A Subcritical Bifurcation for a Nonlinear Reaction–Diffusion System

In this paper the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms is investigated. Through a linear stability analysis we show that the cross-diffusion term allows the pattern formation. To predict the form and the amplitude of the pattern we perform a weakly nonlinear analysis. In the supercritical case the Stuart-Landau equation is found, which rules the evolution of the amplitude of the most unstable mode. With the increasing distance from the bifurcation value of the cross-diffusion parameter, the weakly nonlinear analysis fails and a Fourier–Galerkin approach is adopted. In the subcritical case the weakly nonlinear analysis must be pushed up to the fifth order, recovering the quintic Stuart-Landau equation for the amplitude of the pattern. The bifurcation diagram of this equation shows a range of the bifurcation parameter in which two qualitatively different stable states coexist (the origin and two large amplitude branches). Therefore the evolution of the pattern corresponds to a hysteresis cycle.