ONSET OF CHAOS IN THE WEAKLY DISSIPATIVE 2-DIMENSIONAL COMPLEX GINZBURG-LANDAU EQUATION

The two-dimensional Ginzburg-Landau (GL) equation in the weakly dissip ative regime (real parts of the coefficients are assumed to be small i n comparison with the imaginary ones) is considered in a square cell w ith reflecting (Neumann) boundary conditions. Following the lines of t he analysis developed earlier for the analogous 1D equation, we demons trate that, near the threshold of the modulational instability, the GL equation can be consistently approximated by a five-dimensional dynam ical system which possesses a three-dimensional attracting invariant m anifold. On the manifold, the dynamics are governed by a modified Lore nz model containing an additional cubic term. By means of numerical si mulations of this approximation, a diagram of dynamical regimes is con structed, in a relevant parameter space. A region of chaos is found Un like the previously studied case of the 1D GL equation, in the present case a blow-up is possible, depending on initial conditions.