Low-dimensional behaviour in the complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined here as the number of Fourier modes required to span the universal attractor. We use concepts from the theory of inertial manifolds to determine rigorous upper bounds on the Fourier spanning dimension, which also establishes the finite dimensionality of the universal attractor. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for some parameter values, computing exactly) the Lyapunov dimension and invoking a recent theorem that asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula with the universal (global) Lyapunov exponents, is an upper bound on the Hausdorff dimension. This study of low dimensionality in the complex Ginzburg-Landau equation allows for an examination of the current techniques used in the rigorous investigation of finite-dimensional behaviour. Contact is made with some recent results for fluid turbulence models, and we discuss some unexplored directions in the area of low-dimensional behaviour in the complex Ginzburg-Landau equation.