SIAM J. Applied Dynamical Systems 8 (2009) 298-347.
Design of parametrically forced patterns and quasipatterns
(1) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK
(2) Department of Engineering Sciences and Applied Mathematics,
Northwestern University, Evanston, IL 60208, USA
The Faraday wave experiment is a classic example of a system driven by parametric forcing, and
it produces a wide range of complex patterns, including superlattice patterns and quasipatterns.
Nonlinear three-wave interactions between driven and weakly damped modes play a key role in
determining which patterns are favored. We use this idea to design single and multifrequency
forcing functions that produce examples of superlattice patterns and quasipatterns in a new model
PDE with parametric forcing. We make quantitative comparisons between the predicted patterns
and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very
close to onset. The reason that the range of validity is limited is that the theory requires strong
damping of all modes apart from the driven pattern-forming modes. This is in conflict with the
requirement for weak damping if three-wave coupling is to influence pattern selection effectively. We
distinguish the two different ways that three-wave interactions can be used to stabilize quasipatterns,
and we present examples of 12-, 14-, and 20-fold approximate quasipatterns. We identify which
computational domains provide the most accurate approximations to 12-fold quasipatterns and
systematically investigate the Fourier spectra of the most accurate approximations.
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