Stability and pattern formation in a coupled arbitrary order of autocatalysis system

Spatiotemporal structures arising in two identical cells, each governed by arbitrary order autocatalator kinetics and coupled via the diffusive interchange of a reactant, are discussed. The stability of two homogeneous steady states is obtained with the use of linear stability analysis. By studying the linearized equations, it is found that two steady states, in the uncoupled and coupled system respectively, may give rise to the possibility of bifurcations to spatially nonuniform pattern forms. Further information about Turing bifurcation solutions close to these bifurcation points are obtained by weakly nonlinear theory. It is seen that the coupling leads to bifurcations not present in the uncoupled system which give rise to locally stable nonuniform pattern forms. Finally the stability of the equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches about small coupled system with 0<α≪10<α≪1 and large coupling for α≫1α≫1.