Why do some systems organise themselves into well ordered patterns with astonishing symmetry and regularity, while other superficially similar systems produce defects and disorder? In systems where two different length scales are unstable, the nonlinear interaction between the different modes is key: steady complex patterns can be stabilised when the modes act together to reinforce each other. But, if the two types of pattern compete with each other, the outcome can be considerably more complicated: a time-dependent disordered mixture of patterns constantly shifting and changing. In a small domain, the nature of the interaction between a small number of modes on each length scale can readily be computed. In a large domain, each mode can interact with hundreds of other modes, but the overall behaviour still appears to be guided by small-domain considerations.

The classic Faraday wave experiment consists of a horizontal layer of fluid that spontaneously develops a pattern of standing waves on its surface as it is driven by vertical oscillation with amplitude exceeding a critical value. Faraday wave experiments have consistently produced patterns with remarkably high degrees of symmetry. Quasipatterns, which are quasiperiodic in any spatial direction, are particularly interesting since there is, as yet, no satisfactory theoretical understanding of their formation. We use multi-frequency parametric forcing to investigate the formation of patterns and approximate quasipatterns in a model partial differential equation, which plays the same role for the Faraday wave experiment that the Swift--Hohenberg equation plays for convection. We exploit three-wave resonant interactions to design forcing functions that ought to produce complex patterns, and make quantitative comparisons between weakly nonlinear predictions and the solutions of the PDE. This comparison reveals the limitations of the theory, and we explore ways in which these limitations can be addressed. Based on: Design of parametrically forced patterns and quasipatterns, by A.M. Rucklidge and M. Silber. SIAM J. Applied Dynamical Systems 8 (2009) 298-347.