An orchestra plays without a conductor, and the tempo keeps speeding up and slowing down. You have only a recording of the sound, not the score. Can you tell from that recording which string to pluck to make the orchestra play more steadily? That is the challenge facing scientists who study complex systems such as climate or biological rhythms: they see only a time series of observations, while the equations governing the system remain hidden.

The authors of the work build an approximate model directly from the data. They turn the observed trajectory into a finite chain of transitions that approximates the true dynamics. If measurements are incomplete, a special trick is applied: hidden variables are reconstructed from a single measured quantity, just as the shape of an object can be guessed from its shadow on a wall. Then an optimization problem is solved: find the small perturbation that most strongly changes the frequency or stability of the oscillations. For clarity, a vector field of optimal response is constructed, showing where to act at each point.

Using El Niño climate data as an example, it turned out that the optimal perturbation for increasing frequency leads the cycle by a quarter period, while the one for strengthening correlations is in phase with it. Previously, it was unclear how to find the best perturbation when the equations are unknown. Now this can be done directly from the time series. The method could help assess the influence of external factors on El Niño and control chaotic systems where experiments are impossible.

Returning to the orchestra: now, from a recording alone, one can tell which instrument and at what moment needs a nudge to even out the tempo. Mathematics provides an exact map of possible perturbations.