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

The classical approach required knowing those equations, but for the real climate or a living organism, you cannot write them down. So the authors build an approximate model directly from data: using kernel smoothing, they turn the observed trajectory into a finite Markov chain that approximates the Koopman operator (a mathematical object describing the evolution of functions of the state). If observations are incomplete, Takens' delay embedding is used, which reconstructs hidden variables from a single measured quantity, much as the shape of an object can be guessed from its shadow on a wall.

Next, an optimization problem is formulated: find a perturbation matrix with zero row sums and constraints on allowed transitions that maximizes the derivative of the modulus or argument of a chosen eigenvalue. The solution is given by an explicit formula involving the left and right eigenvectors. For clarity, a vector field of optimal response is constructed: it shows in which direction in observation space one should act at each point.

In the example of nonuniform rotation of a circle, the optimal perturbation to speed up the cycle turned out to be tangential to the trajectory and inversely proportional to the current speed: where motion is slow, the nudge must be stronger. In the Lorenz system, the perturbation that accelerates rotation around the wings is orthogonal to the level lines of the argument of the complex eigenvector, not along the flow. In the El Niño climate example, the optimal perturbation to increase frequency leads the cycle by a quarter of a period, while that for strengthening correlations is in phase. A connection with the phase of the Indian Ocean Dipole was found: negative IOD during El Niño buildup and positive IOD during La Niña.

Previously, it was unclear how to find the best small perturbation for a nonlinear system when only observational data are available and the equations of dynamics are inaccessible. Now one can compute the optimal perturbation that maximally changes the frequency or stability of near-cycles and near-invariant sets directly from the time series. The method could be applied in climatology to assess the influence of external factors on El Niño, and in controlling chaotic systems where experiments are impossible or too costly.

Returning to the orchestra: now, from a recording alone, one can say which instrument and at what moment should be nudged to even out the tempo. And although a real conductor is still needed for interpretation, mathematics puts in their hands a precise map of possible interventions.