Hopf bifurcation in general Brusselator system with diffusion

  • Gai-hui Guo (郭改慧)Email author
  • Jian-hua Wu (吴建华)
  • Xiao-hong Ren (任小红)


The general Brusselator system is considered under homogeneous Neumann boundary conditions. The existence results of the Hopf bifurcation to the ordinary differential equation (ODE) and partial differential equation (PDE) models are obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results. At the same time, the positive steady-state solutions and spatially inhomogeneous periodic solutions are graphically shown to supplement the analytical results.

Key words

general Brusselator system Hopf bifurcation diffusion stability 

Chinese Library Classification


2010 Mathematics Subject Classification



  1. [1]
    Prigogene, I. and Lefever, R. Symmetry breaking instabilities in dissipative systems II. Journal of Chemical Physics, 48, 1665–1700 (1968)CrossRefGoogle Scholar
  2. [2]
    Brown, K. J. and Davidson, F. A. Global bifurcation in the Brusselator system. Nonlinear Analysis: Theory, Methods and Applications, 24(12), 1713–1725 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    You, Y. Global dynamics of the Brusselator equations. Dynamics of Partial Differential Equations, 4, 167–196 (2007)MathSciNetzbMATHGoogle Scholar
  4. [4]
    Peng, R. and Wang, M. X. Pattern formation in the Brusselator system. Journal of Mathematical Analysis and Applications, 309, 151–166 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Cheng, M., Shi, J. P., Wang, J. F., and Wang, Y. W. Qualitative analysis of chemical reaction system of Brusselator type (in Chinese). Natural Sciences Journal of Harbin Normal University, 26(2), 7–9 (2010)Google Scholar
  6. [6]
    Li, B. and Wang, M. X. Diffusion-driven instability and Hopf bifurcation in Brusselator system. Applied Mathematics and Mechanics (English Edition), 29(6), 825–832 (2008) DOI 10.1007/s10483-008-0614-yMathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Ghergu, M. Non-constant steady-state solutions for Brusselator type systems. Nonlinearity, 21, 2331–2345 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Peng, R. and Wang, M. X. On steady-state solutions of the Brusselator-type system. Nonlinear Analysis: Theory, Methods and Applications, 71, 1389–1394 (2009)zbMATHCrossRefGoogle Scholar
  9. [9]
    Yi, F. Q., Wei, J. J., and Shi, J. P. Diffusion-driven instability and bifurcation in the Lengyel-Epstein system. Nonlinear Analysis, 9, 1038–1051 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Yi, F. Q., Wei, J. J., and Shi, J. P. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. Journal of Differential Equations, 246, 1944–1977 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Wang, M. X. Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion. Mathematical Biosciences, 212(2), 149–160 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Du, Y. H., Pang, P. Y. H., and Wang, M. X. Qualitative analysis of a prey-predator model with stage structure for the predator. SIAM Journal on Applied Mathematics, 69(2), 596–620 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Lu, Q. S. Qualitative Method and Bifurcation of Ordinary Differential Equations (in Chinese), Beijing Aviation and Spaceflight University Press, Beijing (1989)Google Scholar
  14. [14]
    Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H. Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge (1981)Google Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Gai-hui Guo (郭改慧)
    • 1
    Email author
  • Jian-hua Wu (吴建华)
    • 2
  • Xiao-hong Ren (任小红)
    • 1
  1. 1.College of ScienceShaanxi University of Science and TechnologyXi’anP. R. China
  2. 2.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anP. R. China

Personalised recommendations