Bifurcation in Multiparametric Systems, Dokladi akademii nauk [Reports of the Academy of Sciences], 2008, vol. 423, no. 5, pp. 1–4 (in Russian).Vyshinsky, A. A., Ibragimova, L. S., Murtazina, S. A. and Yumagulov, M. G. An Operator Method for Approximate Investigation of a Regular Bifurcation in Multiparameter Dynamical Systems, Ufa Mathematical Journal, 2010, vol. 2, no. 4, pp. 3–26 (in Russian).Vyshinsky, A. A., Ibragimova, L. S., Murtazina, S. A. and Yumagulov, M. G. An Operator Method for Approximate Investigation of a Regular Bifurcation in Multiparameter Dynamical Systems, Ufa Mathematical Journal, 2010, vol. 2, no. 4, pp. 3–26 (in Russian).