Imagine a pendulum that is pushed in time with the seasons: harder in winter, softer in summer. But this pendulum has a quirk: a switch that operates with a delay. When the pendulum passes through zero, the switch clicks and changes the direction of the push. That is exactly how the simplified El Niño model built by mathematicians works.

In the original Gil–Zaliapin–Thompson (GZT) model, the feedback is described by a smooth function: the hyperbolic tangent. But solving such equations analytically is nearly impossible. The authors replaced the smooth function with a sign function: it takes only two values, plus or minus, like a switch. Meanwhile, the seasonal forcing remained smooth, in the form of a cosine. The result is a pendulum pushed periodically, with the switch acting after a delay.

For such a pendulum, explicit solutions were found. It turns out that it performs periodic oscillations, and one can precisely calculate for which parameters these oscillations are stable. If the amplitude of the seasonal forcing is large enough, the oscillations are stable; as the amplitude decreases, they lose stability through a torus bifurcation or a fold.

Comparison with numerical calculations of the original GZT model showed excellent agreement. The bifurcation curves and rotation numbers almost coincide, especially for large values of the coupling and delay parameters. This means that the crude simplification does not distort the essence of the phenomenon, but rather makes it accessible to precise analysis.

The switchback pendulum, pushed in time with the seasons, turned out to be more than just a toy. It allows one to rigorously explain why El Niño sometimes behaves regularly and sometimes falls out of rhythm. And now this simple mechanism can be used as a benchmark for testing complex climate models.