A pendulum swings steadily until gusts of wind disturb it. Each push shifts the rhythm slightly, and soon the beats no longer match the metronome. Natural oscillators live the same way: the heart, populations, climate cycles—they all oscillate, but randomness constantly interferes.
The new study considers a system that, without noise, would circle a perfect ring. Noise adds random kicks. The author decomposes the motion into two parts: one along the ring, the other across it. The first part accumulates random shifts, like a drunkard walking around a circle: he keeps moving forward, but his direction wobbles. The second part decays quickly, as if an elastic band pulled the point back to the ring.
For the simple case where the ring emerges from a fixed point, the author derived exact formulas. They show how quickly the signal loses correlation with itself and how the peak frequency shifts under the influence of noise. Numerical simulations confirmed that the formulas work when the noise is weak.
The method was applied to El Niño data—a climate oscillation in the Pacific Ocean. The results gave a period of about 4.2 years and a rate of trajectory divergence consistent with earlier estimates. Thus, a pendulum swaying in the wind turned out to be described by formulas that can be fitted to real data.
El Niño, despite its irregularity, has a characteristic period of about 4 years, which is close to the estimate obtained in this work.