We revisit the classical Suarez–Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide analytical–numerical evidence for the existence of two-dimensional inertial manifolds in the model. This allows to suggest a complete qualitative description of the dynamics in the region of linear stability. We show that there are two subregions corresponding to the existence of hidden or self-excited attracting periodic orbits. These subregions must be separated by a curve on which homoclinic “figure eights”, bifurcating into a single one or a pair of unstable periodic orbits, should exist. We relate the observed hidden oscillations and homoclinics to the irregularity theories of ENSO and provide numerical evidence that chaotic behavior may appear if a small periodic forcing is applied to the model. We also use parameters from the Suarez–Schopf model to discover hidden and self-excited asynchronous periodic regimes in a ring array of coupled lossless transmission lines studied by J. Wu and H. Xia.