Comparison of inertial manifolds and application to modulated systems

We consider two dissipative systems having inertial manifolds and give esti mates which allow us to compare the flows on the two inertial manifolds. As an example of a modulated system we treat the Swift-Hohenberg equation par tial derivative(tau)w = - (1 + partial derivative(y)(2))(2) w + epsilon(2)w - w(3), w(tau, y) is an element of IR, with periodic boundary conditions o n the interval (0, l/epsilon). Recent results in the theory of modulation e quation show that the solutions of this equation can be described over long time scales by those of the associated Ginzburg-Landau equation partial de rivative(t)v = 4 partial derivative(x)(2)v + v - 3 \v\(2)v, v(t, x) is an e lement of C, with suitably generalized periodic boundary conditions on (0, l). We prove that both systems have an inertial manifold of the same dimens ion and that the flows on these finite dimensional manifolds converge again st each other for epsilon --> 0.