Every few years, a warm current off the coast of South America alters weather across the entire planet. This is El Niño, and its behavior resembles a pendulum that sometimes swings too far. In 1986, scientist John Vallis described this process with three equations. The equations turned out to be chaotic: the slightest change in initial conditions leads to completely different trajectories. But in that model, the ocean remembered only the current instant, as if yesterday's water had no influence on today's.

In the new work, the equations were rewritten using fractional derivatives. An ordinary derivative shows how quickly a quantity changes right now. A fractional derivative adds memory to this: it accounts for what happened to the system in the past, with recent events mattering more than distant ones. This is similar to how a person makes decisions based not only on the present moment but also on their entire experience. The ocean also remembers: warm water that arrived last year continues to affect currents and winds.

The researchers numerically solved the fractional system for different values of the parameter p, which describes the average influence of equatorial winds on the ocean. At α=1, chaos is observed for |p|≤2.45. When α decreases to 0.985, the chaotic range narrows to |p|≤1.5. At α=0.975, chaos remains only in a narrow band near p=0. And at α=0.965, chaos disappears completely. This means that sufficiently strong ocean memory can quench the chaotic oscillations of El Niño.

Previously, it was known that the integer-order Vallis model is chaotic and related to El Niño, but the influence of fractional order on chaos remained unexplored. Now it has been established that decreasing the fractional order narrows the chaotic region in the parameter p, and below a critical value chaos disappears entirely. This result was obtained through bifurcation analysis and computation of Lyapunov exponents for the fractional system across all values of p and various derivative orders. If memory in the real ocean is strong enough, climate model forecasts should account for fractional order.