## Abstract

Coherence of interacting oscillating entities has importance in biological, chemical, and physical systems. We report experiments on populations of chemical oscillators and verify a 25-year-old theory of Kuramoto that predicts that global coupling in a set of smooth limit-cycle oscillators with different frequencies produces a phase transition in which some of the elements synchronize. Both the critical point and the predicted dependence of order on coupling are seen in the experiments. We extend the studies both to relaxation and to chaotic oscillators and characterize the quantitative similarities and differences among the types of systems.

The collective behavior and synchronization of a population of somewhat dissimilar cyclic processes depend on the dynamics of the individual elements and on the interactions among them. Wiener raised the question of collective synchronization in a discussion of alpha rhythms in the brain (1). Synchronization has been shown to be an important process in the persistence of species (2) and in the functioning of heart pacemaker cells (3,4), yeast cells (5), and neurons in the cat visual cortex (6). Visual and acoustic interactions make fireflies flash (7), crickets chirp (8), and an audience clap in synchrony (9). Applications in engineering may be found in coupled chemical reactions (10, 11), microwave systems (12), lasers (13), and digital-logic circuitry (14). Winfree (4) and Kuramoto (15, 16) made a major advance in the theory of the onset of synchronization in populations with weak global coupling. In the model, each oscillator of an infinite set is described by a single variable, the phase, and is coupled to all other elements with equal strength. The theory predicts a transition with increasing global coupling strength (*K*) at which some of the oscillators with originally different frequencies become coherent, and it predicts the dependence of order on *K* above the critical point. The theory initiated extensive theoretical activity in collective dynamics and extensions to the effects of finite-size populations (17), fluctuations (18), and more complicated waveforms and coupling mechanisms (19,20). For a recent review, see (21). Simulations on arrays of Josephson junctions (14) and of lasers (13) have shown that coherent motion in physical systems can be interpreted using the Kuramoto model.

In this paper, we present results of a laboratory experiment that confirm the phase transition and dependence of order on coupling strength predicted by the theory on smooth limit-cycle oscillators (16). We show a strong enhancement of fluctuations near the critical point that arise in finite-size systems, in accordance with the theory of Daido (17). In addition, we extend the experiments to relaxation and chaotic (22) oscillators, which often occur in physical systems. We investigate the onset of coherence and the dependence of order and finite-size fluctuations on *K*, and we compare the characteristics of the three types of oscillators.

The experimental system is an array of 64 nickel electrodes in sulfuric acid (fig. S1). Current, which is proportional to the rate of metal dissolution, was measured on each electrode at a constant applied potential. Periodic or chaotic oscillations were observed, depending on conditions such as applied potential, acid concentration, and added external resistance (23,24). Inherent heterogeneities on the metal surface produced a distribution of frequencies of the oscillators. *K*was controlled through the use of external series and parallel resistors; the total external resistance was held constant while the fraction dedicated to individual currents, as opposed to the total current, was varied.

Results obtained with periodic oscillators without added global coupling (*K* = 0) are shown in Fig. 1. As seen in two representative individual-current time series (Fig. 1A), the angular velocity is almost constant, although there is a slight slowing down at minimum values of the current. The natural frequency distribution is unimodal with the mean 〈*f*〉 = 0.4526 Hz and the standard deviation σ = 6.54 mHz. We used the Hilbert transform H(*i*(*t*) − 〈*i*〉), where H (*i*(*t*) − 〈*i*〉) =
*i*(*t*) is the current, and 〈*i*〉 is its temporal mean, to obtain the phase (22) of an individual oscillator. The points on a snapshot in H(*i*(*t*) – 〈*i*〉) versus (*i*(*t*) – 〈*i*〉) space for the 64 oscillators are fairly well distributed on the limit cycle (Fig. 1B). The order parameter *r*(*t*), defined as the normalized length of the vector sum of the phase points in Fig. 1B, is a measure of coherence; this parameter is similar to that introduced by Kuramoto in his study of coupled-phase oscillators (18). We confirmed that the specific definition of phase and order (*18*, 22) did not affect the results. The order parameter for the uncoupled set of 64 oscillators is shown in Fig. 1C. In the theory for an infinite set of oscillators, the parameter is constant and has a value of zero for *K* <*K*
_{c}, where *K*
_{c} is the critical point. The nonzero values obtained in the experiment can be attributed to the finite system size. A series of experiments with increasing number of oscillators (*N* = 1, 2, 4, … 64) showed that the mean order parameter is proportional to 1/.

The oscillator frequencies, as functions of their natural frequencies at three values of *K*, are shown in Fig. 2A. For *K* below*K*
_{c}, deviations from the uncoupled case, the 45° line, are small. At *K* just above*K*
_{c} (circles in Fig. 2A), the oscillators with frequencies near the mean become coherent, but those far above and far below the mean remain unsynchronized. This result is consistent with the original predictions of the theory (18). For larger values of *K*, additional oscillators become synchronized; for *K* > 0.1, all of the oscillators have the same frequency and fall on the horizontal line.

The phase transition is quantitatively described with the mean order parameter 〈*r*〉 as a function of *K* (Fig. 2B). For small *K*, 〈*r*〉 ≈ 0.2, 〈*r*〉 was independent of *K* up to a critical value of 0.03. Above 0.03, 〈*r*〉 increased sharply with *K*. Kuramoto has shown analytically, for an infinite set of oscillators with a Lorentzian frequency distribution, that *r* =
*(16*). The dependence of the order parameter (now*r*′) on *K* is shown in Fig. 2C for three different conditions. *r*′ has been rescaled from 0 to 1 to remove the order at zero coupling caused by the finite system size. All of the experiments are in accordance with the theoretical prediction. Below*K*
_{c}, the quantity 1/(1 –*r*′^{2}) is 1, and above*K*
_{c} it increases linearly with *K* with a slope of 1/*K*
_{c}.

The order parameter does vary with time in the experiments. The variance of *r* (Fig. 2D) has a maximum at*K*
_{c} that is consistent with theoretical studies of systems of finite size (17). The order often undergoes large changes with time near *K* =*K*
_{c}; for example, we have seen transitions from an almost unsynchronized behavior at *r* = 0.3 to an almost synchronized state at *r* = 0.8 in some experiments within a time frame of 20 oscillations.

We carried out the same type of experiments at a higher applied potential at which periodic relaxation oscillations are obtained; the frequency distribution (〈*f*〉 = 0.35 Hz, σ = 39 mHz) is flatter and broader than that of the smooth oscillators considered above. There is again a critical coupling point at which the onset of synchronization is observed; *K*
_{c} becomes larger as the circuit potential is increased, and the relaxation nature of the oscillations is more pronounced. The maximum in the variance of*r* (var *r*) at *K*
_{c} is more than twice that seen in Fig. 2D. The transition from unsynchronized to coherent behavior occurs over a small range in coupling strength. Recent theoretical results on infinite-size systems predict a transition (depending on the nature of the relaxation oscillator) that is either less steep or discontinuous (20); the experimental results seem to be in agreement with the latter.

We also did such experiments with (weakly) chaotic oscillators (Fig. 3). The chaotic attractor is low dimensional and phase coherent. There is again a unimodal frequency distribution of the uncoupled 64 oscillators (〈*f*〉 = 1.29 Hz, σ = 18 mHz); time series of two of the elements are shown in Fig. 3A. A snapshot of the uncoupled behavior in the H(*i*(*t*) – 〈*i*〉) versus (*i*(*t*) – 〈i〉) space (Fig. 3B) shows that the points are distributed over the attractor. The effect of coupling is shown in the rest of Fig. 3. A phase transition occurs at *K*
_{c} = 0.04 (Fig. 3C), and the dependence of *r* on *K* follows Kuramoto's theoretical prediction for *K* < 0.15 (Fig. 3D). However, the transition at *K*
_{c} is not as sharp as it is in the periodic case; the somewhat gradual transition may be due to stronger effects of fluctuations (22). In addition, the order goes to 1 only at very large coupling strengths, where*K* > 9. There are deviations from the line in Fig. 3D for larger values of *K*, denoting less order than would be obtained with periodic oscillators. The same sharp maximum is seen for var *r* at *K*
_{c} (Fig. 3E); var*r* scales as |*K* –*K*
_{c}|^{α} with the scaling parameter (α) = –0.4 and –0.3 for *K* <*K*
_{c} and *K* >*K*
_{c}, respectively (Fig. 3F). Daido showed in an analysis of large sets of phase oscillators (17) that α is –1 for *K* < *K*
_{c} and –¼ for *K* > *K*
_{c} but that the fit is very sensitive to finite-size effects, especially for*K* < *K*
_{c}. Our results with 64 chaotic oscillators are consistent with the prediction for*K* > *K*
_{c}. The periodic oscillators also followed a scaling law, although the exponents (α = –0.5 and –1.2 for *K* <*K*
_{c} and *K* >*K*
_{c}, respectively) differ from the theoretical predictions; it may be that finite-size effects are stronger for periodic oscillators.

Thus, we see that results consistent with theory are obtained in laboratory experiments, even with a finite set of oscillators in the presence of unavoidable noise. At a critical coupling threshold, coherence emerges and some of the oscillators (with inherent frequencies near the mean) become synchronized. Above the critical value, the number of synchronized oscillators increases sharply. Finally, we see that the phase transitions and synchronization predicted in the theory are also obtained both for periodic relaxation oscillators and for coupled chaotic elements.

↵* To whom correspondence should be addressed. E-mail: hudson{at}virginia.edu