Rotationally invariant order parameter equations for natural patterns in nonequilibrium systems

We discuss a theoretical description of the formation of cellular patterns exhibiting defects, grain boundaries, and spiral patterns in nonequilibrium large-aspect-ratio systems by means of rotationally invariant order parameter equations. Starting from evolution equations of general form, we show that the order parameter equations which can be derived close to a bifurcation point in general involve nonlinear terms which are nonlocal. We present a suitable approximation scheme of these terms by local ones which is based on a gradient expansion. A truncation of this expansion leads to model equations which are widely used in the theoretical treatment of natural patterns in complex systems. [S1063-651X(99)01003-X].