Attractors for Lyapunov cases of the complex Ginzburg-Landau equation

A special case of the complex Ginzburg-Landau (CGL) equation possessing a Lyapunov functional is identified. The global attractor of this Lyapunov CGL (LCGL) is studied in one spatial dimension with periodic boundary conditions. The LCGL may be viewed as a dissipative perturbation of the nonlinear Schrodinger equation (NLS), a completely integrable Hamiltonian system. The o-limit sets of the LCGL are identified as compact, connected unions of subsets of the stationary points of the flow. The stationary points do not depend on the strength of the perturbation, and so neither do the o-limit sets. However, the basins of attraction do depend sensitively on the perturbation strength. To determine the stability of the o-limit sets, the global Lyapunov functional is studied. Using the integrable NLS machinery, the second variation of the Lyapunov functional is diagonalized. An analysis of the diagonal elements yields that certain LCGL stationary points are stable. We are able to analyze the basins of attraction for a planar toy problem, which like the LCGL, is a dissipative perturbation of a Hamiltonian system. For this problem, almost every phase point is in a basin of attraction of an asymptotically stable stationary point. As the perturbation tends to zero, these basins become intermingled and the event of a fixed phase point being captured into a particular basin becomes probabilistic. Formulas for computing the probabilities of capture are given. These formulas are substantiated through a formal asymptotic analysis and numerical experiments. Such a probabilistic description of the basins of attraction is not completed for the infinite dimensional LCGL