Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg–Landau equation

We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mané projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of three-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet–Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.