Quasipatterns in a parametrically forced horizontal fluid film

We shake harmonically a thin horizontal viscous fluid layer (frequency forcing ΩΩ, 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/2Ω/2, and the mode with frequency ΩΩ. Moreover, at criticality the corresponding wave lengths kckc and kc′ are such that if we define the family of 2q2q equally spaced (horizontal) wave vectors kj on the circle of radius kckc, then kj+kl=kn′, with |kj|=|kl|=kc,|kn′|=kc′.It results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequency Ω/2Ω/2) having a spatially quasiperiodic pattern if q≥4q≥4. Restricting our study to solutions invariant under rotations of angle  2π/q2π/q, gives a kernel of dimension 4.In the 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/2Ω/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/qπ/qof the pattern.

► We study the Faraday instability of a thin viscous layer film.
► We propose a mechanism for the emergence of quasipatterns.
► We derive the normal form of these quasipatterns.
► We show the existence of stable quasipatterns.