Synchronization of higher-dimensional Kuramoto oscillators on networks: from scalar to matrix-weighted couplings

TL;DR

A d-dimensional generalization in which oscillators are represented as unit vectors on the (d-1)-sphere and interact through a matrix-weighted network (MWN), a recently introduced framework where links are endowed with a matrix weight instead of a scalar one.

Abstract

The Kuramoto model is the paradigmatic model to study synchronization in coupled oscillator systems. In its classical formulation, the oscillators move on the unit circle, each characterized by a scalar phase and a natural frequency, by interacting through a sinusoidal coupling. In this work, we propose a d-dimensional generalization in which oscillators are represented as unit vectors on the (d-1)-sphere and interact through a matrix-weighted network (MWN), a recently introduced framework where links are endowed with a matrix weight instead of a scalar one. We derive necessary conditions for global synchronization via a Master Stability Function approach: the existence of a synchronous solution requires identical frequency matrices across nodes and, in the MWN case, a coherence condition on the network structure. Through a suitable change of variables, the stability analysis reduces the full Nd-dimensional problem to a family of d-dimensional eigenvalue problems, each one parametrized by the eigenvalue of a suitable scalar weighted Laplacian, showing that the synchronous solution is locally stable for any positive coupling strength K on any connected network. Analytical results are complemented by numerical simulations.

Figures (4)

• Figure 1: Dynamics for the $d$-Kuramoto model in the "rotated" variables, $\vec{y}_i$. We report the initial conditions, $\vec{y}_i(0)$, (black dots in panel (a)) and the configuration at some future time $t_{fin}$, i.e., $\vec{y}_i(t_{fin})$, (blue dot in (panel (b)) of the solution of Eq. \ref{['eq:dKyvariab']} in the case $d=3$, i.e., on the $2$--sphere. The MWN is build by using an Erdős-Rényi network with $N=50$ nodes and probability $p_{ER}=0.5$ to have a link among a couple of nodes; the matrices $\mathbf{R}_{ij}$ are rotations about the vector $\vec{u}=(0,0,1)^\top$ and the matrix $\Omega$ is the generator of a rotation about the same vector. The synchronous solution (red dot in both panels) is given by $\vec{s}(t)=e^{\Omega t}\vec{s}(0)$, with $\vec{s}(0)=(\sqrt{3}/3,\sqrt{3}/3,\sqrt{3}/3)^\top$. Being $K=0.1$ the system synchronizes as we can appreciate by looking at the order parameter, $R_y(t)$ (panel (c)) that reaches quite fast the value $1$.
• Figure 2: Dynamics for the $d$-Kuramoto model in the original variables, $\vec{x}_i$. We report the same simulation shown in Fig. \ref{['fig:sync_y']} but in the original variables, $\vec{x}_i$, instead of the rotated ones. Panel (a) shows the initial conditions, $\vec{x}_i(0)=\mathbf{O}_{1i}^\top \vec{y}_i(0)$, (black dots), while in panel (b) we report $\vec{x}_i(t_{fin})=\mathbf{O}_{1i}^\top \vec{y}_i(t_{fin})$ (blue dots). Even if the system synchronizes because $K=0.1$, this behaviour cannot be appreciated by using those coordinates as it is confirmed by looking at the order parameter, $R_x(t)$, (panel (c)) that does not grow beyond the value $\sim 0.6$.
• Figure 3: Dynamics for the $d$-Kuramoto model with $\Omega_i = \mathbf{O}_{1i}^\top \Omega\mathbf{O}_{1i}$. We report a simulation for the case $K=0.1$, node dependent $\Omega_i$ matrices and matrices $\mathbf{R}_{ij}$ to be rotations about the axis $\vec{u}=(0,0,1)^\top$. Top panels refer to the "rotated variables", $\vec{y}_i$, while bottom panels to the original $\vec{x}_i$ ones. One can observe that starting from initial conditions, $\vec{y}_i(0)$, (black dots in panel (a)), randomly distributed about $\vec{s}(0)$ they eventually evolve synchronously to $\vec{s}(t)$ (see blue and red dots in panel (b)). Panel (c) show the order parameter $R_y(t)$ and it confirms the emergence of synchronization. The same time series, when plotted in the original variables, does not allow one to draw any conclusion about the presence of synchronization. Indeed random initial conditions (black points in panel (d)) will distribute on a sphere parallel (blue points in panel (e)) and the order parameter, $R_x(t)$ oscillates in time indicating the absence of synchronization. The simulation refers to $N=50$, $3$-Kuramoto oscillators coupled with a MWN built by using an Erdős-Rényi network with probability to have a link given by $p_{ER}=0.5$. The matrix $\Omega$ is a random antisymmetric $3\times 3$ matrix. The synchronous solution (red dot in panels (a), (b), (d) and (e)) is given by $\vec{s}(t)=e^{\Omega t}\vec{s}(0)$, with $\vec{s}(0)=(\sqrt{3}/3,\sqrt{3}/3,\sqrt{3}/3)^\top$.
• Figure 4: Dynamics for the $d$-Kuramoto model in the "rotated" variables, $\vec{y}_i$. We report a simulation for the case $K=-0.1$. Initial conditions, $\vec{y}_i(0)$, (black dots in panel (a)) are randomly distributed about $\vec{s}(0)$; because $K<0$, the vectors $\vec{y}_i(t)$ cannot synchronize and are dispersed on the sphere (blue points in panel (b)). The order parameter, $R_y(t)$ converges to zero, thus testifying the absence of synchronization. The simulation refers to $N=50$, $3$-Kuramoto oscillators coupled with a MWN is build by using an Erdős-Rényi network with probability to have a link given by $p_{ER}=0.5$. The matrices $\mathbf{R}_{ij}$ are rotations about the vector $\vec{u}=(0,0,1)^\top$ and the matrix $\Omega$ is the generator of a rotation about the same vector. The synchronous solution (red dot in both panels) is given by $\vec{s}(t)=e^{\Omega t}\vec{s}(0)$, with $\vec{s}(0)=(\sqrt{3}/3,\sqrt{3}/3,\sqrt{3}/3)^\top$.

Theorems & Definitions (3)

• Remark 1: About the coherence condition
• Remark 2: On the assumption $\Omega_i=\Omega$ for all $i=1,\dots,N$
• Remark 3: The case $K<0$