Convergence properties of the 8, 10 and 12 mode representations of quasipatterns

Abstract

Spatial Fourier transforms of quasipatterns observed in Faraday wave experiments suggest that the patterns are well represented by the sum of 8, 10 or 12 Fourier modes with wavevectors equally spaced around a circle. This representation has been used many times as the starting point for standard perturbative methods of computing the weakly nonlinear dependence of the pattern amplitude on parameters. We show that nonlinear interactions of n such Fourier modes generate new modes with wavevectors that approach the original circle no faster than a constant times n^{-2}, and that there are combinations of modes that do achieve this limit. As in KAM theory, small divisors cause difficulties in the perturbation theory, and the convergence of the standard method is questionable in spite of the bound on the small divisors. We compute steady quasipattern solutions of the cubic Swift--Hohenberg equation up to 33rd order to illustrate the issues in some detail, and argue that the standard method does not converge sufficiently rapidly to be regarded as a reliable way of calculating properties of quasipatterns.