Origin of Frequency Clusters and Robust Triplet Locking in the Kuramoto Model with Inertia

TL;DR

The paper investigates how frequency clusters form in globally coupled identical oscillators with inertia, showing that two clusters originate through homoclinic bifurcations and that three clusters arise in a small seven-oscillator system via both homoclinic and transversal period-doubling bifurcations. It employs cluster-subspace bifurcation analysis in the thermodynamic limit and uses numerical continuation and full-system simulations to characterize longitudinal and transversal stability, including codimension-2 organizing points and triplet locking. The results establish that Hopf bifurcations cannot create frequency clusters and that only global bifurcations can produce such states, linking the three-cluster regime to triplet locking and to phenomena reminiscent of Arnold tongues. The work provides a framework for predicting cluster formation and destabilization in inertial Kuramoto dynamics and suggests directions for extending to larger networks and higher-cluster states.

Abstract

We investigate the origin of frequency clusters - states where multiple groups of oscillators with distinct mean frequencies coexist. We use the Kuramoto model with inertia, where identical oscillators are globally coupled. First, we study the creation of two frequency clusters in the thermodynamic limit. Via numerical bifurcation analysis, we confirm that two frequency clusters are created by homoclinic bifurcations. Both clusters can lose their phase-synchrony in transcritical or period-doubling bifurcations. Furthermore, we investigate the creation of three frequency clusters in a system of seven oscillators. Here, the frequency clusters are destabilized by a longitudinal and a transversal period-doubling bifurcation, and the frequency clusters are also created by homoclinic bifurcations. We find that the emergence of three or more frequency clusters via a homoclinic bifurcation implies the creation of a triplet locked state, where the frequency differences exhibit a rational relation. Besides the creation of frequency clusters via a homoclinic bifurcation, we state that Hopf bifurcations cannot create frequency clusters in phase oscillators, and frequency clusters can only be created by global bifurcations.

Figures (17)

• Figure 1: Two frequency clusters in the 2-cluster subspace described by equations \ref{['eq:KMI_2_cluster_phase_difference_equations']} containing one period. The time series of the phase difference $\Delta \phi$ and the frequency difference $\Delta \dot{\phi}$ are plotted. The parameter values $\rho_1=0.6$ and $\beta=0.4\pi$ were used.
• Figure 2: Phase space of the 2-cluster subspace from equations \ref{['eq:KMI_2_cluster_phase_difference_equations']} for $\rho_1=0.6$ and $\beta=0.4\pi$. The fixed points are represented by the green and purple dots, while the red line represents the limit cycle of two frequency clusters.
• Figure 3: Longitudinal bifurcations of two synchronous frequency clusters in the $\beta-\rho_1$ parameter plane of the 2-cluster subspace of the kmi, described by the equations \ref{['eq:KMI_2_cluster_phase_difference_equations']}. The area with the circular pattern represents the region where 2-cluster states are longitudinally stable. The colored lines highlight the homoclinic bifurcations (hc) involving the corresponding fixed point (FP) that is shown in the legend.
• Figure 4: Transversal bifurcations of the small cluster in the $\beta-\rho_1$ parameter plane of the 2-cluster subspace. The area with the circular pattern indicates the region of synchronous frequency clusters with a stable small cluster. The bifurcation branches are illustrated by the colored lines. The abbreviation hc denotes a homoclinic bifurcation, pd a period-doubling bifurcation, and bp a branching point, respectively, a transcritical bifurcation.
• Figure 5: Transversal bifurcations of the large cluster in the $\beta-\rho_1$ parameter plane of the 2-cluster subspace. The area with the circular pattern indicates the region of synchronous frequency clusters with a stable large cluster. The bifurcation branches are illustrated by the colored lines. The abbreviation hc denotes a homoclinic bifurcation, pd a period-doubling bifurcation, and bp a branching point, respectively, a transcritical bifurcation.