A Parametrically Excited Nonlinear Wave Equation

Abstract

When considering nonlinear waves with periodic parametric forcing the geometry of the spatial domain plays a crucial part. If the spatial domain is a square we find an infinite number of 1 : 1 resonances and in addition accidental resonances. Using Galerkin projection on 2 modes in 1 : 1 resonance we find stable normal mode periodic solutions and unstable periodic solutions in general position; the location in phase-space is characterised as a triple resonance zone. In the limit case of vanishing dissipation we find neutral stability and strong recurrence of the orbits. Interaction of 1 : 1 resonances shows a selection mechanism of the 1 : 1 modes triggered off by the parametric forcing. In addition we analyse a number of prominent accidental resonances produced by the spectrum induced by our choice of a square in space.