  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 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.
