Every year, as December approaches, the water off the coast of Peru becomes anomalously warm or anomalously cold. This is El Niño and La Niña, two phases of the same climate pendulum that swings in the equatorial Pacific. The pendulum is not a simple one: it remembers what happened to it several months earlier, and this memory determines which way it will swing next.

In the 1980s, climatologists proposed describing this pendulum with a delay equation: the rate of change of water temperature depends not only on the current state but also on what happened earlier. The delay arises because waves in the ocean take several months to travel from one shore to the other, not instantaneously. Physicists Suarez and Schopf simplified the model to a single equation that is easy to solve on a computer and analyze by hand.

In the new work, mathematicians added an external periodic forcing to the Suarez–Schopf equation to mimic the changing seasons. They constructed a complete map of how the system's behavior changes for different values of the delay and the forcing amplitude. It turned out that the regions where the pendulum synchronizes with the season are very narrow and barely overlap. This explains why real El Niño does not repeat strictly every year, although it does tend to peak in December.

To understand the details, the authors developed a method for computing the rotation number for delay systems. The rotation number indicates how many turns the trajectory makes on the torus during one period of the external forcing. If this number is rational, the dynamics are periodic; if irrational, the trajectory densely fills the surface. The calculations required about 768,000 model runs, which took roughly 8 hours on a standard computer.

The result is important for understanding climate: the Suarez–Schopf model with seasonal forcing does not reproduce the chaotic irregularity seen in observations. This means that realistic forecasting must account for additional factors, such as the asymmetry between the warm and cold phases or random perturbations. But the method for computing the rotation number itself will be useful in many fields, from laser physics to biological rhythms.