Geodesic
Dynamical properties of a diffusion-coupled system of differential equations with an additional internal coupling
Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 2, pp. 261-274 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the dynamics of a system of differential equations with the diffusion interaction and an additional internal coupling. Such systems are interesting because a slight variation in the coefficient at the additional coupling allows obtaining intricate scenarios of phase rearrangements. For the system under study, we find the critical dependence of the parameters such that zero equilibrium loses stability as two spatially inhomogeneous states appear in one case and a cycle in another case. With the parameter values close to the critical ones, asymptotic formulas are obtained for the regimes that branch off from the zero solution.
Keywords: system of differential equations, zero equilibrium, loss of stability
Mots-clés : bifurcation. 
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L. I. Ivanovskiy. Dynamical properties of a diffusion-coupled system of differential equations with an additional internal coupling. Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 2, pp. 261-274. http://geodesic.mathdoc.fr/item/TMF_2024_220_2_a2/

[1] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Diffuzionnyi khaos i ego invariantnye chislovye kharakteristiki”, TMF, 203:1 (2020), 10–25 | DOI | DOI

[2] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaksatsionnye kolebaniya i diffuzionnyi khaos v reaktsii Belousova”, Zhurn. vychislit. matem. i matem. fiziki, 51:8 (2011), 1400–1418 | DOI | MR

[3] N. F. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986 | MR

[4] S. A. Gourley, J. W.-H. So, J. H. Wu, “Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics”, J. Math. Sci., 4:4 (2004), 5119–5153 | DOI

[5] A. S. Rudyi, “Theoretical fundamentals of the method for thermal diffusivity measurements from auto-oscillation parameters in a system with a thermal feedback”, Internat. J. Thermophys., 14:1 (1993), 159–172 | DOI

[6] A. S. Rudyi, A. Yu. Kolesov, “Auto-oscillation in one-dimensional two-phase system with thermal feedback”, Nonlinear Anal., 31:5–6 (1998), 503–520 | DOI | MR

[7] L. I. Ivanovskii, “Dinamika odnoi sistemy diffuzionno svyazannykh differentsialnykh uravnenii s dopolnitelnoi vnutrennei svyazyu”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 55:3 (2020), 15–30 | DOI

[8] S. D. Glyzin, A. Yu. Kolesov, Lokalnye metody analiza dinamicheskikh sistem, YarGU im. P. G. Demidova, Yaroslavl, 2006

[9] Dzh. Gukenkheimer, F. Kholms, Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, In-t kompyuternykh issledovanii, M.–Izhevsk, 2002 | DOI

[10] V. I. Arnold, Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978

[11] A. B. Katok, B. Khasselblat, Vvedenie v teoriyu dinamicheskikh sistem s obzorom poslednikh dostizhenii, MTsNMO, M., 2005 | Zbl

[12] B. Khessard, N. Kazarinov, I. Ven, Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Mir, M., 1985 | MR | Zbl

[13] S. A. Kaschenko, “O bifurkatsiyakh pri malykh vozmuscheniyakh v logisticheskom uravnenii s zapazdyvaniem”, Model. i analiz inform. sistem, 24:2 (2017), 168–185 | DOI

[14] Yu. A. Mitropolskii, O. B. Lykova, Integralnye mnogoobraziya v nelineinoi dinamike, Nauka, M., 1973 | MR