In this paper we give a description of the structure of compact global attractor (Levinson cen-ter) for monotone Bohr/Levitan almost periodic dynamical system x^′ = f(t, x) (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.