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

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.

Figures (22)

• Figure 1: Dependence of $\chi_1(l,q)$ on $\kappa$ for $l=1$-$7$: (a) $q=1$; (b) $q=2$; (c) $q=3$; (d) $q=4$. It is plotted as the line of which color is black for $l=1$, red for $l=2$, blue for $l=3$, orange for $l=4$, green for $l=5$, purple for $l=6$ and brown for $l=7$.
• Figure 2: Bifurcation diagrams for \ref{['eqn:r']}.
• Figure 3: Dependence of $\bar{\beta}_1$ on $\kappa$. The black, red, blue and orange lines represent the cases of $q=1,2,3$ and $4$, respectively, The dashed lines with the same colors represent $\kappa=\kappa_q$, on which $b_{1q}=0$, for $q\in[4]$.
• Figure 4: Dependence of $\bar{\beta}_{1\sigma}$ on $\sigma$ for $\kappa=0.4,0.5$: The black, red, blue and orange lines represent the cases of $q=1,2,3$ and $4$, respectively, for $\kappa=0.4$, while the green dashed line represents the case of $q\in[4]$ for $\kappa=0.5$. Note that $\bar{\beta}_{1\sigma}$ is independent of $q$ when $\kappa=0.5$.
• Figure 5: Numerical simulation results for the KM \ref{['eqn:dsys']} with $n=1000$, $\kappa=0.4$ and $\sigma=0$: (a) $(q,b_1,b_3)=(1,0.16,1)$; (b) $(1,0.12,1)$; (c) $(2,0.55,0.5)$; (d) $(2,0.51,0.5)$; (e) $(3,0.34,0.5)$; (f) $(3,0.3,0.5)$; (g) $(4,0.49,0.5)$; (h) $(4,0.45,0.5)$. The values of $u_k^n(t)\mod 2\pi$, $k\in[n]$, are plotted as the ordinates. The five pairs of two lines coincide almost completely in Figs. (c), (d), (g) and (h).

Theorems & Definitions (17)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Corollary 2.5
• Remark 2.6
• Theorem 2.7
• Theorem 2.8
• Remark 2.9
• Proposition 3.1