The model. Following the recharge-oscillator idea (Jin 1997), the eastern-Pacific sea-surface temperature anomaly T (a Nino-3.4-like index, in C) and the basin-mean thermocline depth anomaly h (a proxy for equatorial heat content, in m) obey
dT/dt = R T + F h - e T³ + noise and dh/dt = -a T - r h. Positive R is the Bjerknes feedback: warm east, weaker trade winds, less upwelling of cold water, warmer east. F turns a deeper thermocline into warming; -a T says El Nino drains heat from the equatorial band (the discharge), which later makes the thermocline shallow and brings La Nina; r is damping of the heat reservoir.
Exact linear solution. With e = 0 and no noise the system matrix has trace R - r and determinant F a - R r. The eigenvalues are σ ± iω with σ = (R - r)/2 and ω = √(F a - ((R + r)/2)²) when that is real; the period is 2π/ω. The cycle grows if σ > 0 (R above r), decays if below. With the default numbers the period is about 4 years, the right order for observed ENSO (typically 2 to 7 years, irregular).
Nonlinearity and noise. The cubic term caps the amplitude (the growing oscillation settles on a limit cycle near √(R/e) for small damping). Random wind bursts (westerly wind events) are added as white noise scaled by √dt (Euler-Maruyama), which turns a damped oscillator into an irregular, self-renewing one - the leading explanation for why real ENSO does not repeat like a clock.
Cross-section. The picture is schematic: the thermocline tilt flattens when T is positive, the trade-wind arrows weaken, and the eastern SST colour follows T. The thermocline depth drawn is a mean slope (about 150 m west, 50 m east) plus h.
Honest limits. This is a two-box caricature, not a climate model: no seasonal cycle, no spatial structure, no Kelvin or Rossby wave propagation, no asymmetry between El Nino and La Nina beyond the cubic term, and the constants (F, a, R, r, e) are illustrative tuned values, not fits to data. Thresholds of plus or minus 0.5 C follow the usual ONI convention. Observed period and amplitude figures quoted are typical values.