Entrainment of Randomly Coupled Oscillator Networks by a Pacemaker

Entrainment by a pacemaker, representing an element with a higher frequency, is numerically investigated for several classes of random networks which consist of identical phase oscillators. We find that the entrainment frequency window of a network decreases exponentially with its depth, defined as the mean forward distance of the elements from the pacemaker. Effectively, only shallow networks can thus exhibit frequency locking to the pacemaker. The exponential dependence is also derived analytically as an approximation for large random asymmetric networks.

Figure 1

Phases of elements in the entrained state; $N=100$, $p=0.05$, $N_1=3$, $\kappa =100$, and $\mu =1000$.

Figure 2

Dependence of the entrainment threshold on the depth $L$ for an ensemble of random networks with $N=100$ and $p=0.1$. In the inset, respective data for networks with $p=0.06$ and $0.2$ are plotted. Solid lines are the exponential functions $c_p(1+pN{)}^L$ with appropriate fitting parameters $c_p$.

Figure 3

Dependence of the entrainment threshold on the depth $L$ for an ensemble of small-world networks with $N=100$, $k=3$, various $N_1$, $1\le N_1\le 20$, and different randomness $q$. The solid line is the same exponential function as that fitted to the data with $pN=6(=2k)$ in Fig. 2. The dotted line is linear fitting for the regular lattice ($q=0$).