FINITE-DIMENSIONAL MODELS OF THE GINZBURG-LANDAU EQUATION

In this paper we study truncated finite dimensional models of the infinite-dimensional equation describing the evolution of even, space-periodic solutions of the Ginzburg-Landau equation. We derive estimates on the position of the global attractor of the flow, which yield that the magnitude of the Mth mode of the global attractor decays faster than any algebraic power of M-1. The estimates are independent of the dimension of the model. In a numerical section we simulate the flow for three radical low-dimensional models (of two, three and four complex modes); we analyse the influence of the number of modes on the global dynamics. The four-dimensional model exhibits the same intricate flow-characteristics as the 32-dimensional model studied by Keefe.