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We revisit the classical Suarez-Schopf delayed oscillator. Special attentionis paid to the region of linear stability in the space of parameters. By meansof the theory of inertial manifolds developed in our adjacent papers, weprovide analytical-numerical evidence for the existence of two-dimensionalinertial manifolds in the model. This allows to suggest a complete qualitativedescription of the dynamics in the region of linear stability. We show thatthere are two subregions corresponding to the existence of hidden orself-excited periodic orbits. These subregions must be separated by a curve onwhich homoclinic "figure eights", bifurcating into a single one or a pair ofunstable periodic orbits, should exist. We relate the observed hiddenoscillations and homoclinics to the irregularity theories of ENSO and providenumerical evidence that chaotic behavior may appear if a small periodic forcingis applied to the model. We also use parameters from the Suarez-Schopf model todiscover hidden and self-excited asynchronous periodic regimes in a ring arrayof coupled lossless transmission lines studied by J. Wu and H. Xia.

Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO