Invariant manifolds of a weakly dissipative version of the nonlocal Ginzburg–Landau equation

We consider a periodic boundary value problem for a nonlocal Ginzburg–Landau equation in its weakly dissipative version. The existence, stability, and local bifurcations of one-mode periodic solutions are studied. It is shown that in a neighborhood of one-mode periodic solutions there may exist a three-dimensional local attractor filled with spatially inhomogeneous time-periodic solutions. Asymptotic formulas for these solutions are obtained. The results are based on using and developing methods of the theory of infinite-dimensional dynamical systems. In a special version of the partial integro-differential equation considered, we study the existence of a global attractor. Solution in the form of series are obtained for this version of the nonlinear boundary value problem.