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The center manifold theorem for center eigenvalues with non-zero real parts

We define center manifold as usual as an invariant manifold, tangent to the invariant subspace of the linearization of the mapping defining a continuous dynamical system, but the center subspace that we consider is associated with eigenvalues with small but not necessarily zero} real parts. We prove existence and smoothness of such center manifold assuming that certain inequalities between the center eigenvalues and the rest of the spectrum hold. The theorem is valid for finite-dimensional systems, as well as for infinite-dimensional systems provided they satisfy an additional condition. We show that the condition holds for the Navier-Stokes equation subject to appropriate boundary conditions.