  We investigate a number of complex patterns driven by the electro-convection
instability in a planarly aligned layer of a nematic liquid crystal. They are
traced back to various secondary instabilities of the ideal roll patterns
bifurcating at onset of convection, whereby the basic nemato-hydrodynamic
equations are solved by common Galerkin expansion methods. Alternatively these
equations are systematically approximated by a set of coupled amplitude
equations. They describe slow modulations of the convection roll amplitudes,
which are coupled to a flow field component with finite vorticity perpendicular
to the layer and to a quasi-homogeneous in-plane rotation of the director. It
is demonstrated that the Galerkin stability diagram of the convection rolls is
well reproduced by the corresponding one based on the amplitude equations. The
main purpose of the paper is, however, to demonstrate that their direct
numerical simulations match surprisingly well new experiments, which serves as
a convincing test of our theoretical approach.
