Brusselator模型的扩散引起不稳定性和Hopf分支

Diffusion-Driven Instability and Hopf Bifurcation in the Brusselator System

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.

The Hopf bifurcation for the Brusselator ODE model and the corresponding PDE model are investigated by using the Hopf bifturcation theorem. The stability of the Hopf bifturcation periodic solution was dicussed 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.