Abstract:
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.