Routes to synchrony between asymmetrically interacting oscillator ensembles

We report that asymmetrically interacting ensembles of oscillators follow novel routes to synchrony. These routes seem to be a characteristic feature of coupling asymmetry. We show that they are unaffected by white noise except that the entrainment frequencies are shifted. The probability of occurrence of the routes is determined by phase asymmetry. The identification of these phenomena offers new insight into synchrony between oscillator ensembles and suggest new ways in which it may be controlled.

Figure 1

$\left(B\text{−}\gamma \right)$ bifurcation diagram for $\alpha =0$, $\Delta \omega =1$. The different synchronization regimes are described in the text. The boundary between regimes NS and S/D represents ${\gamma }_{c+}$. The $*s$ represent the numerical bifurcation boundaries; (left) $A^{\left(1\right)}=A^{\left(2\right)}=1$, (right) $A^{\left(1\right)}=1.2$, $A^{\left(2\right)}=1$. Insets show the frequency distributions (also obtained numerically) for the indicated regions; their ordinate axes represent oscillator counts in thousands. Note that the occurrence of perfect synchronization with 2000 (left) and 1000 (right) oscillator groups will not occur throughout all of each indicated region.

Figure 2

Numerical plot of coherence parameters $r^{\left(1\right)}$ (grey), $r^{\left(2\right)}$ (black) and phase difference $\delta \psi$ as a function of time. Here (a) $B=1$, $\alpha =0.23$ and (b) $B=1$, $\alpha =0.47$. (a) and (b) correspond to regions D1 and D2 of Fig. 4b for the same values of parameters as traveling along the line I-II. Note that the order parameters display no synchronization in the D1 region.

Figure 3

Histograms of the oscillator frequencies in the D1 region. Left: First ensemble, right: Second ensemble, for the corresponding parameter values in Fig. 2a. Note that the first frequency component in both the ensembles have two indistinguishably different subcomponents which may be considered as one. This occurs due to the discrepancy between numerics and analytics.

Figure 4

$\left(B\text{−}\alpha \right)$ bifurcation diagram for (a) $A^{\left(1\right)}=1.2$, $A^{\left(2\right)}=1$, (b) $A^{\left(1\right)}=1.8$, $A^{\left(2\right)}=1.4$, and $\Delta \omega =1$, $\gamma =0.5$. Note that the synchronization regimes S1 and D1 are greatly reduced in the presence of $p$ asymmetry. The $*s$ denote the numerical bifurcation boundary between the synchronized and incoherent states. The discrepancy between numerical and analytical boundaries occurs due the $p$ asymmetry (unlike Fig. 1) that affects region S2 and hence the thresholds for $r^{\left(1\right)}$ and $r^{\left(2\right)}$.

Figure 5

Numerical plot of coherence parameters $r^{\left(1\right)}$ (grey), $r^{\left(2\right)}$ (black), and phase difference $\delta \psi$ as a function of time. (a) $B=0.7$, $\alpha =0.2$, (b) $B=0.7$, $\alpha =\pi ∕4$ (near ${\alpha }_c$), (c) $B=0.7$, $\alpha =1.2$ $(\alpha >{\alpha }_c)$. (a), (b), and (c) correspond to regions D2, near the NS/S2 boundary and NS of Fig. 4a for the same values of parameters.

Figure 6

(Color online) The shift in the entrainment frequencies ${\Omega }_{\pm }$ with $\Delta K$. Here $A^{\left(1\right)}=A^{\left(2\right)}$, $B=0.5$, $\alpha =0$, $\overline{\omega }=0$, and $\Delta \omega =1$. $N$ represents oscillator count in thousands.