带扩散低浓度三分子模型的球对称结构研究

Spherically Symmetric Structures of Triple-molecular Model with Low Concentration

文章利用分歧理论研究了带扩散低浓度三分子模型的球对称结构。首先,对模型进行了线性稳定性分析,结果表明:三雄球对称情形与一维情形仅在系统进入定态周期解的不稳定区的必要条件有所不同;开放系统的反应扩散过程中在远离平衡区出现耗散结构。其次,分析了分歧点和耗散结构,说明在Dirichlet边界条件下,三维球对称解和一维解类似,都会出现空间耗散结构。但球对称中心是一高浓度区。最后,应用Hopf分歧理论计算了带扩散低浓度三分子模型的球对称含时解,表明在Dirichlet边界条件下,其含时解是空间的衰减振荡函数;而一维含时解是空间的周期函数。

In this paper, spherically symmetric structures of the triple-molecular model with low concentration are discussed in detail by using the bifurcation theory. First of all, the linear stability of the triple-molecular model with low concentration is analyzed. According to Prigogine's dissipative structure theory, as the system depart far from equilibrium the states begin to become unstable, the bifurcation branch starts. The bifurcation paints and dissipative structure are analyzed. The reaction diffusion equation in three-dimensional space is simiclar as in one-dimensional. A simple solution in three dimensions are given then, there is a high concentration area in the centre of sphere. Spherically symmetric time-periodic solution for triple-molecular model with low concentration is calculated by using the hopf bifurcation, too. The results show that under Dirichelet boundary conditions, the periodical solutions are oscillatory functions of attenuation of the space, and the one--dimensional solutions periodical function of the space.