Feedback control of twisted states in the Kuramoto model on nearest neighbor and complete simple graphs

TL;DR

This paper investigates how feedback control can stabilize twisted states in the Kuramoto model on various graph structures, analyzing bifurcations and stability in the continuum limit with numerical validation.

Contribution

It extends the understanding of feedback control effects on twisted states in the Kuramoto model to both nearest neighbor and complete graphs, including bifurcation analysis.

Findings

Twisted solutions can be stabilized on both graph types.

Bifurcations lead to oscillating twisted solutions depending on phase-lag.

Numerical simulations confirm theoretical predictions.

Abstract

We study feedback control of twisted states in the Kuramoto model (KM) of identical oscillators defined on deterministic nearest neighbor graphs containing complete simple ones when it may have phase-lag. Bifurcations of such twisted solutions in the continuum limit (CL) for the uncontrolled KM defined on nearest neighbor graphs that may be deterministic dense, random dense or random sparse were discussed very recently by using the center manifold reduction, which is a standard technique in dynamical systems theory. In this paper we analyze the stability and bifurcations of twisted solutions in the CL for the KM subjected to feedback control. In particular, it is shown that the twisted solutions exist and can be stabilized not only for nearest neighbor graphs but also for complete simple graphs. Moreover, the CL is shown to suffer bifurcations at which the twisted solution becomes unstable and a stable one-parameter family of modulated or oscillating twisted solutions is born, depending on whether the phase-lag is zero or not. We demonstrate the theoretical results by numerical simulations for the feedback controlled KM on deterministic nearest neighbor and complete simple graphs.