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J. Phys. Chem. B 2001, 105, 7366-7373
Periodic Forcing of Arrays of Chaotic Electrochemical Oscillators Wen Wang, B. J. Green, and J. L. Hudson* Department of Chemical Engineering, 102 Engineers’ Way, UniVersity of Virginia, CharlottesVille, Virginia 22904-4741 ReceiVed: March 15, 2001; In Final Form: May 3, 2001
Experiments on chaotically oscillating arrays of nickel electrodes in sulfuric acid were carried out. A global periodic forcing is added to a base state of 64 weakly coupled elements by means of variations in the applied potential. Intermittent, unstable chaotic clusters are observed at low forcing amplitudes. Stable clusters, first chaotic and then periodic, form as the forcing amplitude is increased. Stable cluster states consisting of two, three, and four different clusters are observed at forcing amplitudes below that at which synchronization occurs.
Introduction Populations of interacting chaotic oscillators can exhibit a rich dynamical behavior. The collective dynamics of the population depend not only on the features of the individual oscillators but also on both the type and strength of the coupling among the elements. The influence of external forcing on the dynamics of a single chaotic oscillator has been studied by several investigators. Weak periodic perturbations applied as driving forces to chaotic oscillators have been shown to suppress chaos.1-7 Resonance control using periodic forcing of an individual chaotic oscillator has been carried out in experiments in lasers,8-10 a discharge plasma,11 a periodically driven pendulum,12 a microwave driven spin-wave system,13 a magnetoelastic beam experiment,14 and both a homogeneous chemical reaction15 and an electrochemical reaction.16 Sets of oscillators and distributed systems have, of course, also been studied in several contexts. The suppression of spatiotemporal chaos via external forcing has been studied by several investigators. Synchronization of two identical and nonidentical chemical oscillators by external forcing has been achieved in simulations.17 Phase synchronization of sets of coupled differential equations by periodic forcing has been seen in simulations.18 Organization of patterns19 and suppression of spatiotemporal chaos have also been shown.20 Experiments have also been carried out in which global coupling,21 forcing,22-24 and feedback 25 were applied to (nonchaotic) continuous distributed chemical reaction systems. The synchronization of small (usually two) sets of chaotic oscillators26-30 has been investigated in several theoretical studies. Experiments on the transition to synchronization of coupled chaotic chemical oscillators have also been reported.31-34 The emphasis of the present paper is on larger sets of chaotic oscillators, particularly on the formation of ordered states involving clustering. Several theoretical studies on coupled maps35-38 and coupled differential equations39-42 have shown the rich behavior of sets of chaotic oscillators. Turbulent, ordered, and partially ordered states have been seen. However, many of these effects have not been tested in experiments. In a * To whom correspondence should be addressed. E-mail: hudson@ virginia.edu.
recent experimental paper on arrays of globally coupled chaotic electrochemical oscillators, we have shown the existence of both stable and intermittent clusters.43 This global coupling was added by means of resistors and is identical to the global coupling in the theoretical studies in which each element is equally influenced by all the elements. The specific system being investigated is the electrodissolution of nickel in sulfuric acid.48 Because the current of each of the electrodes can be measured independently, the experimental system can be used to explore the behavior of coupled systems that cannot be seen in experiments in other types of systems. In addition, an array of discrete electrodes has been shown to behave the same as a single, larger continuous electrode (at length scales larger than that of the size of the individual electrodes),46 and thus, the experiments can be used to study spatiotemporal dynamics of electrochemically reacting systems. There is, of course, some inherent coupling in the electrochemical system even in the absence of imposed coupling, largely through the potential field.44-47 This inherent coupling is minimized, i.e., made significantly smaller than the coupling added by the external circuitry. In the present paper, we explore the effect of adding periodic forcing to the array. The periodic forcing is applied to a system that has some degree of local and global coupling but that is (by itself) too weak to produce clustering or synchronization. The application of periodic forcing affects the dynamics of the individual oscillators and thus changes the collective dynamics, making condensation possible. Experimental Section The experiments were carried out in a conventional threecompartment electrochemical cell. The schematic of the apparatus including a 64-electrode array working electrode is shown in Figure 1. (Most experiments reported here were carried out with 64 electrodes; one set of experiments was done with eight electrodes consisting of a single row of the array.) The potentials of all the electrodes in the array are held at the same value relative to a Hg/Hg2SO4/K2SO4 reference electrode with a potentiostat (EG&G Princeton Applied Research, model 273). The counter electrode is Pt mesh. The electrodes are connected to the potentiostat through individual parallel resistors and through one collective series resistor. Zero resistance ammeter
10.1021/jp010968x CCC: $20.00 © 2001 American Chemical Society Published on Web 07/07/2001
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J. Phys. Chem. B, Vol. 105, No. 30, 2001 7367 of collective resistance, Rcoll, to total resistance, Rtot. Further details can be found in previous papers.43,49 In each experiment, we first hold the applied potential at a constant value, V0. After the system reaches its stationary chaotic state, a sinusoidal signal is superimposed onto the applied potential with frequency ω and amplitude A so that the applied potential becomes
Vapp(t) ) V0 + A sin(2πωt)
(1)
The forcing signal is superimposed on the applied potential through the analogue output of the Keithly data acquisition board. Data collection begins before the periodic forcing is started and continues throughout the forcing and for a short time after the periodic forcing is turned off. Figure 1. Schematic of apparatus with external forcing.
Results (ZRA) circuitry was inserted between the individual and collective resistors to measure the current of each electrode. The voltage from the ZRAs was digitized using a transputer (ADWin Pro, Keithley). The transputer has its own CPU and memory for fast data acquisition. The data acquisition frequency is 100 Hz for the 64-electrode array and 200 Hz for the eightelectrode array. Testpoint software was used to visualize and save the experimental data on a PC. Experiments were carried out in 4.5 M H2SO4 solution diluted from 5 M sulfuric acid. The cell was held at 11 °C. We have developed a method of altering the strength of global coupling while holding other parameters constant. It is achieved by holding the total resistance inserted into the circuit constant and changing the global coupling parameter , which is the ratio
A. Single Oscillator and Array without Forcing. A chaotic time series of a single electrode and an attractor constructed from it are shown in parts a and b of Figure 2, respectively. The chaos can be obtained via a period doubling sequence with changes in the parameter (applied potential); it has an information dimension of approximately 2.2. Because we are applying a periodic forcing to this signal, the fundamental frequency of the unforced signal is important; as can be seen in the fast Fourier transform (FFT) shown in Figure 2c, the dominant peak (ω0) occurs at approximately 1.3 Hz. We now turn to the full 64-electrode array without periodic forcing. Currents of all 64 electrodes are shown as a space/ time gray scale plot in Figure 3a. Currents are shown as gray
Figure 2. Behavior of a single electrode without forcing (Vapp ) 1.35 V, Rtot ) 14.2 Ω): (a) time series of current; (b) attractor constructed using time delay ) 0.1 s; (c) FFT of time series.
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Figure 3. Behavior on a 64-electrode array without external forcing ( ) 0.56, Vapp ) 1.35 V, Rtot ) 14.2 Ω): (a) space/time plot of individual currents (current shown as gray scale with dark denoting high current); (b) time series of an electrode; (c) attractor.
scale with dark denoting a high current. The current from a single electrode and an attractor in state space constructed from it are shown in parts b and c of Figure 3, respectively. The behavior of each electrode is still chaotic. (The time series and attractors of the other 63 electrodes are similar.) It can be seen from the space/time plot that the elements are not synchronized under these conditions. Nevertheless, the state is not fully turbulent, and there is some order among the elements. The behavior shown was obtained by adding some global coupling to the array through the resistors. A complete description of the effect of the added global coupling can be found in an earlier publication.43 Of course in any experiment, there is also some intrinsic short-range coupling among the elements. However, with the arrangement used here, this local coupling is small compared to the imposed global coupling.43
The behavior seen in Figure 3 is used as a base state for the forcing experiments. We take it as the base state because forcing over the frequency and amplitude ranges employed here does not produce clustering or synchronization without added global coupling. An objective of the experiments is to explore the formation of clusters and the transition into synchronized states, and so, a state close to a clustered state is chosen as the starting point. In addition, we used this base state for some experiments in which feedback was added to the array, and we use the same starting conditions for these experiments with periodic forcing so that the results can be better compared.50 The dominant frequency (ω0 ) 1.3 Hz) of the chaotic behavior of a single electrode varies by less than 2% as the electrodes are coupled in arrays of sizes up to 64 and as the global coupling is imposed while holding other experimental
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Figure 4. System response to external forcing as a function of forcing frequency and amplitude. Panel a shows data from the eight-electrode array, ) 0.225, V0 ) 1.35 V, Rtot ) 113 Ω, ω0 ) 1.33 Hz. The notation pn denotes a period equal to n times that of the forcing period. Panel b shows data from the 64-electrode array, ) 0.56, V0 ) 1.35 V, Rtot ) 14.2 Ω. ω0 ) 1.3 Hz.
conditions unchanged. We thus use a value of ω0 ) 1.3 Hz in interpreting the results on the arrays with forcing. B. The Effect of Added Periodic Forcing. Dependency on Amplitude and Frequency. A schematic of the dependency on forcing amplitude and frequency is seen in Figure 4a. This diagram was made with results using 8 electrodes rather than the full 64 that were used for all other results in this paper. The resolutions of the step sizes in the forcing frequency and amplitude are not fine enough to capture all the dynamics; nevertheless, the diagram does show the complex nature of the system. A synchronized periodic state is obtained at sufficiently high forcing amplitudes in a region near ω/ω0 ) 1.0. Synchronized chaotic states are also observed but only at values of ω/ω0 * 1.0. Even with this small array of eight elements, cluster states containing two and also three stable (periodic) clusters are observed. The behavior of the 64-electrode array is seen in Figure 4b. We again vary the amplitude of the forcing over a range but consider only three forcing frequencies, one equal to, and two others close to, the fundamental unforced frequency, ω0. One sees several general trends as the forcing amplitude is increased. At low amplitudes, the behavior is still unsynchronized chaos. With further increase in amplitude, the dynamics are altered and several transitions occur. The global periodic forcing produces synchronization and both chaotic and periodic clustered states. The sequence observed at the fundamental forcing frequency ω/ω0 ) 1.0 (ω ) 1.3 Hz) is unsynchronized chaos, chaotic clusters, periodic clusters, and periodic synchronization; the latter periodic behavior is p1; that is, the period is equal to that of the forcing. The number of clusters in stable configurations varied from two to four; these clusters are described in more detail below. The experiments were done only in steps of 5 mV in the amplitude, and so, some intermediate states were certainly missed. Although only the results obtained at ω/ω0 ) 1.0 will be reported in detail, some comments can be made about the behavior at the other two frequencies investigated. The transition to the synchronized state occurs at a higher forcing amplitude at ω/ω0 * 1.0. Synchronized period-one states were not seen up to the maximum applied forcing amplitude; the synchronized periodic state seen at ω/ω0 ) 1.15 has a period equal to twice that of the forcing signal. Clustering and Synchronization. We now show in more detail the observed behavior as a function of amplitude at the forcing
Figure 5. Space/time plots of individual currents (64-electrode array, ) 0.56, V0 ) 1.35 V, Rtot ) 14.2 Ω, ω ) 1.3 Hz): (a) unsynchronized chaos, A ) 5 mV; (b) intermittent clusters, A ) 25 mV; (c) stable chaotic clusters, A ) 30 mV; (d) periodic synchronization, A ) 50 mV.
frequency ω/ω0 ) 1.0. All results presented were obtained with the 64-element array. Results obtained at a series of forcing amplitudes at ω/ω0 ) 1.0 are shown in space/time plots in Figure 5. The individual currents again are denoted in gray scale with high current being dark. The differences between currents on a pair of the 64 electrodes are shown in Figure 6 for the same experiments. At 5 mV forcing amplitude (Figures 5a and 6a), there is already a tendency for clustering and synchronization. The clustered states, however, are not stable. Note that in Figure 6a, the two electrodes shown are almost synchronized for several seconds at around t ) 80 s. As the forcing amplitude is increased to 25 mV (Figures 5b and 6b), intermittent clustering can be
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Wang et al. position of the pairs on the attractor such as in Figure 7) between each electrode pair of the 64 elements as a function of time. The pair distance is defined as di,j(t) )
x(I(i,t) - I(j,t))2 + (I(i,t-∆t) - I(j,t-∆t))2 + (I(i,t-2∆t) - I(j,t-2∆t))2
Figure 6. Differences between currents on pairs of electrodes (64electrode array, ) 0.56, V0 ) 1.35 V, Rtot ) 14.2 Ω, ω ) 1.3 Hz): (a) unsynchronized chaos, A ) 5 mV; (b) intermittent clusters, A ) 25 mV; (c) stable chaotic clusters, A ) 30 mV.
clearly seen; clusters form more often and remain intact for longer times before breaking up. The clusters are still unstable but can hold together for times up to approximately 15 oscillation cycles. Stable chaotic clusters formed as the forcing amplitude was increased to 30 mV (Figures 5c and 6c). The difference in the currents of two elements in the same cluster, electrodes 31 and 32, is approximately zero at all times as shown in Figure 6c; the difference of the currents of two elements not in the same cluster (not shown) is the same magnitude as the variations in a single element, that is, each cluster moves over the attractor and the two are sometimes close, sometimes at a distance equal to the size of the attractor. In the stable chaotic cluster region, two clusters are always observed. The number of elements in each cluster depends, of course, on the initial conditions. The cluster arrangement from which the data of Figures 5c and 6c are obtained is (26,38); that is, there are 26 elements in one cluster and 38 in the other. As the amplitude is increased further from the stable chaotic cluster region, stable periodic clusters form. Two to four stable clusters are seen, depending on conditions. These will be discussed in more detail below. Additional increase to a forcing amplitude of 50 mV results in a period-one synchronized state as seen in Figure 5d. Attractors constructed from the data of Figures 5 and 6 are shown in Figure 7. Note that at forcing amplitudes below that at which clustering occurs, Figure 7a,b, the attractors are somewhat more complicated than that obtained for a single electrode, Figure 2b. In the stable cluster region, Figure 7c, the two-band nature of the attractor is seen. In Figure 7d, the periodone limit cycle corresponding to the synchronized region is shown. Order Parameter. We have calculated the average pair distances in three-dimensional state space (the difference in
(There are (64)(63)/2 ) 2016 such distances.) An order parameter can be defined as a fraction of the number of pairs whose distance in the three-dimensional state space is less than some value,39 here taken to be 0.06 mÅ. A temporal mean order parameter is calculated by using mean pair distances; this mean parameter has a value of approximately zero for uncoupled chaotic oscillators and of one in the synchronized state. We show this mean order parameter as a function of forcing amplitude in Figure 8. The order parameter is somewhat above 0.5 in the stable cluster region where cluster sizes of approximately equal size are attained. As the forcing amplitude is increased further, the chaotic cluster state changes to a region in which periodic clusters are obtained, and as we shall see below, these can be of disparate sizes, so the order parameter decreases as the forcing amplitude is increased. Finally, however, the forcing amplitude is strong enough to synchronize the entire array, and the order parameter goes to a value of 1.0. Cluster Configurations. We now turn to a more detailed description of the transitions in the region between the stable chaotic clusters and the synchronized state, i.e., between a forcing amplitude of A ) 30 mV and A ) 50 mV. Representative cluster arrangements at 5 mV increments of forcing amplitude and time series for three intermediate values are shown in Figures 9 and 10, respectively. The stable chaotic cluster region always exhibits two clusters. A representative arrangement in which the two clusters contain 26 and 38 elements is shown in Figure 9a. As the forcing amplitude is increased to 35 mV, four periodic clusters form. The four-cluster configuration shown in Figure 9b is (23,25,11,5). The time series of all four clusters are shown in Figure 10a. This arrangement can be divided into two groups. The two large clusters contain 23 and 25 elements, respectively, their time series are shown in the top two panels of Figure 10a. Their amplitudes of oscillation are somewhat greater than those of the two small clusters of size 11 and 5 that are shown in the bottom two panels. The two larger clusters are periodic with a period 4 times that of the forcing; we denote these as p4 oscillations. These two clusters are approximately the same size and have similar time series offset by a phase difference corresponding to one forcing cycle. The other two clusters of size 11 and 5 have less well-defined dynamics. It is clear, however, that the amplitudes are smaller than those of the two larger clusters. In addition, the two time series do not appear to be similar series offset by some phase lag. This may be due to the imbalanced configuration. The behavior at a somewhat higher forcing amplitude, A ) 40 mV, is shown in Figures 9c and 10b,c. After imposition of forcing, the system attains a three-cluster state shown in the left portion of Figure 9c; the cluster configuration is (16,17,31). Time series for this three-cluster state are shown in Figure 10b. (Because electrodes 50 and 54 are in the same cluster containing 31 elements, their time series are approximately the same for time t < 55 s. The currents of four electrodes are shown because the cluster containing electrodes 50 and 54 shall break up.) The three clusters consist of two types of oscillations: two clusters exhibit large amplitude periodic (p4) oscillations and one cluster (the 16 -element group) exhibits periodic oscillations with smaller amplitude.
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Figure 7. Attractors of individual currents (64-electrode array, ) 0.56, V0 ) 1.35 V, Reqtot ) 14.2 Ω, ω ) 1.3 Hz): (a) unsynchronized chaos, A ) 5 mV; (b) intermittent clusters, A ) 25 mV; (c) stable chaotic clusters, A ) 30 mV; (d) periodic synchronization, A ) 50 mV.
Figure 8. Mean order parameter based on mean pair distances as a function of forcing amplitude (64-electrode array, ) 0.56, V0 ) 1.35 V, Req ) 906 Ω, ω ) 1.3 Hz).
The three-cluster configuration undergoes a slow transition to a four-cluster configuration. The cluster containing 31 elements breaks up into two clusters of 27 and 4 elements. The resulting four-cluster configuration is shown in the right of Figure 9c. To see this transition in more detail, consider the two electrodes #50 and #54 that were originally in the same cluster of 31 elements but that end up in the clusters of four elements and 27 elements, respectively. The time series of these two electrodes differ after the transition as can be seen in Figure 10b for the time 85 < t < 100 s. Electrode #50 exhibits small amplitude oscillations for t > 85 s. Thus, part of the original cluster of 31 now seems to be more closely aligned with the original cluster of 16 (as represented by electrode #56), which is more or less unchanged. Thus we see a breakup of one cluster of a three-cluster state so that a portion of it becomes associated with another preexisting cluster. The difference between the currents of the two electrodes #50 and #54 that went from a single cluster to two different clusters is shown in Figure 10c. It can be seen that the amplitude of the difference grows. However, the two electrodes remain in phase. The changes and particularly the
Figure 9. Representative stable cluster arrangements (64-electrode array, ) 0.56, V0 ) 1.35 V, Rtot ) 14.2 Ω, ω ) 1.3 Hz): (a) (26,38) chaotic clusters, A ) 30 mV, O 26, b 38; (b) (23,25,11,5) periodic clusters, A ) 35mV, O 23, b 25, horizontally hatched circle 11, vertically hatched circle 5; (c) transient from (31,17,16) to (27,17,16,4) periodic clusters (four elements from (31) cluster form a new cluster after approximate 60 oscillations), A ) 40 mV, O 31, b 17, horizontally hatched circle 16 f O 27, b 17, horizontally hatched circle 16, vertically hatched circle 4; (d) (26,38) periodic clusters, A ) 45 mV, O 26, b 38; (e) periodic synchronization, A ) 50 mV.
differences between the time series of the two electrodes are then not discernible on a space/time plot during the transition. Finally, however, for t > 85 s, the differences can be seen both in the time series and on a space/time plot; the elements are
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Figure 10. Time series of individual currents (64-electrode array, ) 0.56, V0 ) 1.35 V, Rtot ) 14.2 Ω, ω ) 1.3 Hz). The symbols correspond to those in Figure 9. Panel a represents the four period clusters (23,25,11,5) and corresponds to Figure 9b, A ) 35 mV. Panel b represents the period clusters that are transient from a three-cluster to a four-cluster state. Electrodes 50 and 54 which were originally in the same cluster (31) go to (4) and (27) clusters, respectively. Panel b corresponds to Figure 9c, A ) 40 mV. Panel c shows the difference between currents on two electrodes showing the transition of the clusters in Figure 9c, A ) 40 mV. Panel d represents the two periodic clusters and corresponds to Figure 9d, A ) 45 mV.
clearly now in two different clusters. The cause of this transition in which the cluster breaks up into two smaller clusters is not completely known. A likely possibility is that this is a very slow transient. This is not the type of transient (normally much faster) that occurs at the beginning of an experiment in which all elements go from some arbitrary initial condition to their final, clustered configuration. Rather, four of the elements form a new cluster while all other 60 elements remain approximately unchanged. The three-cluster state, from which the transition originates, is unbalanced in the following sense: As can be seen in Figure 10b, the two clusters in which electrodes #52 and #54 are located are balanced; they undergo out of phase periodfour oscillations. The remaining third cluster, in which electrode #56 is located, has a different dynamics. After the transition, four elements constitute a new cluster that balances the dynamics of the previously unbalanced cluster. It is possible, of course, that a slow drift in the experimental conditions produces the transition. All experiments, even the best-controlled, exhibit
some drift. However, this seems less likely to be the cause in this experiment. We have carried out such experiments under several conditions and with several types of coupling and observed no effect of drift on cluster configurations in any other case. As the forcing amplitude was further increased to 45 mV, the three- and four-cluster region changed to a two-cluster region. Two periodic clusters were observed as seen in Figure 9d. The time series of the two clusters are shown in Figure 10d. The (26,38) state consists of two groups having different amplitudes and shapes. For higher forcing amplitudes, 50 mV, a synchronized period-one oscillation was observed. Concluding Remarks A global periodic forcing was imposed on a set of electrochemical chaotic oscillators. Without forcing, the individual chaotic oscillators are weakly coupled. The frequency of forcing
Chaotic Electrochemical Oscillators (ω) was chosen to be at and near the fundamental frequency (ω0) of the unforced chaotic oscillators. At ω/ω0 ) 1.0, an increase in the forcing amplitude converts a system in which the oscillators had almost independent dynamics to one in which the oscillators were synchronized and thus more ordered. However, the order in the system does not increase monotonically with increasing forcing amplitude. Intermediate states consisting of intermittent, unstable clusters and stable chaotic clusters occur. We have observed only conditions with two stable chaotic clusters although, of course, many such configurations exist at the same parameter value. In these stable chaotic cluster states, the elements all fall into one of two groups and the dynamics of the elements in a given group are the same. Periodic clusters occur at somewhat higher forcing amplitudes. Two, three, and four cluster states are observed. The sizes of the groups differ, but in each case, the elements within a given cluster undergo the same dynamics. When four clusters are formed, there is a tendency for the clusters to group; that is, two of the clusters have similar time series but with a phase lag and the other two clusters then have a different time series. Thus, two of the clusters are on the same (periodic) attractor or limit cycle, and the other two have a different cycle. When only two clusters exist, the limit cycles of the two clusters differ and thus the two do not just differ by a phase lag. The fourcluster state gives way to a two-cluster state as the forcing amplitude is increased. Although we do not have a sufficiently high resolution in the parameter (amplitude of forcing), it appears that the transition from a state with four clusters to one with two clusters may occur by a mechanism in which the sets of clusters with the same time series (but different phases) merge through a loss of the phase difference. There is some very weak inherent local and long-range coupling (through the electrolyte) and also moderate global coupling (controlled by system electronics) in the system to which the forcing is applied. This intrinsic coupling was too weak to produce clustering or synchronization, but it did furnish interactions among the elements and contributed to the collective dynamics of the system. For a related discussion, see a recent theoretical paper.51 The application of the periodic forcing changes the dynamics of the individual elements, and clustering and synchronization become possible with the same strength of intrinsic coupling. There are some differences between the collective dynamics obtained with periodic forcing and that obtained in previous experiments in which global coupling43 and global feedback50 were applied to the array of chaotic oscillators. Two of these differences deserve particular mention: first, states with up to four clusters were observed here with forcing, and second, increasing the forcing amplitude above a value at which chaotic clusters occur decreases the order in the system because a state with two clusters is changed to one with four clusters. Both local and global interactions can affect the dynamics of sets of nonlinear oscillators. The dynamics of a set of oscillators can be altered through direct global coupling, through global feedback, and, as in this study, through external forcing. Although many theoretical studies of collective dynamics have been carried out with examples in biology, physics, and chemistry, far fewer experimental studies are known. The electrochemical arrays are ideal for such experiments because the dynamics can be measured at each site and the coupling can be carefully controlled. Acknowledgment. This work was supported by the National Science Foundation. We thank I. Kiss for helpful discussions.
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