A Framework to\nIdentify Supercritical and Subcritical\nTuring Bifurcations: Case Study of Soai Reaction Featuring Asymmetric\nAutocatalysis

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