  We consider a highly idealized model for El Nino/Southern Oscillation (ENSO)
variability, as introduced in an earlier paper. The model is governed by a
delay differential equation for sea surface temperature in the Tropical
Pacific, and it combines two key mechanisms that participate in ENSO dynamics:
delayed negative feedback and seasonal forcing. We perform a theoretical and
numerical study of the model in the three-dimensional space of its physically
relevant parameters: propagation period of oceanic waves across the Tropical
Pacific, atmosphere-ocean coupling, and strength of seasonal forcing. Phase
locking of model solutions to the periodic forcing is prevalent: the local
maxima and minima of the solutions tend to occur at the same position within
the seasonal cycle. Such phase locking is a key feature of the observed El Nino
(warm) and La Nina (cold) events. The phasing of the extrema within the
seasonal cycle depends sensitively on model parameters when forcing is weak. We
also study co-existence of multiple solutions for fixed model parameters and
describe the basins of attraction of the stable solutions in a one-dimensional
space of constant initial model histories.
