On Fourier parametrization of global attractors for equations in one space dimension

For the dissipative equations of the form $ u_{t}-u_{x x}+f(x,u,u_x)=0$ we prove that the global attractor can be parametrized by a finite number of Fourier modes and that the number of modes is algebraic in parameters. This improves our earlier result [15], where the number of required modes is exponential. The method extends to equations of order higher than two.