🌀 nonlinear sciences

Higher-order interactions for controlling time-delayed Kuramoto model

This paper proposes a delay-free, higher-order approximation framework for the time-delayed Kuramoto model that, through analytical reduction and numerical validation, enables more accurate prediction of collective dynamics and the control of complex states like bistability and intermediate synchronization compared to conventional pairwise approaches.

Original authors: Narumi Fujii, Martin Moriamé, Maxime Lucas, Hiroya Nakao, Timoteo Carletti

Published 2026-07-14

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Narumi Fujii, Martin Moriamé, Maxime Lucas, Hiroya Nakao, Timoteo Carletti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive dance floor filled with thousands of dancers, each trying to find their own rhythm. In the real world, these dancers don't just hear each other instantly; there's a tiny delay, like a lag in a video call, before they can react to a neighbor's move. This delay is the "time delay" in the Kuramoto model, a famous mathematical recipe for how groups of things (like fireflies flashing or neurons firing) sync up.

For a long time, scientists tried to control these dance floors by pretending the delay didn't exist or by only looking at how two dancers interacted at a time. But the authors of this paper, a team of researchers from Japan and Belgium, say: "Hold on, that's not the whole story." They argue that ignoring the delay or only looking at pairs of dancers misses a crucial secret: time delays actually act like secret group conversations.

The Big Discovery: It's Not Just a Pair, It's a Trio

The paper's main finding is that you can control these delayed dance floors without getting bogged down in impossible math. Instead of trying to solve the messy, infinite-dimensional equations caused by the delay, the team proposed a clever trick: approximate the delay as a "higher-order" interaction.

Think of it this way: In a normal dance, you only watch your partner. But with a time delay, your reaction to your partner is actually influenced by a third dancer you saw a moment ago. The authors show that you can model this by pretending the dancers are having three-way conversations (or even more complex group chats) instead of just one-on-one chats. This turns a terrifyingly complex problem into a manageable one-dimensional equation—a single line that tells you exactly how synchronized the whole group is.

What They Ruled Out

The paper explicitly argues against relying on the "conventional pairwise approximation." This is the old-school method where scientists pretend the delay is just a simple phase shift between two people. The authors found that this old method is like trying to predict a traffic jam by only looking at two cars; it fails to predict the "bistability" (where the system can get stuck in two different states) and the "intermediate synchronization" (a half-synced, messy middle ground) that actually happen in the real delayed system. If you use the old pairwise method, you simply won't see these fascinating behaviors.

The Control Knob: Taming the Chaos

The researchers didn't just stop at observation; they built a control framework. They tested two ways to act as the "dance floor manager" to steer the group:

Linear Feedback: Imagine a manager who gently nudges the dancers based on how far off-beat they are. This can stabilize the group into either total chaos (everyone dancing alone) or perfect unison (everyone moving as one).
Nonlinear Feedback: This is the fancy version. The manager uses a more complex rule that allows them to lock the group into any specific level of synchronization they want. They can make the group 50% synced, 70% synced, or anywhere in between.

How Sure Are They?

The authors didn't just guess; they ran numerical simulations to prove their point. They simulated the original, messy time-delayed system with 300 oscillators (dancers) and compared the results to their new, simplified "higher-order" math.

The Results: Their new method predicted the behavior much more accurately than the old pairwise method. For example, with a time delay of τ = 2.9, the error in their new model was about 4.8 × 10⁻², while the old method was off by 4.4 × 10⁻¹. Even with a larger delay of τ = 6.8, their method remained superior, though the error grew slightly to 2.0 × 10⁻¹.
The "Bistability" Proof: In their simulations, they successfully created a "bistable" state where the group could end up either fully synced or fully chaotic, depending on how they started. This is a behavior the old pairwise math completely missed.
Intermediate States: They also showed they could stabilize the group at a specific, intermediate synchronization level (like R = 0.29 or R = 0.71) using their nonlinear control.

The Takeaway

The paper concludes that by viewing time delays as higher-order interactions (like group dynamics rather than just pairs), we can create a "delay-free" control system that is both easier to calculate and much more accurate. While the math gets a bit tricky when the delay gets very large, the simulations show that this approach is a powerful new tool for taming the rhythms of complex networks, from power grids to biological systems. It's a way to turn a laggy, confusing signal into a clear, controllable dance.

Technical Summary: Higher-order interactions for controlling time-delayed Kuramoto model

Problem Statement
The collective synchronization of coupled oscillator systems is fundamental to various engineering and biological applications, including neuronal dynamics, power grids, and swarm robotics. However, real-world networks often involve time delays due to finite signal propagation speeds. These delays render the system infinite-dimensional (described by delay-differential equations), making analysis and control significantly challenging. While time delays can induce complex behaviors such as bistability and hysteresis, standard mathematical modeling often neglects them or relies on first-order phase-lag approximations (e.g., Sakaguchi–Kuramoto models). These conventional pairwise approximations fail to reproduce characteristic dynamical features of delayed systems, specifically the bistability between synchronized and incoherent states.

Methodology
The authors propose a delay-free control framework for the time-delayed Kuramoto model by interpreting time-delayed pairwise interactions as effective higher-order (nonpairwise) interactions. The methodology proceeds through the following steps:

Higher-Order Approximation: Starting with a globally coupled system of $N$ phase oscillators with time-delayed interactions and a control input, the authors approximate the delayed terms for small coupling strength $\epsilon$ . This yields a delay-free Kuramoto model containing both pairwise and effective three-body (higher-order) interaction terms, where the time delay $\tau$ appears as phase lags.
Dimensional Reduction: The authors apply the Ott–Antonsen (OA) ansatz to the probability density function of the oscillator phases. Assuming a Lorentzian distribution of natural frequencies (reducing to identical oscillators in the limit), they derive a two-dimensional dynamical system for the complex order parameter $z(t) = R(t)e^{i\Psi(t)}$ .
Averaging: To isolate the slow dynamics of the synchronization degree $R$ , the authors employ the second-order averaging method. This eliminates the fast phase dynamics, resulting in a one-dimensional reduced ordinary differential equation (ODE) for the order parameter $R$ . This reduced equation explicitly retains terms of order $O(\tau\epsilon^2)$ , which capture the effects of the higher-order interactions.
Control Design: Two mean-field feedback control schemes are designed based on the reduced equation:
- Linear Feedback: $u(t) = C R(t) \cos \Psi(t)$ .
- Nonlinear Feedback: $u(t) = C R(t) (R(t) - E) \cos \Psi(t)$ , introducing an additional parameter $E$ to target specific synchronization levels.
Validation: The stability diagrams derived from the higher-order reduced equation are compared against those from a conventional pairwise approximation (which neglects $O(\tau\epsilon^2)$ terms). These predictions are validated through numerical simulations of the original time-delayed delay-differential equations.

Key Contributions and Results

Superior Predictive Accuracy: Numerical simulations demonstrate that the higher-order approximation predicts the dynamics of the original time-delayed system more accurately than the conventional pairwise approximation. This is quantified using Global Root Mean Squared Error (RMSE), which is consistently lower for the higher-order model across different delay values ( $\tau$ ).
Realization of Bistability: The study successfully reproduces the bistability between the incoherent state ( $R=0$ ) and the fully synchronized state ( $R=1$ ) using the linear feedback control. Crucially, this bistable region is captured only by the higher-order reduced equation; the pairwise approximation fails to predict this phenomenon, showing only monostable regions.
Stabilization of Intermediate States: The nonlinear feedback control scheme enables the stabilization of arbitrary intermediate synchronization levels ( $0 < R < 1$ ). By tuning parameters $C$ and $E$ , the system can be stabilized at a desired reference value $R_{ref}$ .
Complex Bistable Regimes: The framework reveals a bistable regime where the fully synchronized state ( $R=1$ ) coexists with an intermediate synchronized state ( $R=R_{ref}$ ). This complex dynamic structure is accurately predicted by the higher-order model but is absent in the pairwise approximation.

Significance
The paper claims that interpreting time delays as finite-dimensional higher-order interactions provides a powerful methodology for analyzing and controlling delayed coupled oscillator systems. The proposed framework offers a delay-free reduced model that retains the essential predictive accuracy of the original infinite-dimensional system. This approach facilitates faster numerical simulations and enables the analytical design of feedback control strategies that can realize diverse synchronization states—including bistability and intermediate synchronization—which are otherwise difficult to achieve or predict using standard pairwise models. The authors suggest this perspective could be extended to other classes of time-delayed dynamical networks where delay effects complicate analysis and control.