From Delay to Inertia and Triadic Interactions: A Reduction of Coupled Time-Delayed Oscillators

TL;DR

Time-delayed phase-oscillator networks exhibit rich collective behavior that first-order reductions fail to capture. We develop a universal second-order reduction for time-delayed Kuramoto–Daido networks via a multiple-time-scale expansion, yielding a delay-free system with inertia $\tau\frac{d^2\phi_j}{dt^2}$ and delay-generated triadic interactions, as captured by $\tau\frac{d^2\phi_j}{dt^2}+\frac{d\phi_j}{dt}= [\bar{\eta}_j(t)+\frac{\varkappa}{N}\sum_k F_{jk}(\phi_k-\phi_j-\varpi\tau)]\left[1-\frac{\tau\varkappa}{N}\sum_k F'_{jk}(\phi_k-\phi_j-\varpi\tau)\right]+\tau\frac{d\bar{\eta}_j}{dt}$. The reduction reveals that delay acts as an effective inertia and generates triadic interactions, enabling accurate prediction of splay, chimera, and cyclops states across global, nonlocal, and small-world topologies, including biharmonic coupling. This framework unifies delayed-oscillator analysis with higher-order phase reductions and provides a compact, parameter-explicit tool for predicting delay-controlled collective dynamics in diverse networks.

Abstract

Time-delayed phase-oscillator networks model diverse biological and physical systems, yet standard first-order phase reductions cannot adequately capture their high-dimensional collective dynamics. In this Letter, we develop a second-order reduction for a broad class of time-delayed Kuramoto-Daido networks, transforming the original delayed system of one-dimensional phase oscillators into a delay-free network of two-dimensional rotators. The resulting model shows that coupling delay generates inertial terms in the intrinsic dynamics and higher-order (triadic) interactions, and it accurately predicts the emergence of complex collective patterns such as splay, cyclops, and chimera states. The reduction further reveals a qualitative division of roles: time delay acts primarily as effective inertia for higher-dimensional dynamics, including splay states, whereas the induced triadic interactions are decisive for lower-dimensional patterns such as chimeras. The method applies to networks with arbitrary topology, higher-harmonic coupling, and intrinsic-frequency heterogeneity, yielding a compact, parameter-explicit reduced model. This universal reduced description of time-delayed oscillator networks opens the door to systematic prediction and analysis of nontrivial collective dynamics in delay-coupled systems.

Figures (4)

• Figure 1: (a) Probabilities of convergence to full synchrony (white) and generalized splay states (orange) for the time-delayed model with global, single-harmonic coupling and its first-order \ref{['eq:reduced-model-1']} and second-order \ref{['eq:reduced-model-2c']} reductions, estimated from $5\times10^4$ simulations with random initial conditions (uniform phases; for the delayed system, random phase histories on $[-\tau,0]$). (b) Probability density functions of $r_2=|Z_2|$ for realized splay states in the time-delayed system (cyan histogram), compared with the first-order reduction (red curve) and the second-order reduction (green triangles). Parameters in (a,b): $N=11$, $\varkappa=0.1$, $\tau=1$, $\varpi=1$, $\alpha=0.62$ (gray dot in (c)). (c) Stability regions of full synchrony (white) and splay states with $r_2\le0.8$ (orange) for the time-delayed model in the $(\tilde{\varkappa},\tilde{\alpha})$ plane, where $\tilde{\varkappa}=\tau\varkappa$ and $\tilde{\alpha}=\alpha+\varpi\tau$. The hatched region indicates multistability of synchrony and generalized splay states. The solid cyan curves $\Gamma_1$ and $\Gamma_2$ are the stability boundaries for synchrony and splay states in the delayed system (the analytic curve $\Gamma_1$ corresponds to the condition $\tilde{\varkappa}^*=\tilde{\alpha}-\pi/2$, $\Gamma_2$ is obtained numerically). The dashed cyan curve $\Gamma_3$ and dashed green curve $\Gamma_4$ show the numerically computed stability boundaries for generalized splay states with $r_2=0$ in the delayed system and in the second-order reduced model \ref{['eq:reduced-model-2c']}, respectively.
• Figure 2: Dynamics of the order parameter $r_1$ for the time-delayed system \ref{['eq:kd-model']} with Kuramoto–Battogtokh nonlocal coupling (central panel), compared with the first-order reduction \ref{['eq:reduced-model-1']} (top) and the second-order reduction \ref{['eq:reduced-model-2c']} (bottom). Each panel shows results from 250 simulations with random initial phases $\phi_j(0)\in[-\pi,\pi]$ (and zero initial velocities $\dot\phi_j(0)=0$ for the second-order model). The black curve highlights a representative chimera trajectory; the inset shows the corresponding phase distribution $\varphi_j=\phi_j-\phi_{N/2}$ aligned to the central oscillator, with the gray band indicating the synchronized cluster. Right panels display the probabilities $\mathcal{P}$ of convergence to full synchrony and to chimera states. Parameters: $N=1024$, $L=1.0$, $\kappa=5.2$, $\tau=0.08$, $\varpi=5.125$, $\alpha=1.047$.
• Figure 3: (a) Adjacency matrix of the time-delayed small-world network ($N=100$, mean degree $\langle\kappa\rangle=16$, rewiring probability $p=0.08$). (b) Probabilities $\mathcal{P}$ of realizing different stable regimes for random initial conditions in the delayed network and in the corresponding first- and second-order reduced models (2,000 runs; initial phases drawn as random constants). Triangle and diamond markers indicate bins corresponding to the examples shown in (d) and (g). (c,f) Time series of the order parameter $r_1$ for selected regimes, comparing the delayed system (cyan), the second-order reduction (green), and the first-order reduction (red) under identical initial conditions. (d,g) Instantaneous phase snapshots, and (e,h) phase-density profiles (PDFs), illustrating partially synchronous nonuniform twisted states. Parameters: $\varkappa=0.05$, $\varpi=0.94$, $\tau=1$.
• Figure 4: Cyclops, breathing and switching cyclops states in the delayed bi-harmonic KD model (cyan) and its second-order reduction (green). Panels show the time evolution of the first ($r_1$, solid) and second ($r_2$, dashed) Kuramoto order parameters for (a) a stationary cyclops state ($\alpha_1=1.9$, $\alpha_2=-2.4$), (b,d) a breathing cyclops state ($\alpha_1=0.42$, $\alpha_2=-2.3$), and (c,e) a switching cyclops state ($\alpha_1=0.6$, $\alpha_2=-0.92$). Insets in (a,d,e) display instantaneous phase snapshots for the delayed system (colored circles) and the second-order reduction (crosses), illustrating the match of cluster structure and solitary oscillator position. Initial conditions are uniformly random constant phases. Parameters: $N=9$, $\tau=1$, $\varkappa=0.1$, $K_1=10$, $K_2=0.5$, $\varpi=1.2$.