Local bifurcations and a global attractor for two versions of the weakly dissipative Ginzburg–Landau equation

We consider the periodic boundary value problem for two variants of a weakly dissipative complex Ginzburg–Landau equation. In the first case, we study a variant of such an equation that contains the cubic and quintic nonlinear terms. We study the problem of local bifurcations of traveling periodic waves under stability changes. We show that a countable set of two-dimensional invariant tori arises as a result of such bifurcations. Both types of bifurcations are possible in the considered formulation of the problem, soft (postcritical) and hard (subcritical) ones, depending on the choice of the coefficients in the equation. We obtain asymptotic formulas for the solutions forming the invariant tori. We also study the periodic boundary value problem for the equation that is called the nonlocal Ginzburg–Landau equation in physics. We show that the boundary value problem in the considered variant has an infinite-dimensional global attractor. We present the solutions forming such an attractor.