The Kuramoto model emerged from the study of the synchronization among coupled oscillators. It has long been know that when When simple harmonic oscillators are coupled with one another, complicated collective behaviors emerge. The simplest example is the “odd kind of sympathy” between pendulum clocks described by Huygens in 1673. More complicated biological examples range from pacemaker cells in the heart to the formation of circadian rhythm in the brain to synchronized flashing of fireflies. It has since found many other interesting applications that fueled several decades of active research.
A network of oscillators can be modeled as a swarm of points on 2-dimensional plane circling the origin while pulling and pushing on one another. It models a wide range of physical systems in science and engineering, and was first studied by Wiener in the 1950s Two important simplifications enabled several decades of active research that followed.
First, Winfree noted, based on intuition, that for weakly coupled and nearly identical oscillators, there is a a natural separation of timescales: On the short timescale, oscillators are closely approximated by their limit cycles and thus can be represented by their phases. Kuramoto derived the system of differential equations
\[\dot{\theta}_i = \omega_i - \sum_{j=0}^n \Gamma_{ij}(\theta_i - \theta_j) \quad\text{for } i = 0,\ldots,n\]governing the network of $n+1$ oscillators, in which $\theta_i \in [0,2\pi)$ represent the phases of the oscillators, $\omega_i$ are their natural frequencies (i.e., their limit cycle frequencies), and the interaction function $\Gamma_{ij}$ keeps track of the nonlinear interactions among the original limit cycles.
The second simplification came when Kuramoto singled out the simplest case with $\Gamma_{ij} (\theta_i - \theta_j) = \frac{K}{n+1} \sin(\theta_i - \theta_j)$ for a constant $K$ that quantifies the coupling strength, which produces the further simplified model governed by equations
\[\dot{\theta}_i = \omega_i - \frac{K}{n+1} \sum_{j=0}^n \sin(\theta_i - \theta_j) \quad\text{for } i = 0,\ldots,n,\]This is the Kuramoto model.
This model is simple enough to be analyzed in detail yet exhibit interesting emergent behaviors, and thus kick-started an active research field.
The original Kuramoto model only considers uniform and all-to-all coupling, i.e., cases where the underlying graph is the complete graph One generalization is to consider potentially sparse or non-uniform couplings, which can be modeled as a weighted graph $G = (V,E,K)$, where the vertices $V$ and edges $E$ represent the oscillators and their connections, respectively, and symmetric weights $K = {k_{ij}}$ encodes the coupling strengths. With this, we get the generalized model
\[\dot{\theta}_i = \omega_i - \sum_{j \in \mathcal{N}_G(i)} k_{ij} \sin(\theta_i - \theta_j) \quad\text{for } i = 0,\ldots,n,\]where $\mathcal{N}_G(i)$ is the set of vertices adjacent to $i$ in the graph $G$.
Another well-studied generalization is the Kuramoto model with phase delays, given by the differential equations
\[\dot{\theta}_i = \omega_i - \sum_{j \in \mathcal{N}_G(i)} k_{ij} \sin(\theta_i - \theta_j + \delta_{ij}) \quad\text{for } i = 0,\ldots,n.\]In this, each oscillator $i$ is influences by a delayed phase $\theta_j - \delta_{ji}$ for each of its connected oscillator $j$.
One core problem that can be studied algebraically is the analysis of “frequency synchronization” (not to be confused with “phase synchronization” or “phase cohesion”). Frequency synchronization is achieved when the tendency of oscillators relaxing to their limit cycles and the influence of their neighbors reach an equilibrium, and the oscillators are all tuned to their mean frequency. That is, a frequency synchronization configuration is a configuration at which $\dot{\theta}_i = \overline{\omega}$ Such a synchronization configuration correspond to an equivalent class of solutions to the nonlinear system of equations
\[\overline{\omega} = \omega_i - \sum_{j \in \mathcal{N}_G(i)} k_{ij} \sin( \theta_{i} - \theta_{j} + \delta_{ij} ), \quad\text{for } i = 0,\dots,n.\]The problem of finding synchronization configurations is thus reduced to the problem of solving this nonlinear system.
Interestingly, even though this system is derived from a simplification of the oscillators model, its usefulness extends far beyond this narrow setting. In electric engineering, for example, it coincide with a special case of the ``power flow’’ equations, derived from % Ampère’s and Kirchhoff’s laws of AC circuits which governs the interactions of alternating-current circuits
Though the equilibrium equations are not algebraic, it can be made algebraic through a change of variables. For example, through the substitutions $x_i = e^{\mathfrak{i} \theta_i}$ the systems of equations above can be transformed into the system of Laurent polynomials
\[\overline{\omega} = \omega_i - \sum_{j \in \mathcal{N}_G(i)} \frac{k_{ij}}{2 \mathfrak{i}} \left( x_i x_j^{-1} - x_j x_i^{-1} \right) \quad\text{for } i = 0,\ldots,n.\]This, of course, is not the only way to transform the Kuramoto equations into an algebraic system.