On the bandwidth of stable nonlinear stripe patterns in finite size systems

Abstract

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share 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 finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift–Hohenberg (SH) model and the universal Newell–Whitehead–Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for the Turing pattern in cells. In addition, we explain why below a certain system width, stripe patterns behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.

Slow passage through the Busse balloon – predicting steps on the Eckhaus staircase

“…It is therefore a natural starting point to understand the evolution of periodic patterns through the Busse balloon. In the Ginzburg-Landau equation, the boundary of the Busse balloon is determined by the Eckhaus instability; see [14,iVA1a(ii)] for background, [36,55] for a study of the dynamics of the instability, [29,56] for the effect of noise, [51] for finite-size effects, and Section 2 below for a basic review. We allow the linear coefficient μ to vary slowly in time, with time scale ε −1 , and consider the problem on bounded domains with periodic boundary conditions.…”