Brusselator模型的扩散引起不稳定性和Hopf分支

李波; 王明新

Brusselator模型的扩散引起不稳定性和Hopf分支

李波^1,2,
王明新¹

1.
东南大学数学系,南京 210018;2.徐州师范大学数学科学学院,江苏徐州 221116

基金项目: 国家自然科学基金资助项目(10771032);江苏省自然科学基金资助项目(BK2006088)

详细信息

作者简介:
李波(1971- ),女,河北徐水人,博士生(联系人.E-mail:libo5181923@163.com).

中图分类号: O175.26
计量
- 文章访问数: 3030
- HTML全文浏览量: 174
- PDF下载量: 867
- 被引次数: 0
出版历程
- 收稿日期: 2007-12-21
- 修回日期: 2008-04-21
- 刊出日期: 2008-06-15

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;

摘要

摘要: 研究了Brusselator常微分系统和相应的偏微分系统的Hopf分支,并用规范形理论和中心流形定理讨论了当空间的维数为1时Hopf分支解的稳定性．证明了:当参数满足某些条件时,Brusselator常微分系统的平衡解和周期解是渐近稳定的,而相应的偏微分系统的空间齐次平衡解和空间齐次周期解是不稳定的;如果适当选取参数,那么Brusselator常微分系统不出现Hopf分支,但偏微分系统出现Hopf分支,这表明,扩散可以导致Hopf分支．
- Brusselator模型 /
- Hopf分支 /
- 稳定性 /
- 扩散导致Hopf分支
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

HTML全文

参考文献(13)

[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