Onset of chaos in the weakly dissipative two-dimensional complex Ginzburg-Landau equation

The two-dimensional Ginzburg-Landau (GL) equation in the weakly dissipative regime (real parts of the coefficients are assumed to be small in comparison with the imaginary ones) is considered in a square cell with reflecting (Neumann) boundary conditions. Following the lines of the analysis developed earlier for the analogous 1D equation, we demonstrate that, near the threshold of the modulational instability, the GL equation can be consistently approximated by a five-dimensional dynamical system which possesses a three-dimensional attracting invariant manifold. On the manifold, the dynamics are governed by a modified Lorenz model containing an additional cubic term. By means of numerical simulations of this approximation, a diagram of dynamical regimes is constructed, in a relevant parameter space. A region of chaos is found Unlike the previously studied case of the 1D GL equation, in the present case a blow-up is possible, depending on initial conditions.