Kuramoto Ring Oscillator Networks

• Kuramoto ring is a one-dimensional oscillator network arranged on a cycle with periodic boundary conditions that supports phase locking and quantized winding numbers.
• The model exhibits diverse dynamics including synchronized, traveling-wave, and chimera states influenced by coupling strength, delay, and topology.
• Variations in coupling (directed, nonlocal, delayed) and network heterogeneity lead to rich bifurcation structures, critical thresholds, and stability regimes.

Searching arXiv for recent and foundational papers on Kuramoto rings to ground the article. Using arXiv search tool with keywords: "Kuramoto ring nearest-neighbor winding number chimera delay". A Kuramoto ring is a Kuramoto or Kuramoto-type oscillator network on a one-dimensional periodic lattice, usually a cycle graph with periodic boundary conditions, in which each oscillator interacts with neighbors or with a ring-based local field. In the standard finite nearest-neighbor form,

$$\dot{\theta}_i=\omega_i+\frac{K}{2}\Big[\sin(\theta_{i+1}-\theta_i)+\sin(\theta_{i-1}-\theta_i)\Big],\qquad i=1,\dots,N,$$

with $\theta_{N+1}\equiv\theta_1$ and $\theta_0\equiv\theta_N$, synchronized states are sought as $\theta_i(t)=\omega t+\phi_i$, so that the ring closure constrains the allowed phase differences around the cycle (Roy et al., 2011). Across locally coupled, directed, delayed, and nonlocal variants, the ring topology supports phase-locked rotating waves classified by winding number, multiple coexisting attractors, and, in appropriate parameter regimes, chimera-like coexistence of coherent and incoherent domains (Ochab et al., 2009).

1. Canonical ring formulations

The nearest-neighbor ring is the basic local one-dimensional Kuramoto model. In one common sign convention,

$$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$

with periodic boundary conditions, while for identical oscillators one often writes

$$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$

These formulations differ only by convention and parameterization, and both encode local coupling on a closed ring (Ochab et al., 2009).

The literature also studies directed rings, in which each oscillator interacts only with its immediate neighbor in a directed manner: $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$ This model changes the synchronization condition qualitatively, because the closure constraint enters through a one-sided coupling law rather than a symmetric nearest-neighbor balance (Saha et al., 2014).

Ring geometry is also used with nonlocal kernels. In a discrete-time Möbius-map model of the Kuramoto–Battogtokh chimera, oscillators are placed at

$x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$

and the local field is

$$U_j = R_j e^{i\Theta_j} = \frac1N\sum_{m=1}^N g_{jm} e^{i\varphi_m},$$

with, for the cosine kernel,

$g_{jm}=1+B\cos(x_j-x_m).$

In this setting the ring is encoded by periodic positions and a spatial kernel depending on distance along the circle (Gong et al., 2020).

A further generalization places oscillator populations on a ring with finite-range top-hat coupling,

$\theta_{N+1}\equiv\theta_1$0

in an Ott–Antonsen reduced Kuramoto–Sakaguchi model. Here the ring supports coherent twisted states, traveling waves, partially synchronized states, modulated states, and incoherence, with the phase-lag parameter $\theta_{N+1}\equiv\theta_1$1 as the principal control parameter (Li et al., 2024).

2. Phase-locked states, winding number, and twisted configurations

For finite undirected rings, synchronized states are written as

$\theta_{N+1}\equiv\theta_1$2

where the common frequency is forced to be the mean frequency

$\theta_{N+1}\equiv\theta_1$3

Introducing phase differences such as $\theta_{N+1}\equiv\theta_1$4, the closure condition around the ring is

$\theta_{N+1}\equiv\theta_1$5

where $\theta_{N+1}\equiv\theta_1$6 is the winding number (Roy et al., 2011).

The winding number labels how many times the phase pattern winds around the cycle. In the local one-dimensional model, stable synchronized solutions must satisfy

$\theta_{N+1}\equiv\theta_1$7

with

$\theta_{N+1}\equiv\theta_1$8

The authors note that there can be at most

$\theta_{N+1}\equiv\theta_1$9

synchronized solutions distinguished by different winding numbers (Ochab et al., 2009).

Because each inverse-sine phase difference can lie on different branches, synchronized states on the finite ring are refined further by a second integer $\theta_0\equiv\theta_N$0, the number of phase differences lying in the left half-circle $\theta_0\equiv\theta_N$1. In this notation, the $\theta_0\equiv\theta_N$2 solutions are the generic stable synchronized branches for large $\theta_0\equiv\theta_N$3 (Roy et al., 2011).

For large coupling, these $\theta_0\equiv\theta_N$4 states approach traveling phase waves. In the $\theta_0\equiv\theta_N$5 limit,

$\theta_0\equiv\theta_N$6

with arbitrary constant $\theta_0\equiv\theta_N$7. For identical oscillators, the same structure appears as uniform phase-shift states with

$\theta_0\equiv\theta_N$8

which are the classical twisted or traveling-wave patterns of the Kuramoto ring (Roy et al., 2011).

The continuum analogue is explicit on the circle graph: the classical twisted states are

$\theta_0\equiv\theta_N$9

with discrete versions

$\theta_i(t)=\omega t+\phi_i$0

Later work on graph approximations of the Sierpinski gasket identifies stable equilibria that serve as generalizations of these classical twisted states on ring networks (Medvedev et al., 15 Jun 2025).

3. Existence, critical coupling, and linear stability

For heterogeneous nearest-neighbor rings, synchronization requires a sufficiently large coupling. A central existence condition is

$\theta_i(t)=\omega t+\phi_i$1

for all partial sums. Since the partial sums of random frequency deviations behave like a one-dimensional random walk,

$\theta_i(t)=\omega t+\phi_i$2

the critical coupling diverges as $\theta_i(t)=\omega t+\phi_i$3. Accordingly, the infinite one-dimensional locally coupled Kuramoto system does not synchronize, although finite rings can synchronize (Ochab et al., 2009).

The ring boundary condition affects synchronization thresholds nontrivially. In a comparison between a ring and a matched chain with the same frequencies and initial data, stable phase-locked states exist only below topology-dependent locking thresholds $\theta_i(t)=\omega t+\phi_i$4 and $\theta_i(t)=\omega t+\phi_i$5. The paper shows that the intuitive inequality $\theta_i(t)=\omega t+\phi_i$6 is false in general for finite systems: there are cases with $\theta_i(t)=\omega t+\phi_i$7. For large $\theta_i(t)=\omega t+\phi_i$8, however, the asymptotic implication $\theta_i(t)=\omega t+\phi_i$9 is recovered (Ottino-Loffler et al., 2016).

Linear stability is commonly organized by the location of phase differences on the circle. Using Gershgorin’s theorem, one finds that if all phase differences lie in the right half-circle $$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$0, the nontrivial eigenvalues are nonpositive and the synchronized solution is orbitally stable, whereas if all phase differences lie in the left half-circle the solution is unstable (Roy et al., 2011). In the local one-dimensional formulation the corresponding sufficient criterion is: if $$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$1, the solution is stable; if $$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$2, it is unstable (Ochab et al., 2009).

For identical oscillators, the stability bound becomes especially transparent. The Jacobian eigenvalues are

$$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$3

so stability requires

$$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$4

equivalently

$$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$5

Thus only sufficiently low-winding twisted states are stable (Dénes et al., 2018).

4. Collective frequency, heterogeneity, and order parameters

In the standard undirected nearest-neighbor ring, the common locked frequency is the mean natural frequency. Summing the locked-state equations yields

$$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$6

so in a rotating frame with zero mean frequency, the synchronized state rotates with $$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$7 (Roy et al., 2011).

The directed ring behaves differently. Assuming a synchronized state with common frequency $$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$8,

$$\dot{\theta_i}(t)=\omega_i-\frac{k}{2}\left[\sin(\theta_i(t)-\theta_{i-1}(t))+\sin(\theta_i(t)-\theta_{i+1}(t))\right],\qquad i=1,\ldots,N,$$9

and summing the inverse-sine relations around the ring gives the synchronization condition

$$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$0

For a symmetric natural-frequency distribution $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$1, all odd moments vanish and the synchronized frequency is the mean frequency, $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$2, in the chosen rotating frame. For an asymmetric distribution, odd moments do not vanish, and the collective frequency generally shifts away from the mean (Saha et al., 2014).

For slight asymmetry, the shift is estimated perturbatively as

$$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$3

The paper states that the shift is largest if the asymmetry enters through $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$4, that a sharper distribution gives a larger shift, and that the shift grows as $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$5 decreases (Saha et al., 2014).

A recurrent feature of one-dimensional rings is that frequency locking does not necessarily imply phase coherence. The usual order parameter

$$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$6

measures phase coherence, but in the local one-dimensional model synchronized states may have $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$7 even though they are frequency-locked. In particular, only the zero-winding state can achieve $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$8 in the large-coupling limit; nonzero winding-number solutions can remain phase-incoherent (Ochab et al., 2009).

5. Multistability, bifurcations, and pattern selection

Finite Kuramoto rings are generically multistable. The local one-dimensional model may have several synchronized attractors with different winding numbers, and numerical evidence shows that the basin of attraction depends on both stability and winding number. As $$\dot{\theta_i}=\omega_0+K\big[\sin(\theta_{i-1}-\theta_i)+\sin(\theta_{i+1}-\theta_i)\big].$$9 increases, low-winding solutions tend to dominate, while solutions with larger $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$0 become marginal (Ochab et al., 2009).

Above the synchronization threshold, the ring can support a rich branch structure confined to a solvability region in the $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$1 plane. New phase-locked solutions appear inside this solvability region as $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$2 increases, and the paper distinguishes two main families: type I solutions, which emerge from the lower solvability boundary, and type II solutions, which emerge near the upper boundary. Stable branches are typically tangent to the solvability boundary (Tilles et al., 2011).

Even with positive coupling, increasing coupling does not always monotonically improve stability. On ring networks with positive edge couplings, there exist choices of natural frequencies and couplings for which two branches of phase-locked solutions collide as the overall coupling parameter $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$3 increases. For every $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$4 the paper constructs such a bifurcation, and for every $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$5 it constructs examples where a stable phase-locked branch collides with a branch of $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$6-saddles. The authors conjecture that the bifurcation is generically locally subcritical and globally an $$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$7-curve (Ferguson, 2020).

For identical locally coupled rings, unstable stationary points are not merely auxiliary objects. Besides the stable winding-number states, there are alternating and mixed symmetry-breaking saddles, and these saddles shape trajectories in phase-shift space. The final winding-number distribution obtained from random initial conditions is approximately Gaussian with standard deviation scaling as

$$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$8

and the presence of saddle points limits forecasting of the final stationary state from early-time dynamics (Dénes et al., 2018).

Small rings can also show onset scenarios more intricate than simple fixed-point birth. In the three-oscillator ring studied in detail, the first synchronized state appears through a tangent bifurcation at

$$\dot{\theta_i}=\omega_i + K \sin(\theta_{i+1}-\theta_i), \qquad i=1,2,\dots,N.$$9

and just below $x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$0 the system exhibits intermittent chaos with laminar-phase duration scaling

$x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$1

consistent with standard tangent-bifurcation scaling (Roy et al., 2011).

6. Delay, nonlocality, chimera states, and related extensions

Time delay adds a second organizing mechanism to the ring: the topology still quantizes winding, but the delay modifies both existence and basin structure. For identical oscillators with delayed nearest-neighbor coupling,

$x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$2

symmetric phase-locked states with constant neighbor phase shift satisfy

$x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$3

The resulting stability maps are $x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$4-periodic in $x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$5, and numerical basin studies show that the most probable state is typically the one with the smallest mismatch $x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$6 (Dénes et al., 2020).

A complementary delayed regular-ring study identifies fully synchronized states, helical phase-locked states, random phase-locked or glassy states, incoherent states, and chimera states. For the fully synchronized branch,

$x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$7

The ring supports synchrony-possible regions, a synchrony-forbidden region in which $x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$8 for all tested random initial conditions, and transition windows with moving-turbulent chimera states (Ameli et al., 2 Feb 2025).

Nonlocal ring coupling is a standard route to chimeras. In the Möbius-map formulation of Kuramoto–Battogtokh dynamics on a ring, the local Möbius kick is driven by the ring-based field $x_j=\frac{2\pi j}{N},\qquad j=1,\dots,N,$9, reproducing classic chimera patterns for cosine and square kernels and adding discrete-time effects such as overshoot, period-doubling of chimera amplitude, and synchronization transitions for strong negative coupling (Gong et al., 2020). In a laboratory implementation with 32 Wien-bridge oscillators on a ring, circuit-level nearest-neighbor wiring yields an effective exponentially decaying interaction around the ring, and for sufficiently large Sakaguchi phase lag $$U_j = R_j e^{i\Theta_j} = \frac1N\sum_{m=1}^N g_{jm} e^{i\varphi_m},$$0 the system exhibits chimera-like coexistence of synchronized and drifting regions, both traveling and stationary (English et al., 2017).

Ring organization also underlies metastable dynamics in population models. In the Ott–Antonsen reduced multi-population Kuramoto–Sakaguchi ring, coherent twisted states with winding number $$U_j = R_j e^{i\Theta_j} = \frac1N\sum_{m=1}^N g_{jm} e^{i\varphi_m},$$1, traveling waves, partially synchronized states, modulated states, and incoherent states coexist. Around $$U_j = R_j e^{i\Theta_j} = \frac1N\sum_{m=1}^N g_{jm} e^{i\varphi_m},$$2, the model shows the most frequent metastable transitions between coherent states and partially synchronized states, whereas closer to $$U_j = R_j e^{i\Theta_j} = \frac1N\sum_{m=1}^N g_{jm} e^{i\varphi_m},$$3 the transitions occur between partially synchronized and modulated states (Li et al., 2024).

Several later constructions use the ring as a reference topology rather than as the final model itself. Stable equilibria on graph approximations of the Sierpinski gasket are described as generalizations of the classical twisted states on ring networks (Medvedev et al., 15 Jun 2025). By contrast, a single nonlinear MEMS device can realize an effective fully connected Kuramoto-Sakaguchi-type network rather than a ring adjacency matrix (Houri et al., 2022), and oscillator networks built from knot diagrams are cycle-like generalizations with crossings and region structure rather than simple rings (Sparavigna, 2011). These developments suggest that the Kuramoto ring functions both as a concrete nearest-neighbor model and as a benchmark topological motif for broader synchronization theory.