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Four-dimensional Brusselator model with periodical solution. (English) Zbl 1453.34066

Summary: In the paper, a four-dimensional model of cyclic reactions of the type Prigogine’s Brusselator is considered. It is shown that the corresponding dynamical system does not have a closed trajectory in the positive orthant that will make it inadequate with the main property of chemical reactions of Brusselator type. Therefore, a new modified Brusselator model is proposed in the form of a four-dimensional dynamic system. Also, the existence of a closed trajectory is proved by the DN-tracking method for a certain value of the parameter which expresses the rate of addition one of the reagents to the reaction from an external source.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

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References:

[1] Adomian G., “The diffusion Brusselator equation”, Comput. Math. Appl., 29:29 (1995), 1-3 · Zbl 0827.35056 · doi:10.1016/0898-1221(94)00244-F
[2] Alqahtani A. M., “Numerical simulation to study the pattern formation of reaction-diffusion Brusselator model arising in triple collision and enzymatic”, J. Math. Chem., 56 (2018), 1543-1566 · Zbl 1394.92153 · doi:10.1007/s10910-018-0859-8
[3] Azamov A. A., “DN-tracking method for proving the existence of limit cycles”, Abstr. of the Int. Conf. Differential Equations and Topology dedicated to the Centennial Anniversary of L.S. Pontryagin (Russia. Moscow: MSU, June 17-22, 2008), 2008, 87-88 (in Russian)
[4] Azamov A. A., Ibragimov G., Akhmedov O. S., Ismail F., “On the proof of existence of a limit cycle for the Prigogine brusselator model”, J. Math. Res., 3:4 (2011), 983-989 · Zbl 1297.34042 · doi:10.5539/jmr.v3n4p93
[5] Azamov A. A., Akhmedov O. S., “Existence of a complex closed trajectory in a three-dimensional dynamical system”, Comput. Math. Math. Phys., 51:8 (2011), 1353-1359 · Zbl 1249.37035 · doi:10.1134/S0965542511080033
[6] Azamov A. A., Akhmedov O. S., “On existence of a closed trajectory in a three-dimensional model of a Brusselator”, Mech. Solids, 54:2 (2019), 251-265 · Zbl 1469.92141 · doi:10.3103/S0025654419030038
[7] Bakhvalov N. S., CHislennye metody [Numerical methods], Mir, Moscow, 1977 (in Russian)
[8] Boldo S., Faissole F., Chapoutot A., “Round-off error analysis of explicit one-step numerical integration methods”, IEEE 24th Symposium on Computer Arithmetic (ARITH), July 24-26, 2017, London, UK, IEEE Xplore, 2017, 82-89 · doi:10.1109/ARITH.2017.22
[9] Butcher J. C., Numerical Methods for Ordinary Differential Equations, 3rd ed., John Wiley & Sons Ltd., New York, 2016, 538 pp. · Zbl 1354.65004
[10] Cartan H., Calcul Différentiel. Formes Differentielles, Hermann, Paris, 1967 · Zbl 0156.36102
[11] Elyukhina I., “Nonlinear stability analysis of the full Brusselator reaction-diffusion model.”, Theor. Found. Chem. Eng., 48:6 (2014), 806-812 · doi:10.1134/S0040579514060025
[12] Glansdorff P., Prigogine I., Thermodynamic Theory of Structure, Stability and Fluctuations, John Wiley & Sons Ltd., New York, 1971 · Zbl 0246.73005
[13] Guckenheimer J., “Dynamical systems theory for ecologists: a brief overview”, Ecological Time Series, Springer, Boston, MA, 1995, 54-69 · doi:10.1007/978-1-4615-1769-6_6
[14] Hairer E., Nõrsett S., Wanner G., Solving Ordinary Differential Equations I. Non-stiff Problems, Springer Ser. Comput. Math., 8, Springer-Verlag, Berlin, Heidelberg, 1993, 528 pp. · Zbl 0789.65048
[15] Holmes M. H., Introduction to Numerical Methods in Differential Equations, Texts Appl. Math., 52, Springer-Verlag, New York, 2007, 239 pp. · Zbl 1110.65001 · doi:10.1007/978-0-387-68121-4
[16] Kehlet B., Logg A., “A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equation”, Numer. Algorithms, 76:48 (2017), 191-210 · Zbl 1379.65056 · doi:10.1007/s11075-016-0250-4
[17] Li Y., “Hopf bifurcations in general systems of Brusselator type”, Nonlinear Anal. Real World Appl., 28 (2016), 32-47 · Zbl 1329.35053 · doi:10.1016/j.nonrwa.2015.09.004
[18] Ma M., Hu J., “Bifurcation and stability analysis of steady states to a Brusselator model”, Appl. Math. Comput., 236 (2014), 580-592 · Zbl 1334.37047 · doi:10.1016/j.amc.2014.02.075
[19] Ma S. J., “The stochastic Hopf bifurcation analysis in Brusselator system with random parameter”, Appl. Math. Comput., 219 (2012), 306-319 · Zbl 1297.34070 · doi:10.1016/j.amc.2012.06.021
[20] Nicolis G., Prigogine I., Self-Organization in Nonequilibrium Systems, John Wiley & Sons Ltd., New York, 1977, 491 pp. · Zbl 0363.93005
[21] Nikol’skii M. S., Pervyj pryamoj metod L.S. Pontryagina v differencial’nyh igrah [Pontryagin’s First Direct Method for Differential Games], Moscow State Univ., Moscow, 1984, 65 pp. (in Russian) · Zbl 0595.90100
[22] Peña B., Pérez-Garc\'{i}a C., “Stability of Turing patterns in the Brusselator model”, Phys. Rev. E, 64 (2001), 156-213 · doi:10.1103/PhysRevE.64.056213
[23] Pontryagin L. S., Foundations of Combinatorial Topology, Graylock press, Rochester, New York, 1952, 99 pp. · Zbl 0049.39901
[24] Prigogine I., Lefever R., “Symmetry breaking instabilities in dissipative systems II”, J. Chem. Phys., 48:4 (1968), 1695-1700 · doi:10.1063/1.1668896
[25] Prigogine I., From Being to Becoming: Time and Complexity in the Physical Sciences, W.H. Freeman & Co. Ltd, New York, San Francisco, 1980, 272 pp.
[26] Rubido N., Stochastic dynamics and the noisy Brusselator behaviour, 2014, 7 pp., arXiv:
[27] Tucker W., “A rigorous ODE solver and Smale’s 14th problem”, Found. Comput. Math., 2 (2002), 53-117 · Zbl 1047.37012 · doi:10.1007/s002080010018
[28] Twizell E. H., Gumel A. B., Cao Q., “A second-order scheme for the “Brusselator” reaction-diffusion system”, J. Math. Chem., 26 (1999), 297-316 · Zbl 1016.92049 · doi:10.1023/A:1019158500612
[29] Tyson J. J., “Some further studies of nonlinear oscillations in chemical systems”, J. Chem. Phys., 58 (1973), 3919-3930 · doi:10.1063/1.1679748
[30] Tzou J. C., Ward M. J., “The stability and slow dynamics of spot patterns in the 2D Brusselator model: The effect of open systems and heterogeneities”, Phys. D: Nonlinear Phenomena, 373 (2018), 13-37 · Zbl 1392.35039 · doi:10.1016/j.physd.2018.02.002
[31] You Y., “Global dynamics of the Brusselator equations”, Dyn. Partial Differ. Equ., 4:2 (2007), 167-196 · Zbl 1158.37028 · doi:10.4310/DPDE.2007.v4.n2.a4
[32] You Y., Zhou Sh., “Global dissipative dynamics of the extended Brusselator system”, Nonlinear Anal. Real World Appl., 13:6 (2012), 2767-2789 · Zbl 1253.35192 · doi:10.1016/j.nonrwa.2012.04.005
[33] Yu P., Gumel A. B., “Bifurcation and stability analysis for a coupled Brusselator model”, J. Sound and Vibration, 244:5 (2001), 795-820 · Zbl 1237.80021 · doi:10.1006/jsvi.2000.3535
[34] Zhao Z., Ma R., “Local and global bifurcation of steady states to a general Brusselator model”, Adv. Differ. Equ., 2019, no. 491, 1-14 · Zbl 1487.35048 · doi:10.1186/s13662-019-2426-4
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