Renormalization group approach to oscillator synchronization

We develop a renormalization group method to investigate synchronization clusters in a one-dimensional chain of nearest-neighbor coupled phase oscillators. The method is best suited for chains with strong disorder in the intrinsic frequencies and coupling strengths. The results are compared with numerical simulations of the chain dynamics and good agreement in several characteristics is found. We apply the renormalization group and simulations to Lorentzian distributions of intrinsic frequencies and couplings and investigate the statistics of the resultant cluster sizes and frequencies, as well as the dependence of the characteristic cluster length upon parameters of these Lorentzian distributions.

Figure 1

A small piece of the chain around the strongest coupling labeled here as $K_n$. All the neighboring $K_i$ and all the local ${\omega }_i$ are assumed to have much smaller magnitude than $K_n$.

Figure 2

A small piece of the chain around the oscillator with the largest frequency labeled here as ${\omega }_n$. All the neighboring $K_i$ and all the neighboring ${\omega }_i$ are assumed to have much smaller magnitude than ${\omega }_n$.

Figure 3

Comparisons between cluster structure of dynamic frequencies found in numerical simulations of a 10 000-oscillator chain with those predicted by RG over a representative number of lattice sites. (a)–(c) The top row plots the dynamic frequencies found for 500 oscillators in the simulations and demonstrate how some oscillators maintain large frequencies. The dashed boxes in the centers of (a)–(c) encircle 50 oscillators magnified in panels (d)–(f), respectively. In (d)–(f), average frequencies from simulations given by Eq. (29) are denoted by solid squares, while RG predictions are shown as open circles. The differences between RG and simulations are shown in panels (g)–(i). Each column shows results for a different coupling width: [(a), (d), and (g)] $\mu =1.25$, [(b), (e), and (h)] $\mu =3.75$, and [(c), (f), and (i)] $\mu =7.5$.

Figure 4

Number of clusters of a particular frequency divided by the total length of the chain $\left(N={10}^6\right)$. The plot compares the RG prediction (solid lines) and the simulation results (open diamonds) for three coupling widths and four cluster sizes. Each panel has a fixed coupling width and cluster size. Frequencies, on the horizontal axes, are grouped into bins of size $\Delta \overline{\omega }=0.1$. The rows are each a different cluster size: (a)–(c) $m=1$, (d)–(f) 2, (g)–(i) 3, and (j)–(l) 4 from top to bottom. Each column is a different coupling width: $\mu =1.25$ left, $\mu =3.75$ center, and $\mu =7.5$ right.

Figure 5

Number of clusters over a range of sizes. The plot compares RG predictions (open circles) with numerical integrations of Eqs. (7) (solid squares). The curves are spread out for clarity by multiplying each successive data set as $\mu$ increases by 0.1 (a linear shift in the semilog plot). The solid lines denote the fits to the RG data over the domain described in the text.

Figure 6

Characteristic cluster length $\xi$ given by the fits to data in Fig. 5. (a) Results of simulation (solid squares), the numerical RG of Sec. 2 (open circles), and an enhanced numerical RG with strain check (diamonds) described in Sec. 4B. The solid line is a simple analytic prediction $\xi \left(\mu \right)=-1/\text{log}\left[1-\wp \left(\mu \right)\right]$, where $\wp \left(\mu \right)$ is defined in Eq. (33). The dashed line is a power-law fit to the solid squares for $7.5>\mu >0.625$ and has an exponent of 0.47; similar fits to the diamonds and circles in the same range of $\mu$ both give exponents of 0.48. (b) Log-log plot of the same data excluding the analytic prediction.

Figure 7

(Color online) Fixed points and flow diagram of the oscillator chain. The short-range oscillator chain has only two fixed points: the unstable fully synchronized fixed point at infinite interactions (or zero disorder) $\mu \to \infty$ and the stable unsynchronized fixed point. The crossover flow between the two points is associated with a correlation length captured by the average cluster size $\xi \left(\mu \right)$.