Delayed feedback control of synchronization in weakly coupled oscillator networks

We study control of synchronization in weakly coupled oscillator networks by using a phase-reduction approach. Starting from a general class of limit-cycle oscillators we derive a phase model, which shows that delayed feedback control changes effective coupling strengths and effective frequencies. We derive the analytical condition for critical control gain, where the phase dynamics of the oscillator becomes extremely sensitive to any perturbations. As a result the network can attain phase synchronization even if the natural interoscillatory couplings are small. In addition, we demonstrate that delayed feedback control can disrupt the coherent phase dynamic in synchronized networks. The validity of our results is illustrated on networks of diffusively coupled Stuart–Landau and FitzHugh–Nagumo models.

Figure 1

Control of synchronization in the SL network (18) for the mismatches $\mathrm{\Delta }{T}_i=0$. (a), (b) Local periods; (c), (d) Kuramoto order parameter; (e), (f) DFC force applied to the first unit $F_1\left(t\right)=K[x_1(t-T_1)-x_1\left(t\right)]$. The vertical dashed red line shows the time when the control is turned on. (a), (c), (e) DFC causes synchronization in the network with parameters $\varepsilon =9\times {10}^{-4}$ and $K=-0.3$, since ${\varepsilon }^{\text{eff}}>{\varepsilon }_{\text{th}}$. (b), (d), (f) DFC causes desynchronization in network with parameters $\varepsilon =5\times {10}^{-2}$ and $K=4$, since ${\varepsilon }^{\text{eff}}<{\varepsilon }_{\text{th}}$.

Figure 2

The synchrony-asynchrony transition in the SL network for the control regime $\mathrm{\Delta }{T}_i=0$. A snapshot of the local periods versus the control gain calculated numerically for different coupling strengths: (a) $\varepsilon =9\times {10}^{-4}$ and (b) $\varepsilon =5\times {10}^{-2}$. The vertical red line shows the analytically derived critical value of the control gain.

Figure 3

Kuramoto order parameters of SL network (18) depicted on a semilog plot. The thin blue line reproduces results from Fig. 1, while the thick green line is calculated in the full-phase-synchronization regime. The vertical dashed red line shows the time when the control is turned on.

Figure 4

Topology of the FitzHugh–Nagumo oscillator network (20).

Figure 5

Control of synchronization in the FHN network (20) for the mismatches $\mathrm{\Delta }{T}_i=0$. (a), (b) Local periods; (c), (d) DFC force applied to the first unit $F_1\left(t\right)=K[x_1(t-T_1)-x_1\left(t\right)]$. The vertical dashed red line shows the time when the control is turned on. (a), (c) DFC causes synchronization in network with parameters $\varepsilon =5\times {10}^{-5}$ and $K=-0.09$, since ${\varepsilon }^{\text{eff}}>{\varepsilon }_{\text{th}}$. (b), (d) DFC causes desynchronization in network with parameters $\varepsilon ={10}^{-3}$ and $K=0.5$, since ${\varepsilon }^{\text{eff}}<{\varepsilon }_{\text{th}}$.