Universality in the one-dimensional chain of phase-coupled oscillators

We apply a recently developed renormalization-group (RG) method to study synchronization in a one-dimensional chain of phase-coupled oscillators in the regime of weak randomness. The RG predicts how oscillators with randomly distributed frequencies and couplings form frequency-synchronized clusters. Although the RG was originally intended for strong randomness, i.e., for distributions with long tails, we find good agreement with numerical simulations even in the regime of weak randomness. We use the RG flow to derive how the correlation length scales with the width of the coupling distribution in the limit of large coupling. This leads to the identification of a universality class of distributions with the same critical exponent $\nu$. We also find universal scaling for small coupling. Finally, we show that the RG flow is characterized by a universal approach to the unsynchronized fixed point, which provides physical insight into low-frequency clusters.

Figure 1

Average frequency along the chain when $\omega$ and $K$ are both drawn from (a) rectangular and (b) triangular distributions. The RG predictions (open circles) are compared with simulation results (dashed lines, solid squares). The coupling width $\lambda$ is (a) 1 and (b) 10.

Figure 2

Cluster mass distribution when $\omega$ and $K$ are both drawn from (a) rectangular and (b) triangular distributions. The RG predictions (open circles) are compared with simulation results (solid squares). The plots for different values of the coupling width $\lambda$ are offset for visibility.

Figure 3

Correlation length $\xi$ vs coupling width $\lambda$ when $\omega$ and $K$ are both drawn from (a) rectangular and (b) triangular distributions. The RG predictions (open circles) are compared with simulation results (solid squares). The lines show the power-law fits to the solid squares with the exponent $\nu =0.67$. Here, $\xi$ is defined as the average cluster mass.

Figure 4

(Color online) Flow diagram of the RG showing the unsynchronized stable fixed point at $\lambda =0$ and synchronized unstable fixed point at $\lambda =\infty$.

Figure 5

An example of when the initial coupling distribution (dashed-dotted line) is wider than the initial frequency distribution (dashed line), i.e., when the coupling width $\lambda >1$. Rectangular distributions are shown here. The RG does mostly strong coupling decimation until some energy $E^{\ast }$, which corresponds to a length scale $\xi$ at which the system looks unsynchronized due to the emergence of fast oscillators.

Figure 6

Correlation length $\xi$ vs coupling width $\lambda$ for small $\lambda$ for Gaussian (circles), triangular (triangles), rectangular (squares), and Lorentzian (plus signs) distributions calculated from simulations. The prediction from analytical RG is plotted for weak randomness (solid line) and Lorentzian (dotted line). Lorentzian is in a different universality class from the others. Here, $\xi$ is defined according to Eq. (30).

Figure 7

As the RG progresses, the chain is punctured by more and more $K=0$ bonds. At the start of the RG (a), all oscillators are coupled to their neighbors. Near the end of the RG (c), it is unlikely to have two nonzero bonds in a row. We study the properties of pairs of oscillators that are still connected with a nonzero bond.

Figure 8

Distribution of oscillator frequency (top) and bond energy (bottom) at different stages of the RG: (a,d) 50%, (b,e) 25%, and (c,f) 3% of original oscillators left in chain. The results for different initial distributions of $\omega$ and $K$ are shown: Lorentzian $\lambda =7.5$ (black, solid), triangular $\lambda =1.25$ (gray, solid), triangular $\lambda =7.5$ (black, dotted). The RG approaches the fixed point regardless of the initial chain: the distribution of $\omega$ becomes rectangular, while the distribution of $\tilde{K}$ approaches a delta function at zero.

Figure 9

Distribution of frequency of unisolated oscillators (top) and nonzero bond energies (bottom) at different stages of the RG: (a,d) 50%, (b,e) 25%, and (c,f) 3% of original oscillators left in chain. The results for different initial distributions of $\omega$ and $K$ are shown: Lorentzian $\lambda =7.5$ (black, solid), triangular $\lambda =1.25$ (gray, solid), triangular $\lambda =7.5$ (black, dotted). These are the perturbations that decay as the RG approaches the fixed point. The $\omega$ perturbation becomes quadratic in the middle and flat otherwise, while the $\tilde{K}$ perturbation becomes triangular. This shows that the system approaches the fixed point in the same way regardless of the initial chain.

Figure 10

Illustration of the flow of tail pairs according to Eq. (42). The distribution is constant on a triangular region in $\left|\Delta \omega \right|,\tilde{K}$ space and zero elsewhere. As energy decreases, the triangular region shrinks but keeps the same shape. Both existing tails and incoming tails at a given energy are uniformly distributed on the same triangular region.