Onset of chaos in the generalized Ginzburg-Landau equation

The complex Ginzburg-Landau equation is considered in the weak-dissipation regime. The equation is supplemented by a periodic boundary condition, admitting a minimum wave number close to the threshold of the Benjamin-Feir instability. We demonstrate that the original equation can be consistently approximated by a three-dimensional dynamical system, which, depending on values of parameters, either coincides with the Lorenz model or differs from it in the sign of one coefficient. For the latter case, a diagram of dynamical regimes is constructed by numerical methods, and a region of chaos is found.