Low-dimensional Dynamics for the Complex Ginzburg-landau Equation

A method is presented which results in low-dimensional dynamical systems for situations in which low-dimensional attractors are known to exist. The method is based on the use of the Karhunen-Loeve procedure for the determination of an optimal basis and the subsequent use of the Galerkin procedure to generate the dynamical system. The method is applied to two problems for the Ginzburg-Landau equation for which large databases have, been obtained. In each instance a dynamical system is generated which has roughly twice the number of degrees of freedom as the Hausdorff dimension of the exact case. It is also demonstrated that the approximations are robust in that they are accurate over a wide range of parameter space.