Pattern selection in bistable systems

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Abstract
Pattern selection in reaction-difiusion systems exhibiting bistability of homoge- neous steady states is discussed. In agreement with recent experimental results, we obtain new bifurcation diagrams involving large-amplitude structures that arise from the coupling of the spatial critical modes with a quasi-neutral homogeneous mode. An ever-increasing set of chemical and physical systems give rise to macroscopic patterns, dissipative structures, resulting from difiusive instabilities. They range from chemical re- actors (1)-(3) (Turing structures) to electron-hole plasmas (4), semiconductor devices (5), gas discharges (6), materials irradiated by energetic particles or light (7), optics (8), ::: Experiments recently revealed that the same sequence of structures occurs in these physically dissimilar systems when they exhibit bistability between homogeneous steady states (h.s.s.). (9)-(12). We here analyze the formation and selection of these difiusion-driven patterns. It is indeed amazing to note that such studies for multistable systems are not very advanced although scores of works have been devoted to the properties of the fronts linking two h.s.s. that may also arise (4), (13)-(15). We show that pattern selection is dominated by the coupling of the active spatial modes with a quasi-neutral homogeneous mode generated by the bifurcations inducing the bistability. We also discuss the existence of isolated branches of patterned solutions even when the h.s.s. are stable with respect to non-uniform perturbations. We consider a general two-variable reaction-difiusion model characterized by the competi- tion between an activator u that stimulates its own variations and a controlling inhibitor v: 8 > > < > > : @u @t = f(u;v;a;∞) + 5 2 u;

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