A Framework to Identify Supercritical and Subcritical Turing Bifurcations: Case Study of Soai Reaction Featuring Asymmetric Autocatalysis

In this work, we focus on an autocatalytic reaction–diffusion model and carry out multiple-scale weakly nonlinear analysis. A cubic and a quadratic autocatalytic reaction system is analyzed. We develop a framework to identify the critical surfaces in parameter space across which the nature of the Turing bifurcation changes from supercritical to subcritical. These are verified by numerical simulations of the system using Chebyshev spectral collocation method. Using weakly nonlinear analysis, we derive the amplitude equations up to the fifth order. This helps determine accurately the limit point of the bifurcating solution. The numerical solutions are in good agreement with the predictions of the weakly nonlinear analysis for supercritical bifurcations. We show that when multiple steady states arise, Turing patterns can coexist with other spatially uniform steady states. Furthermore, we show that our framework can be extended to get different patterns, such as squares, in a two-dimensional domain. This study will help identify operating conditions where Turing patterns can be obtained for reaction–diffusion systems.