Belousov-Zhabotinsky反应斑图形成的图灵不稳定分析  

Turing Instability Analysis on the Pattern Formation in Belousov-Zhabotinsky Reaction

作　　者：戴金东 艾佳莉 孙巍[1] DAI Jindong;AI Jiali;SUN Wei(College of Chemical Engineering,Beijing University of Chemical Technology,Beijing 100029,China)

机构地区：[1]北京化工大学化学工程学院,北京100029

出　　处：《华东理工大学学报（自然科学版）》2021年第2期183-188,共6页Journal of East China University of Science and Technology

基　　金：国家自然科学基金(21878012,21576015)。

摘　　要：基于经典的Tyson模型,对Belousov-Zhabotinsky(BZ)反应进行了图灵不稳定分析,得到了使BZ反应产生图灵斑图的数学条件,并对计算结果进行了数值模拟验证。在分析过程中,利用傅里叶展开法将偏微分方程转化为若干常微分方程的和,通过Routh-Hurwitz判据来判断系统平衡解的稳定性,以得到系统在不考虑扩散项时保持稳定、考虑扩散项时不稳定的参数范围,即产生图灵不稳定的参数范围。在数值模拟过程中,采用有限差分法,将连续区域和近似函数分别替换为离散区域和离散函数,对BZ反应空间演化进行了模拟。所采用的方法与研究结果为包括生物系统在内的非线性系统的研究提供了参考。Pattern dynamics is an important branch of nonlinear theoretical system.When Turing instability exists in the Belousov-Zhabotinsky(BZ)reaction,the system will generate Turing pattern.Turing pattern is a kind of stable and non-uniform spatial structure,resulting from the instability of the equilibrium solution of the reaction-diffusion equation caused by diffusion.It is different from the chemical waves generated by the BZ reaction.Here,based on the classical Tyson model,the Turing instability analysis of BZ reaction was carried out,and the ranges of parameters of the system corresponding to the Turing pattern were obtained.The calculated results were verified by numerical simulation,and the possible morphology of Turing pattern in BZ reaction was also shown.In the work process,Routh-Hurwitz criterion was used to judge the stability of the equilibrium solution,and the ranges of parameters that made the equilibrium solution stable or unstable were obtained.The partial differential equation was transformed into the sum of several ordinary differential equations by the Fourier expansion method.The finite difference method was used to simulate the BZ reaction,in which the continuous region and functions were approximatively replaced by discrete region and functions.Through numerical simulation,not only the correctness of the calculated results was verified,but also the conclusion was obtained,suggesting the bigger the difference of two diffusion coefficient was,the faster the Turing pattern appeared.The methods proposed and the results obtained provide a reference for the study of nonlinear systems including biological systems.

关 键 词：BZ反应 图灵不稳定 图灵斑图 偏微分方程 数值模拟 

分 类 号：TP3[自动化与计算机技术—计算机科学与技术]