Abstract Non-linear flexural vibrations of nearly square plates subject to parametric in-plane excitations are studied. The theoretical results are based on the analysis of a fourth order system of non-linear ordinary differential equations in normal form derived from the dynamic analog of von Karman equations. The equations represent a system with 1:1 resonance and Z 2B + Z 2 or broken D 4 symmetry. Local bifurcation analysis of these equations shows that the system is capable of extremely complex standing as well as travelling waves. A global bifurcation analysis shows the existence of heteroclinic loops which when they break lead to Smale horseshoes and chaotic behavior on an extremely long time scale.