Summary: Nonlinear 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 nonlinear ordinary differential equations in normal form derived from the dynamic analog of von Kármán equations. The equations represent a system with 1:1 resonance and \(Z_ 2\oplus Z_ 2\) 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 of an extremely long time scale.