Dynamics of a nonlinear parametrically-excited PDE: 2-term truncation (Q1352682)

This paper concerns the semilinear wave equation \[ \partial^2_tu- c^2\partial^2_xu+ \varepsilon\beta\partial_tu+ (\delta+\varepsilon\gamma\cos t)u= \varepsilon\alpha u^3,\quad 0<x<\pi \] with boundary conditions \(\partial_x u=0\) at \(x=0\) and \(x=\pi\). An averaged two-term truncation is investigated. The \(\delta\)-\(c\) plane is divided up into many regions (for \(\alpha=1\), \(\beta=0\), \(\gamma=1\), \(\varepsilon=0.1\) there appear 18 regions), each of which has different dynamics than its neighbors and in most of which there is more than one stable periodic solution.