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

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.

💡 Research Summary

This paper investigates the feedback control of twisted states in the Kuramoto model (KM) of identical oscillators placed on deterministic nearest‑neighbor graphs and on complete simple graphs, allowing for a phase‑lag parameter. The authors consider a KM with natural frequency ω, phase‑lag σ∈(−½π,½π), and add both linear (b₁) and cubic (b₃) feedback terms that drive each oscillator toward a prescribed target orbit. The underlying graph Gₙ is described by a weight matrix Wₙ that, as the number of nodes n→∞, converges in L² to a graphon W(x,y). For κ∈(0,½] the graphon corresponds to a κ‑nearest‑neighbor coupling; κ=½ yields the complete simple graph.

The target orbit is chosen as a q‑twisted solution
  (\hat u_nk(t)=2πq k/n+Ω_n^D t)
with q∈ℕ and Ω_n^D defined so that (1.3) solves the uncontrolled KM. Introducing the error variable vₙk(t)=uₙk(t)−(\hat u_nk(t)) leads to a closed system (1.4) that, after applying the continuum limit theory of Medvedev (2014) and subsequent extensions, is approximated by the partial differential equation (1.6):

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