This is a joint work with Médéric Argentina (Nice).
We consider a horizontal fluid film, vertically harmonically shacked (frequency forcing 1, only one harmonic). In addition to the amplitude Γ of the forcing, other parameters of the system are the Reynods number R, the Froude number Fr, and the Bond number Bo. We give the conditions under which the basic time periodic flow losses its stability, via time periodic perturbations of period 4π or 2π with the same critical value Γc of the forcing. These instabilities lead to spatial wave numbers k and k', corresponding to critical circles in the Fourier plane (due to rotational invariance of the system).
Now consider the cases when k'/k=2cos(nπ/q) with n and q integers, and denote by kj (resp. k'j) the wave vectors of length k (resp. k'), making the angle (j-1)π/q with the x axis, then this condition corresponds to kj+kj+2n=k'j+n, for j=1,…2q and these cases occur for values of R and Bo/(Fr)2 depending only on n/q.
Quasipatterns correspond to the cases q≥4.
Searching for solutions invariant under rotations of angle 2π/q, we reduce formally the problem to a system of amplitude equations. In the case where q≥5 is odd, we find 4π - time periodic quasipatterns such that a time shift of 2π is the same as rotating the pattern by π/q (this type of phenomenon is observed experimentally). We give simple conditions for their stability.