Quasipatterns in a parametrically forced horizontal fluid film

We shake harmonically a thin horizontal viscous fluid layer (frequency forcing Ω , only one harmonic), to reproduce the Faraday experiment and using the system derived in Rojas et al. (2010) [34] invariant under horizontal rotations. When the physical parameters are suitably chosen, there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency Ω / 2 , and the mode with frequency Ω . Moreover, at criticality the corresponding wave lengths k c and k c ′ are such that if we define the family of 2 q equally spaced (horizontal) wave vectors k j on the circle of radius k c , then k j + k l = k n ′ ,  with  | k j | = | k l | = k c , | k n ′ | = k c ′ . ults under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequency Ω / 2 ) having a spatially quasiperiodic pattern if q ≥ 4 . Restricting our study to solutions invariant under rotations of angle 2 π / q , gives a kernel of dimension 4. spirit of Rucklidge and Silber (2009) [29] we derive formally amplitude equations for perturbations possessing this symmetry. Then we give simple necessary conditions on coefficients, for obtaining the bifurcation of (formally) stable time-periodic (frequency Ω / 2 ) quasipatterns. In particular, we obtain a solution such that a time shift by half the period, is equivalent to a rotation of angle π / q of the pattern.