We study the behavior of an iterative map as a model for El Nino and the Southern Oscillation (ENSO). This map is derived from a model that combines linear equatorial beta-plane ocean dynamics with a version of the Bjerknes hypothesis for ENSO. It differs from the linear model of Cane et al. only in that the coupling from ocean to atmosphere is idealized as a nonlinear relation tau-(h(e)) between a wind stress tau of fixed spatial form and h(e), the thermocline displacement at the eastern end of the equator. The model sustains finite amplitude periodic and aperiodic oscillations. A period doubling bifurcation leads from a period of less than 2 years to the 3-4 year one observed in nature. Other principal results are: the resulting period depends on the curvature of the function away from the unstable equilibrium at h(e) = 0, and not solely on its linear instability; at least two Rossby modes must be included in the model for aperiodic oscillations to appear; no stochastic term is needed for this aperiodicity, but it appears more readily if the model background state includes an annual cycle.
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