  Models of global climate phenomena of low to intermediate complexity are very
useful for providing an understanding at a conceptual level. An important
aspect of such models is the presence of a number of feedback loops that
feature considerable delay times, usually due to the time it takes to transport
energy (for example, in the form of hot/cold air or water) around the globe. In
this paper we demonstrate how one can perform a bifurcation analysis of the
behaviour of a periodically-forced system with delay in dependence on key
parameters. As an example we consider the El-Nino Southern Oscillation (ENSO),
which is a sea surface temperature oscillation on a multi-year scale in the
basin of the Pacific Ocean. One can think of ENSO as being generated by an
interplay between two feedback effects, one positive and one negative, which
act only after some delay that is determined by the speed of transport of
sea-surface temperature anomalies across the Pacific. We perform here a case
study of a simple delayed-feedback oscillator model for ENSO (introduced by
Tziperman et al, J. Climate 11 (1998)), which is parametrically forced by
annual variation. More specifically, we use numerical bifurcation analysis
tools to explore directly regions of delay-induced resonances and other
stability boundaries in this delay-differential equation model for ENSO.
