Linear and weakly nonlinear analysis of the rotating polygon instability

In this talk we will present new analytic results about the polygonal instability obtained in a cylindrical container with rotating bottom [G. H. Vatistas, J. Fluid. Mech, \textbf{217}, 241, (1990), Jansson et al., Phys. Rev. Lett, \textbf{96}, 174502, (2006)]. In a recent study we showed that this spectacular instability can be explained as a result of wave interaction by introducing a simplified model that allows analytical predictions [L. Toph{\o}j et al., Phys. Rev. Lett, \textbf{110}, 194502, (2013)]. Instability maps of the global stability analysis will be presented here, as well as results of the weakly nonlinear analysis performed on the simple model which lead to the amplitude equations of the resonating free surface waves.