Feedback control of the Kuramoto model defined on uniform graphs I: Deterministic natural frequencies

TL;DR

This work advances the control of Kuramoto oscillators on graphs by proving that, for uniformly spaced frequencies, the controlled model on complete graphs admits exactly $2^n$ synchronized equilibria, with a saddle-node bifurcation creating a stable and an unstable branch as the feedback gain varies; among these, only the branch converging to the desired motion as the gain becomes large remains stable. It then shows that the continuum limit, when it exists, has a uniquely asymptotically stable synchronized solution, and that this CL solution captures the CKM behavior for large $n$ even on random dense or sparse graphs. The analysis links finite homogeneous networks and graphon-based continuum models through precise convergence and stability-transfer results, and is complemented by numerical simulations on three graph classes that confirm the theoretical predictions. Overall, the paper provides a rigorous bridge between finite CKMs and their CLs under uniform graph topologies and demonstrates that large networks on various graph types exhibit CL-predicted synchronization patterns under feedback control.

Abstract

We study feedback control of the Kuramoto model with uniformly spaced natural frequencies defined on uniform graphs which may be complete, random dense or random sparse. The control objective is to drive all nodes to the same constant rotational motion. For the case of node number $n\ge 3$, we establish the existence of exactly $2^n$ synchronized solutions in the controlled Kuramoto model (CKM) and their saddle-node and pitchfork bifurcations, and determine their stability. In particular, we show that only a solution converging to the desired motion in the limit of infinite feedback gain is stable and the others are unstable. Based on the previous results, it is shown that (i) the solution to which the stable synchronized solution in the CKM converge as $n\to\infty$ is always asymptotically stable in the continuous limit (CL) if it exists, and (ii) the asymptotically stable solution of the CL captures the asymptotic behavior of the CKM when the node number is sufficiently large, even if the graphs are random dense or sparse. We demonstrate the theoretical results by numerical simulations for the CKM on complete simple, and uniform random dense and sparse graphs.

Figures (6)

• Figure 1: Function $\bar{\chi}^\sigma(\xi)$ for $n=4$: (a) and (b) $\beta=1$; (c) and (d) $0.1$. In plates (a) and (c) (resp. plates (b) and (d)) the sign '$+$' (resp. '$-$') is taken in \ref{['eqn:con']}. See the text for more details.
• Figure 2: Bifurcation diagrams of equilibria in \ref{['eqn:lem3b']}: (a) $c_1/c_2<0$; (b) $c_1/c_2>0$.
• Figure 3: Pixel pictures of sampled weighted matrices for the random undirected graphs given by $w_{ij}^n=1$, $i,j\in[n]$ with probability \ref{['eqn:Pdg']} and \ref{['eqn:Psg']} for $n=1000$: (b) Dense graph with $p=0.5$; (c) Sparse graph with $p=0.5$ and $\gamma=0.3$. The color of the corresponding pixel is blue if $w_{ij}=1$ and it is light blue otherwise.
• Figure 4: Numerical simulation results of the CKM \ref{['eqn:dsys']} with $n=1000$, $K=0.5$, $V_1,b_0=1$, $V_0=1$ and $b_1=0.2$: (a) $(a,p)=(1,1)$ in case (i); (b) $(0.5,0.5)$ in case (ii); (c) $(a,p,\gamma)=(0.5,0.5,0.3)$ in case (iii). The time-history of every 100th node (from 50th to 950th) is plotted with different colors.
• Figure 5: Deviations of steady-state responses from the desired motion in the CKM \ref{['eqn:dsys']} with $n=1000$, $K=0.5$, $V_1,b_0=1$, $V_0=1$ and $b_1=0.2$: (a) $(a,p)=(1,1)$ in case (i); (b) $(0.5,0.5)$ in case (ii); (c) $(a,p,\gamma)=(0.5,0.5,0.3)$ in case (iii). Here $u_i^n(t)-V(t)$, $i\in[n]$, with $t=100$ are plotted as red dots. The blue line represents the corresponding theoretical predictions computed from the synchronized solution \ref{['eqn:csol1']} in the CL \ref{['eqn:csys']}.

Theorems & Definitions (38)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Corollary 2.5
• Theorem 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 3.1
• proof