[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Hopf Bifurcation in a Delayed Brusselator Model with Network Structure

Journal of Applied Nonlinear Dynamics 11(4) (2022) 895--912 | DOI:10.5890/JAND.2022.12.009

Hongxia Liu, Ranchao Wu, Mengxin Chen

School of Mathematical Sciences, Anhui University, Hefei, 230601, China

Download Full Text PDF

 

Abstract

In this paper, the Brusselator reactor model with time delay and network structure is introduced and investigated. In view of understanding spatiotemporal dynamics of the model, dynamical analysis about the reactor is first carried out, such as stability and bifurcation analysis near the homogeneous steady state via choosing time delay as the critical parameter. It is found that time delay can induce stable and unstable states of the homogeneous steady state, and the model experiences the Hopf bifurcation at such critical value. Then the center manifold reduction and the normal form theory are applied to explore the detailed properties of the Hopf bifurcation. It is noted that such Brusselator model possesses the supercritical and subcritical Hopf bifurcation. Numerical simulations are performed to verify the validity of theoretical analysis.

References

  1. [1]  Prigogene, I. and Lefever, R. (1968), Symmetry breaking instabilities in dissipative systems II, Journal of Chemical Physics, 48, 1665-1700.
  2. [2]  Murray, J.D. (2002), Mathematical Biology I: An Introduction (3rd edition), Springer-Verlag: Berlin.
  3. [3]  Ma, M.J. and Hu J.J. (2014), Bifurcation and stability analysis of steady states to a Brusselator model, Applied Mathematics and Computation, 236, 580-592.
  4. [4]  Li, B. and Wang, M.X. (2008), Diffusion-driven instability and Hopf bifurcation in Brusselator system, Applied Mathematics and Mechanics, 29, 825-832.
  5. [5]  Guo, G.H., Wu, J.H., and Ren X.H (2011), Hopf bifurcation in general Brusselator system with diffusion, Applied Mathematics and Mechanics, 32, 1177-1186.
  6. [6]  Liao, M.X. and Wang, Q.R. (2016), Stability and bifurcation analysis in a diffusive Brusselator-Type system, International Journal of Bifurcation and Chaos, 26, 1-11.
  7. [7]  Zhang, C.H. and He, Y. (2020), Multiple stability switches and Hopf bifurcation induced by the delay in a Lengyel-Epstein chemical reaction system, Applied Mathematics Computation, 378, 1-22.
  8. [8]  Han, B.S. Wang, Z.C. (2018), Turing patterns of a Lotka-Volterra competitive system with nonlocal delay, International Journal of Bifurcation and Chaos, 28, 1-25.
  9. [9]  Guo, Y.X. Jiang, W.H., and Niu, B. (2013), Bifurcation analysis in the control of chaos by extended delay feedback, Journal of the Franklin Institute, 350, 155-170.
  10. [10]  Tian, C.R. and Ruan, S.G. (2019), Pattern formation, synchronism in an allelopathic plankton model with delay in a network, SIAM. Journal of Applied Dynamical Systems, 18, 531-557.
  11. [11]  Wu, R.C. Zhang, C.Y., and Feng, Z.S., (2021), Hopf bifurcation in a delayed single species network system, International Journal of Bifurcation and Chaos, 31, 1-15.
  12. [12]  Ji, Y.S. and Shen, J.W. (2020), Turing instability of Brusselator in the reaction-diffusion network, Complexity, 2020, 1-12.
  13. [13]  Zheng, Q.Q. and Shen, J.W. (2020), Turing instability induced by random network in FitzHugh-Nagumo model, Applied Mathematics Computation, 381, 1-13.
  14. [14]  Chang,L.L., Liu, C., Sun, G.Q., Wang Z., and Jin Z. (2019), Delay-induced patterns in a predator-prey model on complex networks with diffusion, New Journal of Physics, 16, 1-18.
  15. [15]  Z$\acute{\mbox{u}}\tilde{\mbox{n}}$iga-Galindo, W.A. (2020), Reaction-diffusion equations on complex networks and Turing patterns, via p-adic analysis, Journal of Mathematical Analysis and Application, 491, 1-39.
  16. [16]  Idea, Y., Izuharab, H., and Machida, T. (2016), Turing instability in reaction-diffusion models on complex networks. Physica A, 457, 331-347.
  17. [17]  Tyagi, S. Jain, S.K. Abbas, S., Meherrem, S., and Ray R.K. (2018), Time-delay-induced instabilities and Hopf bifurcation analysis in 2-neuron network model with reaction-diffusion term, Neurocomputing, 313, 306-315.
  18. [18]  Petit,J., Asllani, M. Fanelli D., Lauwens B. and Carletti T. (2016), Pattern formation in a two-component reaction-diffusion system with delayed processes on a network, Physica A, 462, 230-249.
  19. [19]  Ruan, S.G. (2001), Absolute stability, conditional stability and bifurcation in Kolmogorov-Type predator-prey system with discrete delays, Quarterly of Applied Mathematics, 6, 159-173.
  20. [20]  Freedman, H.I. and Rao, V.S.H. (1983), The trade-off between mutual interference and time lags in predator-prey-systems, Bulletin Mathematical Biology, 45, 991-1004.
  21. [21]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981), Theory and Applications of Hopf Bifurcation, Cambridge-New York: Cambridge University Press.
  22. [22]  Wu, J.H. (1996), Theory and Applications of Partial Functional Differential Equations, Springer Berlin.
  23. [23]  Watts, D.J. and Strogatz, S.H. (1998), Collective dynamics of small-world networks, Nature, 393, 440-442.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates