Experimental investigation of high-quality synchronization of coupled oscillators

CHAOS VOLUME 10, NUMBER 3 SEPTEMBER 2000

Experimental investigation of high-quality synchronizationof coupled oscillators

Jonathan N. Blakelya) and Daniel J. GauthierDepartment of Physics and Center for Nonlinear and Complex Systems, Duke University,Box 90305, Durham, North Carolina 27708

Gregg Johnson, Thomas L. Carroll, and Louis M. PecoraCode 6343, U. S. Naval Research Laboratory, Washington, DC 20375

~Received 30 November 1999; accepted for publication 2 May 2000!

We describe two experiments in which we investigate the synchronization of coupled periodicoscillators. Each experimental system consists of two identical coupled electronic periodicoscillators that display bursts of desynchronization events similar to those observed previously incoupled chaotic systems. We measure the degree of synchronization as a function of couplingstrength. In the first experiment, high-quality synchronization is achieved for all coupling strengthsabove a critical value. In the second experiment, no high-quality synchronization is observed. Wecompare our results to the predictions of the several proposed criteria for synchronization. We findthat none of the criteria accurately predict the range of coupling strengths over which high-qualitysynchronization is observed. ©2000 American Institute of Physics.@S1054-1500~00!01203-9#

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The once surprising fact that the irregular oscillations oftwo chaotic oscillators can be synchronized is now welestablished. However, deterministic chaos is not the onlysource of irregular oscillations. Recent research showsthat stable periodic systems may oscillate irregularlywhen subject to small random noise†F. Ali and M. Men-zinger, Chaos 9, 348„1999…; Trefethen et al., Science 261,578 „1993…‡. Can a high degree of synchronization beachieved between two systems undergoing irregular oscillations due to small noise rather than deterministicchaos? We address this question here through two experiments on coupled periodic electronic circuits. In eachexperiment, we observe the degree of synchronization between a pair of coupled oscillators as the couplingstrength is increased from zero. In the first experiment,we observe a sudden transition to high-quality synchro-nization at a critical coupling strength. In the second ex-periment, no high-quality synchronization is observedover the range of accessible coupling strengths. In addition, we apply to our experimental systems several pro-posed criteria for high-quality synchronization developedin studies of synchronized chaos. We find that none othese criteria accurately predict the behavior observed inthe experiments. These results may provide some guidance in the development of practical applications of syn-chronized chaos such as secure communication schemwhere a criterion for high-quality synchronization isneeded.

I. INTRODUCTION

It is now well established that the dynamics of a nonlear system can become highly irregular when small rand

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noise is injected into the system. For example, Ali aMenzinger1 recently showed that a globally stable limit cycoscillator subject to small amounts of noise can displayexplosive divergence of trajectories away from the limcycle. The origin of this disproportionate response to smperturbations is the fact that the limit cycle is composedsegments of varying local stability, most of which are stabut some of which are highly unstable as shown schemcally in Fig. 1. In regions of pronounced local instability,perturbation may undergo transient growth before decayasymptotically. A similar behavior is displayed by nonormal linear systems as shown schematically in Fig. 2~a!.2

Non-normal systems are characterized by nonorthogoeigenvectors. A small perturbation to such a systempressed as a linear superposition of such vectors may hlarge coefficients but a small norm due to cancellationdepicted in Fig. 2~b!. As shown in Fig. 2~c!, when the systemevolves in time, the coefficients of the superposition mdecay at different exponential rates so the cancellation is lcausing the norm to increase even though the individualgenvector components are decaying asymptotically. Agthe result is a transient amplification of a perturbation bystable dynamical system. These two examples do not exhthe possible scenarios in which highly irregular oscillatioare generated by large noise amplification in a dynamsystem. An interesting question is whether the irregularcillations occurring in two identical noise-amplifying dynamical systems can be synchronized.

The primary objective of this paper is to present tresults of an experimental investigation of synchronizationnoise-amplifying dynamical systems consisting of nonlineelectronic periodic oscillators. In our experiments, the dnamical behavior of the coupled oscillators is described bset of nonlinear differential equations given by

© 2000 American Institute of Physics

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739Chaos, Vol. 10, No. 3, 2000 Synchronization of coupled oscillators

xm5F@xm#, ~1a!

xs5F@xs#1gK ~xm2xs!, ~1b!

where xm(xs) denotes the position in theN dimensionalphase space of the master~slave! oscillator, F is a vectorfield describing the dynamics of an individual oscillator,K isan N3N coupling matrix, andg is the scalar couplingstrength. Note that the coupling is one-way so that the elution of the master oscillator is unaffected by that of tslave. The synchronized statexs(t)5xm(t) defines anN di-mensional invariant manifold~called the synchronizationmanifold! residing in the 2N dimensional phase space of thcoupled system. We introduce new coordinatesxi5(xm

1xs)/2 andx'5(xm2xs)/2 to separate the dynamics alonand transverse to the synchronization manifold. In this codinate system, synchronization may be defined asx'(t)50.In each of two experiments, we couple two periodic osciltors subject to small input noise and observe the degresynchronization as a function of coupling strength. We uvarious norms ofx'(t) to quantify the degree of synchronzation between the oscillators.

In the first experiment, each oscillator is a ondimensional ~1D! periodically driven system. The phasspace trajectory of a single oscillator brings the system vclose to a threshold beyond which the trajectory chanshape considerably. A small perturbation that pushes thecillator over the threshold is amplified as the oscillator flows the new trajectory. When two such oscillators acoupled and the coupling strength is increased from zerofind that high-quality synchronization occurs suddenly ona critical coupling is reached.

In the second experiment, the oscillators evolve onlimit cycle along which the Jacobian is non-normal atpoints. The non-normality gives rise to transient amplifiction of perturbations as described previously. When two soscillators are coupled and the coupling strength is increafrom zero, we do not observe any threshold for synchrontion. The degree of synchronization increases smoothlyslowly. We do not observe high-quality synchronizationany coupling strength in the range of experimentally accsible coupling strengths.

A second objective of this paper is to explore the psible implications of our experiments with noise-amplifyin

FIG. 1. Schematic diagram of a globally stable limit cycle with varyilocal stability. The segment of the limit cycle that is locally unstable~dottedline! allows perturbations to grow transiently before they decay asympcally in the locally stable segment~solid line!.


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systems on recent experimental studies of synchronizedotic oscillators. Several researchers have reported expments where, in a regime where synchronization waspected, intervals of synchronization are interruptirregularly by large, brief desynchronization events~orbursts!.3–6 This bursting is attributed to the interaction overy small noise or parameter mismatch between oscillaand local variations in the stability of the synchronized staSpecifically, the synchronized state may be locally unstanear unstable periodic orbits embedded in the chaotic atttor on which the oscillators evolve. A perturbation to thsynchronized motion in the neighborhood of such an omay be amplified before decaying asymptotically.3 With thisscenario in mind, several researchers have attempted tovelop criteria for ‘‘high-quality,’’ ‘‘robust,’’ or ‘‘burst-free’’synchronization.3–7 Since none of these criteria explicitly require the oscillators to be chaotic for applicability, we appthem to our experimental systems and compare their pretions with our observations. We find that none of these cteria accurately predicts the behavior observed in our expments.

In the next two sections, we describe our experimensystems. Section II describes a pair of coupled driven oslators. We find that these oscillators display high-quality sy

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FIG. 2. ~a! Schematic diagram of a fixed point in a non-normal linesystem. The system has negative eigenvalues and non-orthogonal eigetors. ~b! A perturbation from the fixed point depicted~in bold! as a super-position of the eigenvectors. The perturbation has a small norm ducancellation of the nonorthogonal eigenvectors. The dotted lines showeigendirections of the fixed point.~c! The same perturbation some time lateBoth eigenvectors have contracted but at different rates. Cancellation isand the norm of the perturbation grows.


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740 Chaos, Vol. 10, No. 3, 2000 Blakely et al.

chronization for coupling strengths above a critical valuSection III describes a pair of coupled limit cycle oscillatocharacterized by highly non-normal Jacobians. These olators do not display high-quality synchronization for aobserved coupling strength. In Sec. III, we review seveproposed criteria for synchronization and apply them tosystem. Finally, we discuss the implications of our expements in Sec. IV.

II. EXPERIMENT 1: 1D DRIVEN OSCILLATORS

The choice of oscillators used in our first experimentmotivated by the noise-induced bursting observed in coup‘‘double scroll’’ chaotic oscillators in a previous study.5 The‘‘double scroll’’ chaotic attractor has a saddle point at torigin separating the two ‘‘scrolls.’’ Consider the case whtwo such oscillators are coupled as master and slave. Wsynchronized, the master and slave follow identical phspace trajectories. However, a perturbation occurring wthe oscillators are synchronously evolving in the neighbhood of the saddle may kick the slave over onto the otscroll for a long excursion away from the master. We mimthis behavior with two well-matched, one-way coupled nolinear RC circuits driven by identical sinusoidal signalNote that these oscillators are not chaotic but displamechanism for noise amplification similar to that foundthe double scroll chaotic system. A single oscillator is shoschematically in Fig. 3 and its behavior is described by

dV

dt5@ f ~V!2V#/RC1I ~ t !/C, ~2!

where

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andI (t)5I 01I d cos(vdt) is a sinusoidal driving current. Thparameter values areR52 kV, C50.047mF, G53.1, V0

50.7 V, I d5449mA, andvd/2p5100 Hz.The dynamical behavior of a single oscillator in the a

sence of noise is shown in Fig. 4~a! for I 052186mA ~solidline!, generated by a numerical simulation. A brief, sufciently large perturbation to the system when the trajectorin the vicinity of V52V0 /G ~dashed line! causes it to un-dergo a large excursion away from the orbit before returnias shown by the dotted line in Fig. 4~a!. Once the trajectorycrosses the threshold, the growth rate of the perturbatiovery large:V(t) increases from20.23 V to ;1 V in ;0.1ms while the period of the driving signal is 10 ms. Thusperturbation during the brief interval when the trajectory is

FIG. 3. Schematic diagram of an individual driven nonlinear oscillator. Tdevice labeledf (V) is a high input impedance nonlinear amplifier.


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the neighborhood of the threshold can be amplified signcantly. This behavior resembles the bursting observedcoupled chaotic double scroll oscillators evolving nearsaddle point at the origin of their phase space, as discupreviously.

In the experiment, the slave oscillator is coupled to tmaster by injecting a currentI sync5gC(Vm2Vs) into theslave circuit at the same node as the drive signal. We bboth oscillators very close to the threshold by settingI 0

52186mA. For this value ofI 0 and the inherent level onoise in the system, the master oscillator never crossesthreshold and remains on the trajectory shown as the sline in Fig. 4~a!. A small Gaussian white noise current~band-width from 10 Hz to 1 kHz, rms current;0.5% of I d! isinjected into the slave oscillator. When there is no coupl~g50!, the slave occasionally crosses the thresholdbursts away from the periodic orbit. For the oscillators tosynchronized, the coupling has to be chosen so that the snever undergoes a burst.

For each of several different values of the couplistrength, we record a long time series of the Euclidean noux'u5uVm2Vsu. To quantify the degree of synchronizatiowe determine from these time series the average distafrom the synchronization manifoldux'urms and the maximumobserved value of the distance from the manifoldux'umax

~Ref. 5! for each coupling strength, as shown in Fig. 4~b!.For coupling strengths between 0.6 and 0.83104 s21,ux'umax is on the order of the size of the orbit~;2 V! eventhoughux'urms is very small~;1% of the orbit size!, imply-ing that there exist large, brief, occasional desynchronizaevents even when the oscillators are synchronized on aage. From the figure, it is seen that the large desynchron

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FIG. 4. Synchronization of the driven nonlinear electronic circuit.~a! Nu-merical simulation of the oscillator trajectory forI 052186mA under noisefree conditions~solid line!. A perturbation to the system that pushes it acrothe threshold~dashed line! causes it to undergo a large excursion~dottedline!. ~b! Two measures of the degree of synchronization as a function ofcoupling strength.


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tion events only cease forg*1.33104 s21, as indicated bythe large drop inux'umax.

To interpret these results, we must keep in mind tthere is no precise way to define synchronization in anperiment. Gauthier and Bienfang referred to synchronizaas high quality whenux'umax,e where e is a small lengthscale ~typically within a few percent of the characteristdimensions of the attractor!.5 This condition inherently de-pends on the choice of a metric and can be violated simby making the noise level large enough even whencoupled system does not amplify perturbations. In lightthis trivial case, this condition should not be taken as a fmal definition but rather as an attempt to quantify the idthat small noise or mismatch in the systems should only grise to small deviations between the master and slave. Wout making an explicit choice fore, we see from Fig. 4~b!that the behavior makes a transition from poor synchrontion, where small noise gives rise to large separationstween master and slave, to burst free synchronization arog'1.33104 s21 where small noise causes only small seprations. Note that the coupling strength at which this trantion occurs increases with increasing level of noise injecinto the slave oscillator.

III. EXPERIMENT 2: NON-NORMAL LIMIT CYCLEOSCILLATORS

The choice of oscillators used in our second experimis motivated by the increasing evidence that noise amplifition due to non-normality plays an important role in machaotic physical systems. For example, non-normality mbe responsible for turbulence in some fluid flows8–10 andchaotic behavior in mode-locked lasers.11 The non-normalityassociated with homoclinic tangencies in nonhyperbolic cotic attractors gives rise to noise-induced attracdeformation,12,13 a phenomenon recently observed expementally in an electronic circuit.14 Almost all chaotic sys-tems of physical interest are nonhyperbolic and therefmay have regions of phase space characterizednon-normality.13 The periodic oscillator used in our seconexperiment evolves on a stable limit cycle on which tJacobian is highly non-normal. The experimental apparaconsists of two one-way coupled, well-matched electrooscillators. An individual oscillator, shown schematicallyFig. 5, is described by the set of equations

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and l5210, andRC50.1 ms. The trajectory of a singloscillator is a limit cycle of radius;4 V in the xy planecentered on the origin.

At each point on the limit cycle, the oscillator’s Jacobiahas two eigenvectors in thexy plane and a third inclined aan anglea with respect to the plane. The eigenvalue assoated with this third eigenvector is negative and equal tol.The nonorthogonality of the eigenvectors~whenaÞ90°! in-dicates the Jacobian is non-normal, giving rise to the pobility that perturbations can undergo significant transiegrowth. Therefore, perturbing the oscillator in thez directionresults in a transient excursion from the limit cycle, the sof which depends on the size of the perturbation anddegree of non-normality. Thus, very small noise can be aplified dramatically by the system with strong non-normali

We note, as an aside, that a change of coordinatesremove the non-normality from the system, but it cannremove the noise sensitivity. For example, consider addinnoise term to the evolution equation for thez variable, asdone in our experiment, and make the change of variabz85z cota in order to reduce the non-normality. In this necoordinate system, the noise term is multiplied by a faccota, which is large in the case of strong non-normaliThus, the sensitivity to noise is independent of the chocoordinate system.

In the experiment, the two oscillators are ‘‘xy’’ coupled~K115K2251, Ki j 50 otherwise!. As before, we quantify thedegree of synchronization by measuringux'umax and ux'urms

for several coupling strengths as shown in Fig. 6. To deonstrate the role of non-normality in amplifying noise, thexperiment is performed once with very small~cota51! andonce with rather large~cota5100! non-normality. Gaussianwhite noise~30 mV rms, dc to 15 MHz! is added to thezcomponent of the slave circuit. The amount of noise addethe slave oscillator is the same in both situations.

For small non-normality~cota51!, it is seen in Fig. 6~a!that the observed distance from the synchronization maniux'umax&0.1 V, or 2.5% of the limit cycle radius, forg.0.23103 s21. Apparently, the perturbations are not ampfied significantly when cota51 since the eigenvectors aralmost orthogonal. We find dramatically different resuwhen the non-normality is increased~cota5100!, as seen inFig. 6~b!. Over the range of experimentally attainable co

FIG. 5. Schematic diagram of an individual oscillator. Component valare R15100 kV, R2510 kV, R35150 kV, R455 kV, R5550V, C51 nF. The diodes are type MV2101. The op amps are type OP-07.


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pling strengths,ux'umax>0.9 V, or 23% of the radius of thelimit cycle, andux'urms>0.35 V, or 9% of the radius of thelimit cycle. Thus a high degree of synchronization is nevobserved. At all accessible coupling strengths, the noisamplified dramatically. In addition, the degree of synchrozation increases slowly and smoothly with no sharp trantion as observed in the previous experiment.

IV. CRITERIA FOR SYNCHRONIZATION

In the experiments described above, we investigateeffects of noise on the synchronization of periodic dynamisystems. These oscillators are designed, however, to meffects previously observed or theorized in chaotic systeSpecifically, the mechanisms for noise amplification buinto these oscillators can also be found in chaotic dynamsystems of physical interest. To further explore the implitions of noise-amplifying dynamics for chaotic systems,apply several criteria for synchronization developed wchaotic systems in mind. Note that none of these criterequires chaos for applicability. In the next few paragrapwe briefly review each of the five criteria. Then we apply tcriteria to our experiments and compare their predictionsthe experimental observation reported in Secs. II and III.

Criterion (1). Fujisaka and Yamada15 claim that the syn-chronized state of noise free, identical coupled chaotic oslators is asymptotically stable when the largest transveLyapunov exponent is negative. The exponents are demined from the solution of the equations

ddx'

dt5Jdx' , ~6!

J[DF@xm~ t !#2gK , ~7!

obtained by linearizing Eq.~1! aboutx'50, wheredx' is asmall perturbation away from the synchronization manifoandDF@xm(t)# is the Jacobian of the vector fieldF evaluatedon the attractor of the master oscillator. Apparently, Fujisaand Yamada believed this criterion is also valid for systesubject to small noise or parameter mismatch: They incor

FIG. 6. Synchronization of the ‘‘non-normal limit cycle’’ electronic circuiTwo measures of the degree of synchronization for~a! small ~cota51! and~b! large ~cota5100! non-normality.


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rated this criterion into a proposed method for determinexperimentally the largest Lyapunov exponent of a dynacal system. This method relies on the coincidence ofcoupling strength at which synchronization becomesstable and the coupling strength at which synchronizatiolost experimentally. Many other researchers have assuthat the asymptotic stability of the synchronization manifois a good predictor of synchronization in real physical stems despite the presence of noise and parameter mismFor example, a demonstration that the largest Lyapunovponent is negative is presented as proof that synchronizaof chaotic oscillators will occur~see, e.g., Refs. 16 and 17!.Note that several researchers have already shown this crion to be unreliable.3–6 We include it here for completenes

Criterion (2). Ashwin, Buescu, and Stewart3 suggest thatbursts of desynchronization events in coupled chaotic oslators are due to transversely unstable invariant sets emded in the transversely stable chaotic attractor on the schronization manifold. When the system is in thneighborhood of such a set, small noise can push it offmanifold resulting in a brief desynchronization event. Aswin, Buescu, and Stewart named this behaviorattractor bub-bling. To avoid attractor bubbling, the largest transveLyapunov exponent characterizing the most unstable invant set must be negative for synchronization in the preseof noise.

Criterion (3). Based on the same idea that the stabilityunstable sets governs the region of high-quality synchrontion, Brown and Rulkov18 suggest an alternative method fodetermining the transverse stability of these sets. They deoped a sufficient, but not necessary, condition forasymptotic stability of the synchronized state using Growald’s theorem. Briefly, they decompose the matrixJ into atime-independentA[^DF&2gK and a time-dependenB(xm ;t)[DF@xm(t)#2^DF& parts, where•& denotes a timeaverage over the driving trajectory. A trajectory is tranversely stable when

2Re@L1#.^iP21@B~xm ;t !#Pi&, ~8!

whereL1 is the largest eigenvalue ofA andP is a matrix ofeigenvectors ofA. The notationi•i denotes a norm whoschoice is arbitrary. The predictions of this criterion depeboth on the choice of norm and metric. Following Brown aRulkov, we use the Frobenius norm. As for the previocriterion, Brown and Rulkov suggest evaluating this criterialong every invariant set embedded in the attractor in orto determine the region of high-quality synchronization.

Criterion (4). Pecora, Carroll, and Heagy19 and Johnsonet al.20 attempt to ensure high-quality synchronization byquiring all eigenvalues of the matrixJ have negative reaparts at all points along the driving trajectoryxm(t). As forthe previous case, this criterion depends on the choicemetric.

Criterion (5). Gauthier and Bienfang5 introduce theLyapunov functionL[udx'(t)u2 to obtain an estimate of theregime of high-quality synchronization. They suggest thhigh-quality synchronization occurs for coupling schemwhere


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As in the two previous cases, this criterion depends onchoice of metric.

Applying the criteria to our first experiment, we find theach predicts synchronization for all coupling strenggreater than a critical value. Although we do see a shtransition to high-quality synchronization in the experimenone of the criteria accurately predict the coupling streng'1.33104 s21 at which this transition is observed. We dnote the critical coupling strength for criteria 1–5 asgFY ,gA , gBR, gJ , and gGB, respectively. Our analysis of thcircuit model reveals thatgFY521.1104 s21. We attributethe negativity ofgFY to the fact that, without the drivingcurrents, the nonlinearRC circuits do not self-oscillateSince the master and slave receive identical driving currethe circuits would behave identically even with zero copling if no noise were present. Criterion~2! is equivalent tocriterion ~1! in this case since the only invariant set in tattractor is the orbit; thus,gA521.13104 s21. In addition,for the criterion proposed by Brown and Rulkov~3!, we findgBR521.13104 s21. From Fig. 4~b!, it is clear that thesethree criteria fail to provide even a ‘‘sufficient’’ condition fohigh-quality synchronization sinceux'umax and ux'urms arelarge for a range of positive coupling strengths belowg'1.33104 s21. On the other hand, criteria~4! and ~5! pre-dict gJ5gGB52.23104 s21, overestimating the couplingstrength needed to obtain high quality synchronization inexperiment for the particular level of noise injected into tslave oscillator.

Applied to our second experiment, each criterion agpredicts a critical coupling strength above which high-quasynchronization should be observed. We first considercase of small non-normality where experimentally we oserve high-quality synchronization at coupling strenggreater than 0.23103 s21. For the first four criteria, we findthat gFY5gA50, gBR50.133103 s21, and gJ50.113103 s21. The fifth criterion predicts a much larger criticacoupling strength ofgGB50.93103 s21. Thus, the first fourcriteria reasonably predict the range of coupling strengover which a high degree of synchronization is observwhile the fifth criterion significantly overestimates the rquired coupling strength. More interesting is the caselarge non-normality where we observe experimentallytransition to high-quality synchronization. Applying the crteria in this case, we find that the first four criteria are indpendent of the non-normality and hence predict high-quasynchronization at the same critical coupling strengths athe previous case despite the fact that the observed degrsynchronization is degraded significantly. On the other hawe find the fifth criterion predictsgGB56.93106 s21, a valuemuch too large to implement using our experimental appatus. We attribute the sensitivity to non-normality in the sy


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tem of criterion ~5! to the above-mentioned fact that thcriterion restricts transient amplification of perturbations.

V. DISCUSSION

From these experiments we draw two conclusions. Fiany physical system that contains some mechanism for tsient growth of perturbations may show bursting behavwhen coupled to an identical system because of the intable presence of noise. In this paper, we have presentedsuch mechanisms for transient growth:~1! a sharp thresholdin phase space that can separate the master and slave, a~2!noise amplification due to non-normality. Note that thesenot the only possible mechanisms for transient growth. Otexamples are attractor bubbling in coupled chaooscillators3 and local stability variations in limit cycles.1

Therefore, a general criterion for synchronization of physisystems must have some sensitivity to transient behavioour experiments, only criterion~5! shows any sensitivity tothe transient growth displayed by the oscillators, but it apears to be overly conservative in its estimate of the coupstrength needed for high-quality synchronization. Secothe details of the bursts, their size and frequency, maypend intimately on the details of the noise. For exampleour first experiment, the frequency of the bursts is direcdetermined by the frequency of perturbations that are blarge enough and time appropriately. We speculate thatther progress in predicting high-quality synchronization wrequire explicitly taking into account the details of the noiin the system.

ACKNOWLEDGMENTS

J.N.B. and D.J.G. gratefully acknowledge the financsupport of the U.S. ARO through Grant Nos. DAAD 19-91-0199 and DAAG55-97-1-0308.

1F. Ali and M. Menzinger, Chaos9, 348 ~1999!.2L. N. Trefethen, inNumerical Analysis, edited by D. F. Griffiths and G. A.Watson~Longman, Birmingham, AL, 1992!, pp. 234–266.

3P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett. A193, 126 ~1994!;Nonlinearity9, 703 ~1996!.

4J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. E52, 1253~1995!.

5D. J. Gauthier and J. C. Bienfang, Phys. Rev. Lett.77, 1751~1996!.6N. F. Rulkov and M. M. Sushchik, Int. J. Bifurcation Chaos Appl. SEng.7, 625 ~1997!.

7S. C. Venkataramani, B. R. Hunt, E. Ott, D. J. Gauthier, and J. C. Bfang, Phys. Rev. Lett.77, 5361 ~1996!; V. Astakhov, A. Shabunin, T.Kapitaniak, and V. Anishchenko,ibid. 79, 1014~1997!; R. Konnur,ibid.79, 3877~1997!; and K. Josic´, ibid. 80, 3053~1998!.

8L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Scien261, 578 ~1993!.

9T. Gebhardt and S. Grossman, Phys. Rev. E50, 3705~1994!.10S. Grossmann, inNonlinear Physics of Complex Systems~Springer, Ber-

lin, 1996!, pp. 10–22.11F. X. Kartner, D. M. Zumbu¨hl, and N. Matuschek, Phys. Rev. Lett.82,

4428 ~1999!.12L. Jaeger and H. Kantz, Physica D105, 79 ~1997!.


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13C. G. Schroer, E. Ott, and J. A. Yorke, Phys. Rev. Lett.81, 1397~1998!.14M. Diestelhorst, R. Hegger, L. Jaeger, H. Kantz, and R.-P. Kapsch, P

Rev. Lett.82, 2274~1999!.15H. Fujisaka and T. Yamada, Prog. Theor. Phys.69, 32 ~1983!; 70, 1240

~1983!.16J. H. Peng, E. J. Ding, M. Ding, and W. Yang, Phys. Rev. Lett.76, 904

~1996!.17K. Pyragas, Phys. Lett. A181, 203 ~1993!.


s.

18R. Brown and N. F. Rulkov, Phys. Rev. Lett.78, 4189~1997!.19L. Pecora, T. Carroll, and J. Heagy, inChaotic Circuits for Communica-

tions, Photonics East, SPIE Proceedings, Philadelphia, 1995@Proc. SPIE2612, 25 ~1995!#.

20G. A. Johnson, D. J. Mar, T. L. Carroll, and L. M. Pecora, Phys. Rev. L80, 3956~1998!.

21T. Kapitaniak, M. Sekieta, and M. Ogorzalek, Int. J. Bifurcation ChaAppl. Sci. Eng.6, 211 ~1996!.


Experimental investigation of high-quality synchronization of coupled oscillators

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