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

TL;DR

This paper analyzes feedback control in the Kuramoto model on various graphs, establishing the existence and stability of synchronized solutions, and connecting finite-node behavior to the continuous limit.

Contribution

It provides a detailed bifurcation and stability analysis of controlled Kuramoto models on different graph structures, including the asymptotic behavior as node number grows.

Findings

Exactly 2^n synchronized solutions exist for n≥3.

Only the solution converging to the desired motion with infinite feedback gain is stable.

Stable solutions in the continuous limit reflect the behavior of large finite networks.

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.