context, by considering the discrete-time dynamical system derived by sampling the continuous-time system at a single phase of the forcing period.

Secondly, the theory applies close to the threshold. Our simulations are carried out far from threshold, so that instabilities can grow on a reasonable timescale and stabilize at an amplitude that can be clearly seen. Our patterns contain modes generated by nonlinear interactions and their existence or stability may result from secondary bifurcations.

Despite these differences, we will see that there is much common ground between the patterns we observe and those predicted by theory.

Problem formulation, governing equations and numerical scheme {#sec:numerics}
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The governing equations for an incompressible two-phase flow can be expressed by a single field formulation: $$\label{NS-Eqs}
\displaystyle{\rho  \left( \frac{\partial \textbf{u}}{\partial t}
+\textbf{u}\cdot\nabla \textbf{u}\right) =  -\nabla P + \rho \textbf{G}
+ \nabla \cdot \mu\left(\nabla\textbf{u} +\nabla\textbf{u}^T \right)
+ \textbf{F}}, \qquad \displaystyle{\nabla\cdot\textbf{u}=0 }$$ where $\textbf{u}$ is the velocity, $P$ is the pressure, $\rho$ is the density, $\mu$ is the dynamic viscosity and $\textbf{F}$ is the local surface tension force at the interface. Here, ${\bf G}$ is an imposed time-dependent radial acceleration: $${\bf G}=-\left(g+ a \cos(\omega t)\right)\frac{r}{R}\textbf{e}_r
\label{eq:grav}$$ where $g$ is a constant acceleration, referred to for simplicity 