How a 1931 Mathematical Trick Is Transforming Control of Chaotic Systems

It sounds like science fiction: a mathematical trick that transforms the writhing complexity of turbulence, the chaotic flutter of a flexible wing, or the nonlinear tangle of a power grid into a problem as tractable as a simple linear system—a set of equations a first-year student could solve. Yet this is precisely what the Koopman operator offers, and a new comprehensive tutorial by Igor Mezić and colleagues, published in 2026, suggests this is no mere theoretical curiosity but a practical toolkit rapidly reshaping how engineers think about control, prediction, and the fundamental structure of dynamical systems (Mezić et al., 2026). The implications cascade from the most abstract corners of functional analysis into garage workshops where hobbyists tune drones, hospital laboratories where clinicians model neural activity, and boardrooms where grid operators orchestrate renewable energy flows. The paper synthesizes decades of theoretical development into a coherent roadmap for practitioners, offering both the mathematical foundations and the software implementations needed to put this powerful idea to work.

The Fundamental Problem: Why Nonlinearity Breaks Our Tools

To understand why the Koopman operator matters, you first need to understand the problem it solves. Most physical phenomena—weather systems, chemical reactors, robotic arms, financial markets—behave according to nonlinear rules. A nonlinear system is one where the relationship between cause and effect isn't proportional. Double the input doesn't double the output. Push a pendulum slightly harder at the bottom of its swing and it responds very differently than if you push it at the top. The mathematics of such systems is notoriously difficult because their behavior can include chaos (extreme sensitivity to initial conditions), multiple equilibria, limit cycles, and other phenomena that simply cannot arise in linear systems.

For over a century, control engineers have dealt with this complexity by doing something clever: they pretended the system was linear. Not completely, of course. They acknowledged that the system was nonlinear, but then they focused on a small neighborhood around some operating point—a specific temperature, a particular angle, a steady-state equilibrium—and pretended the system behaved linearly within that narrow window. This technique, called linearization, works well enough in many contexts. Engineers can apply a vast arsenal of linear control theory—PID controllers, state feedback, optimal control—to stabilize a system near its operating point. Aircraft use linearized models to maintain stable flight; process plants use them to regulate temperature and pressure.

But there's a catch that practicing engineers know intimately: linearized models fail catastrophically outside their narrow valid region. The aircraft that handles beautifully at cruising altitude may become uncontrollable during a steep dive where the linear approximation no longer holds. The chemical reactor that tracks setpoints reliably near equilibrium may oscillate wildly or even explode when asked to transition between operating regimes. The power grid that maintains stability under normal conditions may cascade into blackout when subjected to disturbances the linear model never anticipated. These failures aren't edge cases or engineering incompetence—they're fundamental limitations of a methodology built on the assumption that nonlinear systems behave like linear ones, at least in small neighborhoods.

The alternative approaches have historically required either enormous expert knowledge to craft specialized solutions for specific systems, or computational methods so expensive they could only be applied offline. Neither scales to the modern challenge: controlling increasingly complex systems across wide operating ranges, often with incomplete information, in real-time. The Koopman operator offers a third path.

The Koopman Insight: Seeing Linear Structure in Nonlinear Worlds

The Koopman operator was born in 1931, proposed by Bernard Osgood Koopman in work that drew from developments in quantum mechanics. The fundamental insight is deceptively simple: instead of studying a dynamical system by tracking its state directly, look at how the state transforms quantities you choose to observe. The shift in perspective is subtle but profound.

Consider a simple nonlinear system: the logistic map, which models population dynamics with the rule that the next population depends quadratically on the current one. This system exhibits period-doubling, chaos, and all the hallmarks of nonlinear complexity. If you try to write down a linear equation that captures this behavior, you will fail. There is no 2×2 matrix whose repeated application produces the logistic map's characteristic bifurcation diagram.

But here's what you can do instead: define an observable. An observable is just a function that maps the system's state to some real or complex number. For the logistic map, you might define observables like the current population value, its square, its cube, and so on. Now here's the magic: for any choice of observable, you can ask how that observable's value changes when you apply one step of the logistic map. This transformation—mapping an observable to its evolved value—defines the Koopman operator.

Formally, if your dynamical system maps state x to state F(x), and your observable g maps states to values, then the Koopman operator K acts by composition: Kg = g ∘ F. The key property, and the source of the Koopman operator's power, is that K is linear, even when the original system F is nonlinear. The operator maps functions to functions, and that mapping satisfies all the familiar properties of linearity: K(αg + βh) = αKg + βKh for any observables g and h and any scalars α and β.

This seems almost too good to be true. You've taken a nonlinear system and somehow extracted a linear operator that describes its dynamics. But the linearity comes at a price: the Koopman operator operates on an infinite-dimensional space of functions. While your original system state might live in ℝ^n (say, three numbers describing position and velocity), the Koopman operator acts on spaces of functions—spaces that are inherently infinite-dimensional. You can't represent this operator as a finite matrix; you'd need infinitely many columns and rows.

Nevertheless, the linear structure is genuine and enormously useful. Just as a linear system on ℝ^n can be analyzed through its eigenvalues and eigenvectors—the rates and directions of natural motion—so too can the Koopman operator. The eigenvalues of the Koopman operator reveal the fundamental modes of the system's dynamics. An eigenvalue on the unit circle in the complex plane corresponds to a periodic oscillation; an eigenvalue inside the unit circle corresponds to decaying motion, a stable equilibrium; an eigenvalue outside corresponds to growth, instability. These relationships between eigenvalues and dynamics are exactly the same as in linear systems theory, but they're now available for nonlinear systems.

The eigenfunctions of the Koopman operator—functions φ such that Kφ = λφ for some eigenvalue λ—play a role analogous to coordinate systems. An eigenfunction is an observable that transforms particularly simply under the dynamics: it merely scales by the eigenvalue at each time step. If you know the eigenfunctions and eigenvalues of the Koopman operator, you can decompose any observable into contributions from different modes, each evolving linearly at its own rate. This decomposition, called Koopman Mode Decomposition, is one of the most powerful applications of the theory. It tells you not just whether a system is stable but why—which modes are decaying, which are oscillating, and how they interact.

Mezić and colleagues emphasize a remarkable property: eigenfunctions can be multiplied together, and their product is still an eigenfunction, with eigenvalue equal to the product of the eigenvalues. This algebraic closure means the eigenfunctions form a structure closed under pointwise multiplication. If φ₁ and φ₂ are eigenfunctions with eigenvalues λ₁ and λ₂, then φ₁φ₂ is an eigenfunction with eigenvalue λ₁λ₂. This property underlies many of the practical approximation methods discussed in the paper.

When Linear Tools Meet Nonlinear Reality: The Approximation Problem

The infinite-dimensional nature of the Koopman operator would seem to make it a curiosity, useful for theoretical insight but impractical for engineering application. This is where the recent explosion of work on approximation techniques comes in. The key question becomes: can we find finite-dimensional approximations of the Koopman operator that capture the behavior we care about while remaining computationally tractable?

The answer is yes, but with important caveats that the paper develops carefully. The fundamental approach is to choose a finite-dimensional space of observables—a dictionary of functions—and represent the Koopman operator's action restricted to that space. If you pick your dictionary well, the resulting finite-dimensional operator may closely approximate the true Koopman operator's key properties: its dominant eigenvalues, the structure of its eigenfunctions, its long-term behavior.

A dictionary might consist of polynomial functions: 1, x, x², x₁x₂, and so on. Or it might include trigonometric functions for systems with oscillatory behavior. It might incorporate neural network features or Gaussian processes. The choice of dictionary determines what aspects of the system's behavior can be represented linearly in the finite-dimensional approximation. If the dictionary doesn't contain functions that interact importantly with the system's dynamics, the approximation may fail.

The concept of Koopman invariance formalizes when a dictionary is well-suited for approximation. A subspace of the function space is invariant under the Koopman operator if applying the operator to any function in the subspace keeps you within that subspace. If your dictionary spans an invariant subspace, then the finite-dimensional approximation perfectly captures the Koopman operator's action on that subspace—no approximation error arises from restricting to a finite dictionary. Realistically, perfect invariance is rarely achievable with finite dictionaries. The paper introduces the concept of