Delay-induced remote synchronization in bipartite networks of phase oscillators

We study a system of mismatched oscillators on a bipartite topology with time-delay coupling, and analyze the synchronized states. For a range of parameters, when all oscillators lock to a common frequency, we find solutions such that systems within a partition are in complete synchrony, while there is lag synchronization between the partitions. Outside this range, such a solution does not exist and instead one observes scenarios of remote synchronization—namely, chimeras and individual synchronization, where either one or both of the partitions are synchronized independently. In the absence of time delay such states are not observed in phase oscillators.

Figure 1

Schematic diagram of the bipartite network studied here where the nodes separate into two groups denoted $A$ and $B$. We consider phase oscillators at each node, with coupling of strength $\varepsilon$ and delay $\tau$ only between oscillators belonging to different partitions or groups. In numerical calculations we have taken 32 oscillators in each partition.

Figure 2

Analytically calculated plots of (a) synchronized frequencies ${\Omega }_{\mp }$, and (b) phase difference $\varphi$, shown with symbols (circles and triangles) as a function of delay $\tau$, for $\Delta \omega =0.2$ and $\varepsilon$ = 0.125. Numerically obtained frequencies from the system Eq. (1) are plotted with lines (dashed blue and solid red) in (a) and the phase difference with red dots in (b).

Figure 3

Behavior of the system in the $(\varepsilon ,\tau )$ plane for a fixed frequency mismatch $\Delta \omega =0.2$. The hatched region represents the parameter values where synchronized solutions Eq. (2) do not exist. In the blue (dark gray) region, only the solutions ${\Omega }_{-}$ are observed, and only ${\Omega }_{+}$ frequencies are observed in the red (medium gray) region while both coexist in the multistable region shown in gray (light gray). The dashed vertical line at $\varepsilon =\Delta \omega /2=0.1$ is the limit below which Eq. (9) is not satisfied for any delay value.

Figure 4

Time averaged frequencies of the oscillators from both partitions ${\overline{\Omega }}_A\in A$ (dashed blue) and ${\overline{\Omega }}_B\in B$ (solid red) are plotted as a function of the time delay $\tau$. For regions I and V, frequencies are equal and independent of time. One partition is synchronized while the other remains incoherent in regions II and IV. In region III, both partitions are separately synchronized.

Figure 5

In the right panel, time variation of frequencies of four oscillators $({\Omega }_A^{ij}\in A,{\Omega }_B^{ij}\in B)$, are plotted with blue (lower) curves and red (upper) curves, respectively. The frequencies of the two chosen oscillators from each partition is shown with solid and dashed lines. The corresponding phase dynamics is plotted in the left panel. The frequencies for the two partitions are equal and constant in time for (a) and (e). (b) and (d) indicate chimeras, as only one of the partitions is locked to a time dependent frequency while the other partition is phase incoherent. In (c), the oscillators in both partitions are separately synchronized.

Figure 6

Time evolution of oscillator phases at four different delay values $(N=64)$. When $\tau =3.0$, both partitions are globally synchronized (GS) with equal frequencies and have a nonzero phase difference. For $\tau =4.0$ and $5.5$, we see chimera states (CS) where only one of the partitions is synchronized. Individual synchronization (IS) is observed for $\tau =4.75$ wherein both partitions are independently synchronized.

Figure 7

Variation of the real parts $r_A$ and $r_B$ of the partition order parameters as a function of the time delay $\tau$.

Figure 8

Level sets of the order parameter $r$ as a function of $\varepsilon$ and $\tau$, demarcating the regions of different scenarios of remote synchronization. Red regions (CS) indicate chimera states $(r\approx 0.5)$, and yellow regions (IS) are where both the partitions are independently synchronized $(r\approx 1)$. In black regions (IR) oscillators in both the partitions move incoherently $(r\approx 0)$. Blank white regions (GS) represent globally synchronized solutions Eq. (2). The vertical dashed line at $\varepsilon =\Delta \omega /2=0.1$ indicates the threshold for the existence of GS solutions.

Figure 9

Similar to Fig. 8 for different $\tau$ values, as a function of $\Delta \omega$ and $\varepsilon$. The blank white regions (GS) show global synchronization. Red regions (CS) indicate chimera state $(r\approx 0.5)$, and yellow regions (IS) represent when both the partitions are individually synchronized $(r\approx 1)$. In the black regions (IR) oscillators in both the partitions move incoherently $(r\approx 0)$. The dashed curves show the boundary below which globally synchronized solutions disappear.