__Convergence
Properties of the 8, 10 and 12 Mode __

__Representations
of Quasipatterns__

__ __

__ Alastair Rucklidge__

Department of Applied Mathematics

University of Leeds

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. 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~$n^{-2}$. These close approaches lead to small divisors in the standard
perturbation theory used to compute properties of these patterns, and we show
that the convergence of the standard method is questionable in spite of the
bound on the small divisors.