Physica D **178** (2003) 62-82.
doi:10.1016/S0167-2789(02)00792-3
## Convergence properties of the 8, 10 and 12 mode representations of
quasipatterns

A.M. Rucklidge(1)
and
W.J. Rucklidge(2).

(1) Department of Applied Mathematics,

University of Leeds, Leeds, LS2 9JT, UK

(2) Mountain View, CA 94043, USA

**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.

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