Odiljon S. Akhmedov     (Uzbekistan Academy of Sciences V. I. Romanovskiy Institute of Mathematics, Tashkent, 100041, Uzbekistan)
Abdulla A. Azamov     (Uzbekistan Academy of Sciences V. I. Romanovskiy Institute of Mathematics, Tashkent, 100041, Uzbekistan)
Gafurjan I. Ibragimov     (Department of Mathematics & Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang Selangor, Malaysia)


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.


Chemical reaction, Closed trajectory, DN-tracking method, Discrete trajectory, Numerical trajectory.

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  1. Adomian G. The diffusion Brusselator equation. Comput. Math. Appl., 1995. Vol. 29, No. 5. P. 1–3. 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., 2018. Vol. 56. P. 1543–1566. DOI: 10.1007/s10910-018-0859-8
  3. Azamov A.A. DN-tracking method for proving the existence of limit cycles. In: Abstr. of the Int. Conf. Differential Equations and Topology dedicated to the Centennial Anniversary of L.S. Pontryagin, June 17–22, 2008, Moscow, Russia. Moscow: MSU, 2008. P. 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., 2011. Vol. 3, No. 4. P. 983–989. 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., 2011. Vol. 51, No. 8. P. 1353–1359. 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, 2019. Vol. 54, No. 2. P. 251–265. DOI: 10.3103/S0025654419030038
  7. Bakhvalov N. S. CHislennye metody [Numerical methods]. Moscow: Mir, 1977. (in Russian)
  8. Boldo S., Faissole F., Chapoutot A. Round-off error analysis of explicit one-step numerical integration methods. In: IEEE 24th Symposium on Computer Arithmetic (ARITH), July 24–26, 2017, London, UK. IEEE Xplore, 2017. P. 82–89. DOI: 10.1109/ARITH.2017.22
  9. Butcher J.C. Numerical Methods for Ordinary Differential Equations, 3rd ed. New York: John Wiley & Sons Ltd., 2016. 538 p.
  10. Cartan H. Calcul Différentiel. Formes Differentielles. Paris: Hermann, 1967.
  11. Elyukhina I. Nonlinear stability analysis of the full Brusselator reaction–diffusion model. Theor. Found. Chem. Eng., 2014. Vol. 48, No. 6. P. 806–812. DOI: 10.1134/S0040579514060025
  12. Glansdorff P., Prigogine I. Thermodynamic Theory of Structure, Stability and Fluctuations. New York: John Wiley & Sons Ltd., 1971.
  13. Guckenheimer J. Dynamical systems theory for ecologists: a brief overview. In: Ecological Time Series. Boston, MA: Springer, 1995. P. 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., vol. 8. Berlin, Heidelberg: Springer-Verlag, 1993. 528 p.
  15. Holmes M.H. Introduction to Numerical Methods in Differential Equations. Texts Appl. Math., vol. 52. New York: Springer–Verlag, 2007. 239 p. 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, 2017. Vol. 76. P. 191–210. DOI: 10.1007/s11075-016-0250-4
  17. Li Y. Hopf bifurcations in general systems of Brusselator type. Nonlinear Anal. Real World Appl., 2016. Vol. 28. P. 32–47. 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., 2014. Vol. 236. P. 580–592. 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., 2012. Vol. 219. P. 306–319. DOI: 10.1016/j.amc.2012.06.021
  20. Nicolis G., Prigogine I. Self-Organization in Nonequilibrium Systems. New York: John Wiley & Sons Ltd., 1977. 491 p.
  21. Nikol’skii M.S. Pervyj pryamoj metod L.S. Pontryagina v differencial’nyh igrah [Pontryagin’s First Direct Method for Differential Games]. Moscow: Moscow State Univ., 1984. 65 p. (in Russian)
  22. Peña B., Pérez-García C. Stability of Turing patterns in the Brusselator model. Phys. Rev. E, 2001. Vol. 64. P. 156–213. DOI: 10.1103/PhysRevE.64.056213
  23. Pontryagin L. S. Foundations of Combinatorial Topology. Rochester, New York: Graylock press, 1952. 99 p.
  24. Prigogine I., Lefever R. Symmetry breaking instabilities in dissipative systems II. J. Chem. Phys., 1968. Vol. 48, No. 4. P. 1695–1700. DOI: 10.1063/1.1668896
  25. Prigogine I. From Being to Becoming: Time and Complexity in the Physical Sciences. New York, San Francisco: W.H. Freeman & Co. Ltd, 1980. 272 p.
  26. Rubido N. Stochastic dynamics and the noisy Brusselator behaviour. 2014. P. 1–7. arXiv: 1405.0390 [cond-mat.stat-mech]
  27. Tucker W. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math., 2002. Vol. 2. P. 53–117. 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., 1999. Vol. 26. P. 297–316. DOI: 10.1023/A:1019158500612
  29. Tyson J. Some further studies of nonlinear oscillations in chemical systems. J. Chem. Phys., 1973. Vol. 58. P. 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, 2018. Vol. 373. P. 13–37. DOI: 10.1016/j.physd.2018.02.002
  31. You Y. Global dynamics of the Brusselator equations. Dyn. Partial Differ. Equ., 2007. Vol. 4, No. 2. P. 167–196. 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., 2012. Vol. 13, No. 6. P. 2767–2789. 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, 2001. Vol. 244, No. 5. P. 795–820. 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. Art. No. 491. P. 1–14. DOI: 10.1186/s13662-019-2426-4


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