Cluster-Mediated Synchronization Dynamics in Globally Coupled Oscillators with Inertia

TL;DR

The paper investigates multi-cluster synchronization in globally coupled oscillators with inertia by developing a coarse-grained Kuramoto framework that treats each cluster as a macroscopic oscillator. It reveals three main phenomena: (i) the primary cluster (PC) suppresses secondary cluster (SC) formation during growth via inter-cluster interactions and a dynamically evolving energy landscape; (ii) SCs and higher-order clusters form near the PC's aphelion, where slow motion fosters frequency resonances that generate a Devil's Staircase of rational ratios; (iii) sufficiently large SCs can destabilize and collapse the PC, with post-collapse dynamics ranging from reassembly into SCs to coherent oscillations between SCs. The study derives analytical constructs (Melnikov-based boundaries, ad hoc potentials, and inter-cluster coupling terms) and validates them through numerical simulations, highlighting the importance of cluster-level dynamics in inertial systems. The findings extend synchronization theory beyond mean-field averages and have implications for real systems such as power grids and neural networks, where multi-cluster synchronization and resonance phenomena play key roles.

Abstract

Globally coupled oscillator systems with inertia exhibit complex synchronization patterns, among which the emergence of a couple of secondary synchronized clusters (SCs) in addition to the primary cluster (PC) is especially distinctive. Although previous studies have predominantly focused on the collective properties of the PC, the dynamics of individual clusters and their inter-cluster interactions remain largely unexplored. Here, we demonstrate that multiple clusters emerge and coexist, forming a hierarchical pattern known as the Devil's Staircase. We identify three key findings by investigating individual cluster dynamics and inter-cluster interactions. First, the PC persistently suppresses the formation of SCs during its growth and even after it has fully formed, revealing the significant impact of inter-cluster interactions on cluster formation. Second, once established, SCs induce higher-order clusters exhibiting frequency resonance via inter-cluster interactions, resulting in the Devil's Staircase pattern. Third, sufficiently large SCs can destabilize and fragment the PC, highlighting the bidirectional nature of cluster interactions. We develop a coarse-grained Kuramoto model that treats each cluster as a macroscopic oscillator to capture these inter-cluster dynamics and the resulting phenomena. Our work marks a significant step beyond system-wide averages in the study of inertial oscillator systems, offering new insights into the rich dynamics of cluster formation and synchronization in real-world applications such as power grid networks.

Figures (15)

• Figure 1: Cluster Synchronization and ad hoc Potential (a) Steady-state mean angular velocity $\langle \dot \theta_i \rangle$ versus natural frequency $\omega_i$, showing a Devil's Staircase pattern with distinct plateaus: PC (dark blue, $\langle \dot \theta_i \rangle \simeq 0$), SC (bright red), TC (dark green), and QC (deep red). The relative frequencies of SC, TC, and QC to PC are given by the rational ratios 1:1.5:2. (b) Angular velocity evolution $\dot \theta_i(t)$ for oscillators in different clusters, using the same color scheme as in (a). The inset shows how these clusters form sequentially after the onset of the PC cluster. (c) The evolution of order parameter $R$ illustrates the PC cluster's initial development followed by the SC cluster's emergence. (d)--(i) Evolution of order parameters and corresponding ad hoc potentials: (d,f,h) Time series of the order parameters $R_P$, $R_S$, and $R_T$, respectively, plotted with time increasing from top to bottom along the $y$-axis. These data were obtained using the fourth-order Runge–Kutta (RK-4) method. (e,g,i) Shows the corresponding ad hoc potentials as functions of the respective order parameters, calculated with specific boundaries for PC, SC, and TC. The bright red, dark green and deep red curves represent different boundary formulations (see the main text for details). All calculations were performed with $(m,K)=(8,12)$.
• Figure 2: Evolution of average angular velocities and potential energy landscapes. Upper row: Snapshots of mean angular velocity $\langle \dot \theta_i \rangle$ as a function of $\omega_i$ at different time points [(a) $t=0$, (b) $t=960$, and (c) $t=19200$], where the average is taken over a time window (b) $\pm 16$ and (c) $\pm 1024$ centered at each point. Lower row: Potential energy $\mathcal{U}$ as a function of $\omega_i$ at (d) $t=0$, (e) $t=960$, and (f) $t=19200$. The system develops a potential well for the PC at (e) $t=960$, eventually forming global and local wells corresponding to PC (dark blue), SC (bright red), TC (dark green), and QC (deep red) at (f) $t=19200$. All panels adopt the parameter values $(m,K)=(8,6)$.
• Figure 3: Schematic illustration of HOC formation near the SC. (a) The oscillators in SC and HOC exhibit orbital motion around the PC, with increased velocities near perihelion and decreased velocities near aphelion. (b) The reduced velocity near the aphelion enhances the interaction strength between SC and HOC oscillators, leading to cluster formation in this region. The SC establishes resonance with HOC oscillators, resulting in rotational frequencies that are rational multiples of the SC frequency.
• Figure 4: Devil's Staircase patterns and Arnold's tongues (a)--(c) Mean angular velocity $\langle \dot \theta_i \rangle$ versus natural frequency $\omega_i$ showing the Devil's Staircase patterns for parameter sets: (a) $(m,K)=(5,6)$, (b) $(8,6)$, and (c) $(8,12)$. The data, computed from steady-state time series, reveal distinct clusters: PC (dark blue), SC (bright red), TC (dark green), and QC (deep red) with frequency gaps of 1, 3/2, and 2 from PC, respectively. The cluster sizes vary with $(m,K)$ values. (d)--(f) Corresponding Arnold's tongues showing oscillators belonging to different clusters in the $(K,\omega)$ parameter space, colored according to their cluster membership, for different inertia values: (d) $m=5.0$, (e) $m=6.0$, and (f) $m=8.0$. The stability regions of the Devil's Staircase patterns extend over a broader range of coupling strength $K$ with increasing inertia $m$.
• Figure 5: Evolution of Devil's Staircase and total energy profiles. PC collapse scenario (a)--(f): Formation of a PC (dark blue), two SCs (bright red), and HOC during the early stage (a), followed by PC collapse in the intermediate stage (b) and merging of PC fragments (dark blue) into SCs to form two enlarged SCs in the late stage (c). A Devil's Staircase pattern emerges where a reconstructed SC frequency serves as a new reference frequency, determining the rational relationships between plateaus (d). The total energy $E$ evolution shows distinct stages marked by vertical dashed lines (e). Oscillator angular velocities undergo transition phases, with PC collapse occurring at the vertical dashed line (b), where most PC fragments merge into the upper SC (f). PC non-collapse scenario (g)--(i): Temporal dynamics showing PC fragments whose angular velocities oscillate between the values of two SCs (bright red) (g). The total energy decreases to a steady state, where the oscillation frequency of PC fragment oscillations approximates a rational multiple of the SC angular velocity (h). In the Devil's Staircase pattern of average angular velocities versus $\omega_i$, the PC appears as a plateau while its individual oscillator's actual temporal dynamics exhibit pronounced oscillations (i).