Diffusion-Driven Instability and Hopf Bifurcation in the Brusselator System

LI Bo; WANG Ming-xin

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2008 > 29(6): 749-756

LI Bo, WANG Ming-xin. Diffusion-Driven Instability and Hopf Bifurcation in the Brusselator System[J]. Applied Mathematics and Mechanics, 2008, 29(6): 749-756.

Citation:

LI Bo, WANG Ming-xin. Diffusion-Driven Instability and Hopf Bifurcation in the Brusselator System[J]. Applied Mathematics and Mechanics, 2008, 29(6): 749-756.

Citation:

LI Bo, WANG Ming-xin. Diffusion-Driven Instability and Hopf Bifurcation in the Brusselator System[J]. Applied Mathematics and Mechanics, 2008, 29(6): 749-756.

PDF( 338 KB)

Diffusion-Driven Instability and Hopf Bifurcation in the Brusselator System

LI Bo^1,2,
WANG Ming-xin¹

1.
Department of Mathematics, Southeast University, Nanjing 210018, P. R. China;

Received Date: 2007-12-21
Rev Recd Date: 2008-04-21
Publish Date: 2008-06-15

Abstract

Abstract

The Hopf bifurcation for the Brusselator ODE model and the corresponding PDE model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution was discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. The results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.
- Brusselator system,
- Hopf bifurcation,
- stability,
- diffusion-driven Hopf bifurcation

FullText(HTML)

References(13)

References

[1]	Erneux T,Reiss E.Brusselator isolas[J].SIAM Journal on Applied Mathematics,1983, 43(6):1240-1246. doi: 10.1137/0143082
[2]	Nicolis G.Patterns of spatio-temporal organization in chemical and biochemical kinetics[J].SIAM-AMS Proc,1974,8(1):33-58.
[3]	Prigogene I, Lefever R.Symmetry breaking instabilities in dissipative systems Ⅱ[J].The Journal of Chemical Physics,1968,48(4):1665-1700. doi: 10.1063/1.1668893
[4]	Brown K J,Davidson F A. Global bifurcation in the Brusselator system[J].Nonlinear Analysis,1995, 24(12):1713-1725. doi: 10.1016/0362-546X(94)00218-7
[5]	Callahan T K, Knobloch E. Pattern formation in three-dimensional reaction-diffusion systems[J].Physica D,1999,132(3): 339-362. doi: 10.1016/S0167-2789(99)00041-X
[6]	Rabinowitz P.Some global results for nonlinear eigenvalue problems[J].Journal of Functional Analysis,1971,7(3): 487-513. doi: 10.1016/0022-1236(71)90030-9
[7]	Peng R, Wang M X. Pattern formation in the Brusselator system[J].Journal of Mathematical Analysis and Applications,2005,309(1): 151-166. doi: 10.1016/j.jmaa.2004.12.026
[8]	Yi F Q,Wei J J,Shi J P. Diffusion-driven instability and bifurcation in the Lengyel-Epstein system[J].Nonlinear Analysis,2008,9(8):1038-1051. doi: 10.1016/j.nonrwa.2007.02.005
[9]	Wang M X. Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion[J].Mathematical Biosciences,2008,212(2): 149-160. doi: 10.1016/j.mbs.2007.08.008
[10]	陆启韶.常微分方程的定性分析和分叉[M].北京:北京航空航天大学出版社,1989.
[11]	Wang M X.Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion[J].Physica D,2004,196(1):172-192. doi: 10.1016/j.physd.2004.05.007
[12]	Hassard B D, Kazarinoff N D, Wan Y H.Theory and Application of Hopf Bifurcation[M].Cambridge:Cambridge University Press,1981.
[13]	Crandall Michael G,Rabinowitz Paul H.The Hopf bifurcation theorem in infinite dimensions[J].Archive for Rational Mechanics and Analysis,1977, 67(1): 53-72. doi: 10.1007/BF00280827