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Nonlinear stripe patterns occur in many different systems, from the smallscales of biological cells to geological scales as cloud patterns. They allshare the universal property of being stable at different wavenumbers $q$,i.e., they are multistable. The stable wavenumber range of the stripe patterns,which is limited by the Eckhaus- and zigzag instabilities even in finitesystems for several boundary conditions, increases with decreasing system size.This enlargement comes about because suppressing degrees of freedom from thetwo instabilities goes along with the system reduction, and the enlargementdepends on the boundary conditions, as we show analytically and numericallywith the generic Swift-Hohenberg (SH) model and the universalNewell-Whitehead-Segel equation. We also describe how, in very small systemsizes, any periodic pattern that emerges from the basic state is simultaneouslystable in certain parameter ranges, which is especially important for Turingpattern in cells. In addition, we explain why below a certain system widthstripe pattern behave quasi-one-dimensional in two-dimensional systems.Furthermore, we show with numerical simulations of the SH model in medium-sizedrectangular domains how unstable stripe patterns evolve via the zigzaginstability differently into stable patterns for different combinations ofboundary conditions.

On the band-width of stable nonlinear stripe patterns in finite size systems