(18m) Subdiffusion Symphony: Multiplicity and Spatiotemporal Patterns in Autocatalytic Systems

Many natural phenomena involving physical, chemical and biological processes are subdiffusive in nature. Mathematically these are governed by fractional-order differential equations. The mean square displacement of molecules here scale as 𝑡^𝛾, where 0<𝛾<1. A two-variable system governed by a combination of quadratic and cubic autocatalytic reactions in a porous media is analysed in this work. Here each reactant is involved in the autocatalytic generation of the other. In this work, we investigate the effect of subdiffusion on the stability of the steady-states of the system. We use singularity theory to identify critical surfaces across which the bifurcation diagrams vary. We also develop a methodology in this paper to determine regions in parameter space where Turning patterns can be observed. This approach is based on identifying different critical surfaces across which steady-state stability changes when diffusive effects are included. The behaviour in the different regions is verified by a robust implicit numerical method developed for nonlinear systems based on the 𝐿1 scheme. We find that sub diffusion or fractional-order effects increase the region of dynamic stability of a system. We show that for a subdiffusive system, the region in parameter space, which shows Turing patterns, increases.