具有扩散的广义Brusselator系统的Hopf分歧

郭改慧; 吴建华; 任小红; 于鹏

doi:10.3879/j.issn.1000-0887.2011.09.009

具有扩散的广义Brusselator系统的Hopf分歧

doi: 10.3879/j.issn.1000-0887.2011.09.009

陕西科技大学理学院, 西安 710021;

基金项目: 国家自然科学基金资助项目(10971124;11001160);陕西省自然科学基础研究计划资助项目(2009JQ1007;2011JQ1015)

详细信息

作者简介:
郭改慧(1979- ),女,河南新郑人,讲师,博士(联系人.E-mail:guogaihui@sust.edu.cn).

中图分类号: O175.26
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- 文章访问数: 2045
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- 被引次数: 0
出版历程
- 收稿日期: 2010-11-04
- 修回日期: 2011-06-08
- 刊出日期: 2011-09-15

Hopf Bifurcation in the General Brusselator System With Diffusion

College of Science, Shaanxi University of Science and Technology, Xi'an 710021, P. R. China;
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, P. R. China

摘要

摘要: 在齐次Neumann边界条件下，考虑广义Brusselator系统．首先讨论常微分系统Hopf分歧的存在性,得到渐近稳定的周期解．其次讨论具有扩散的偏微分系统,在扩散系数满足一定的条件下,得到超临界的Hopf分歧,并利用规范形理论和中心流形定理给出空间齐次周期解的渐近稳定性．最后,借助Matlab软件进行数值模拟,证明了定理的结论．同时,正平衡态解和空间非齐次周期解的描绘补充了理论分析结果．
- 广义Brusselator系统 /
- Hopf分歧 /
- 扩散 /
- 稳定性
Abstract: The general Brusselator system was considered under homogeneous Neumann boundary conditions.The existence results of Hopf bifurcation to the ODE and PDE models were obtained.By the center manifold theory and the normal form method,the bifurcation direction and stability of periodic solutions were also established.Moreover,some numerical simulations were shown to support the analytical results.At the same time,the figures of positive steady-state solutions and spatially inhomogeneous periodic solutions were drawn,which supplement the analytical results.
- general Brusselator system /
- Hopfbi furcation /
- diffusion /
- stability

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参考文献(14)

[1]	Prigogene I, Lefever R.Symmetry breaking instabilities in dissipative systems Ⅱ[J].J Chemical Physics, 1968, 48(4): 1665-1700. doi: 10.1063/1.1668893
[2]	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
[3]	You Y. Global dynamics of the Brusselator equations[J]. Dynamics of PDE, 2007, 4(2): 167-196.
[4]	Peng R, Wang M X.Pattern formation in the Brusselator system[J].J Math Anal Appl, 2005, 309(1): 151-166. doi: 10.1016/j.jmaa.2004.12.026
[5]	程铭, 史峻平, 王金凤, 王玉文.Brusselator型化学反应的定性分析[J]. 哈尔滨师范大学自然科学学报, 2010, (2): 7-9.(CHENG Ming, SHI Jun-ping, WANG Jin-feng, WANG Yu-wen.Qualitative analysis of chemical reaction system of Brusselator type[J].Natural Sciences Journal of Harbin Normal University, 2010, (2): 7-9.(in Chinese))
[6]	李波, 王明新. Brusselator模型的扩散引起不稳定性和Hopf 分支[J]. 应用数学和力学, 2008, 29(6): 749-756.(LI Bo, WANG Ming-xin. Diffusion-driven instability and Hopf bifurcation in Brusselator system[J].Applied Mathematics and Mechanics(English Edition), 2008, 29(6): 825-832.)
[7]	Ghergu M. Non-constant steady-state solutions for Brusselator type systems[J]. Nonlinearity, 2008, 21(10): 2331-2345. doi: 10.1088/0951-7715/21/10/007
[8]	Peng R, Wang M X.On steady-state solutions of the Brusselator-type system[J]. Nonlinear Analysis: TMA, 2009, 71(3/4): 1389-1394. doi: 10.1016/j.na.2008.12.003
[9]	Yi F Q, Wei J J, Shi J P.Diffusion-driven instability and bifurcation in the lengyel-epstein system[J].Nonlinear Analysis: RWA, 2008, 9(3): 1038-1051. doi: 10.1016/j.nonrwa.2007.02.005
[10]	Yi F Q, Wei J J, Shi J P.Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system[J].J Differential Equations, 2009, 246(5): 1944-1977. doi: 10.1016/j.jde.2008.10.024
[11]	Wang M X.Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion[J].Math Biosci, 2008, 212(2): 149-160. doi: 10.1016/j.mbs.2007.08.008
[12]	Du Y H, Pang P Y H, Wang M X.Qualitative analysis of a prey-predator model with stage structure for the predator[J].SIAM J Appl Math, 2008, 69(2): 596-620. doi: 10.1137/070684173
[13]	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.(LU Qi-shao.Qualitative Method and Bifurcation of Ordinary Differential Equations[M].Beijing: Beijing Aviation and Spaceflight University Press, 1989.(in Chinese))
[14]	Hassard B D, Kazarinoff N D, Wan Y H.Theory and Application of Hopf Bifurcation[M].Cambridge: Cambridge University Press, 1981.