We analyze the dynamical processes behind delayed double ionization of atoms subjected to strong laser pulses. Using reduced models, we show that these processes are a signature of Hamiltonian chaos which results from the competition between the laser field and the Coulomb attraction to the nucleus. In particular, we exhibit the paramount role of the unstable manifold of selected periodic orbits which lead to a delay in these double ionizations. Among delayed double ionizations, we consider the case of recollision excitation with subsequent ionization (RESI) and, as a hallmark of this mechanism, we predict oscillations in the ratio of RESI to double ionization yields versus laser intensity. We discuss the significance of the dimensionality of the reduced models for the analysis of the dynamical processes behind delayed double ionization.