Reentrant synchronization and pattern formation in pacemaker-entrained Kuramoto oscillators

We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We analytically derive the entrainment frequency for arbitrary interaction range and the entrainment threshold for all-to-all couplings. For intermediate couplings our numerical results show a reentrance of the synchronization transition as a function of the coupling range. The origin of this reentrance can be traced back to the normalization of the coupling strength. In the second part we consider a system of oscillators with an initial gradient in their natural frequencies, extended over a one-dimensional chain or a two-dimensional lattice. Here it is the oscillator with the highest natural frequency that becomes the pacemaker of the ensemble, sending out circular waves in oscillator-phase space. No asymmetric coupling between the oscillators is needed for this dynamical induction of the pacemaker property nor need it be distinguished by a gap in the natural frequency.

Figure 1

Normalized entrainment window as a function of the interaction range parametrized via $\alpha$, for a one-dimensional lattice with $N+1$ sites, periodic boundary conditions, and for interaction strength between the oscillators $i$ and $j$ suppressed by the power $\alpha$ of their distance $r_{j,i}$. The main plot indicates the reentrance of the synchronization transition. The upper inset shows the size dependence of ${\alpha }_m$, for which the entrainment window becomes minimal, it saturates at $\alpha \sim 3$. The lower inset shows the monotonic dependence on $\alpha$ for constant coupling strength $K∕N$. The figure was obtained for homogeneous initial conditions. The qualitative features remain the same for random initial conditions.

Figure 2

Normalized entrainment window as function of the number of neighbors $k$ (a) and of the probability $p$ for adding shortcuts in a small-world topology (b) for a one-dimensional lattice with periodic boundary conditions and $N+1$ sites. Average values in (b) are taken over 50 realizations, error bars correspond to the standard deviation. Again the nonmonotonic behavior indicates reentrance of the synchronization transition.

Figure 3

Critical threshold for a one-dimensional lattice as function of its linear size $N$, full line corresponds to the analytic prediction, full dots to numerical results for $N⩾2$, the symmetry breaking parameter is chosen as $\gamma =0$.

Figure 4

Natural frequencies ${\omega }_i$ (full lines) and measured, stationary frequencies $f_i$ (full dots) as a function of the position $i$ of the oscillators on a one-dimensional lattice, for $N=100$, periodic boundary conditions, and four values of the maximal natural frequency $\Delta \omega$. The oscillator with maximal natural frequency $\Delta \omega$ is at position $s=0$. Only half of the oscillators up to index 50 are shown because of the symmetric arrangement. The symmetry breaking parameter is chosen as $\gamma =0$.

Figure 5

Bifurcation in cluster-frequency space above a critical slope in the natural frequencies, parametrized by $\Delta \omega$, the maximal natural frequency. Numerically measured frequencies $f_i$ (normalized such that $f_i∕\Delta \omega ∊[0,1]$) as a function of $\Delta \omega$ for a one-dimensional lattice with $N=20$ and periodic boundary conditions. The symmetry breaking parameter is chosen as $\gamma =0$.

Figure 6

Phase differences ${\psi }_i={\phi }_i-{\phi }_0$ (from the oscillator at $i=0$) as a function of the oscillator index $i$ for a one-dimensional lattice, with $N=100$ and periodic boundary conditions, for different values of the maximal natural frequency $\Delta \omega$ and $\gamma =0$. The difference depends nonlinearly on the distance from the oscillator at $i=0$. The “steps” in these plots are due to the projection of the variables ${\psi }_i$ onto the interval $[0,2\pi )$.

Figure 7

Pattern formation for symmetry breaking parameter $\gamma =0$ on a two-dimensional lattice with $N=100$ oscillators per side. The oscillator $s$ with the maximal natural frequency is placed at the center of the lattice. We plot $\sin {\psi }_i$ for every oscillator $i$. The colors vary between white, corresponding to $-1$, and black, corresponding to 1. The plots are realized after $2\times {10}^4$ integration steps and for different values of $\Delta \omega$ [therefore after different time in natural units of $1∕\Delta \omega$]: (A) 0.1 $\left[{10}^2\right]$, (B) 0.5 $[5\times {10}^2]$, (C) 1.0 $\left[{10}^3\right]$, (D) 5.0 $[5\times {10}^3]$, (e) 10 $\left[{10}^4\right]$, and (F) 50 $[5\times {10}^4]$. A cross section through the pattern of (a) would reveal a frequency distribution as in Fig. 4(B).

Figure 8

Time evolution of the “disorder” in Fig. 7(f) for symmetry breaking parameter $\gamma =0$. All plots are realized for $\Delta \omega =50$, after a different number of integration steps [time in natural units of $1∕\Delta \omega$]: (A) 9 [22.5], (B) 41 [102.5], (C) 150 [375], (D) 393 [982.5], (E) 490 [1225], and (F) 1995 [4987.5]. Only the pattern of (F) is stationary on a global scale, i.e., up to local fluctuations in the phases.

Figure 9

Natural and measured frequencies for symmetry breaking parameter $\gamma =2$. The frequency of the synchronized cluster decreases and the cluster size shrinks, until all oscillators keep their natural frequency in a completely desynchronized state.

Figure 10

Bifurcation in cluster-frequency for symmetry breaking parameter $\gamma =2$. Above the synchronization threshold the size of the original synchronized cluster shrinks with $\Delta \omega$. Oscillators outside this cluster stay isolated in contrast to Fig. 5.

Figure 11

Pattern formation for symmetry breaking parameter $\gamma =2$. We use the same parameter choice as in Fig. 7. Now (A), (B), and (C) correspond to full synchronization, while one-dimensional cross sections through (D), (E), and (F) would reveal distributions of $f_i$ as in Figs. 9(C) and 9(D).

Figure 12

Time evolution for symmetry breaking parameter $\gamma =2$, otherwise the same as in Fig. 8. This time the plots are realized after 9 [22.5] (A), 48 [120] (B), 160 [400] (C), 277 [692.5] (D), 386 [965] (E), and 2926 [7315] (F) time steps [time in units of $1∕\Delta \omega$]. Only the disordered pattern of (F) is stationary up to local fluctuations in the phases.

Figure 13

Average frequency and variance as defined in the text (both divided by $\Delta \omega$) as a function of the maximal natural frequency $\Delta \omega$, in the case of a one-dimensional lattice with $N=100$ oscillators and for values of the symmetry breaking parameter $\gamma =0$ and $\gamma =2$. The raise in the variance from zero indicates the transition to the desynchronized phase.