  We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling
$\kappa$, and propagation period $\tau$ of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the $(b,\tau)$ plane at constant $\kappa$. The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling $\kappa$ increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.
