In this talk we present the behavior of the solutions of nonlinear parabolic and elliptic problems under a specific kind of singular perturbation of domain. In some cases, we combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problem and analyze the convergence properties of the solutions and of the attractors of the evolutionary equations.