Experimental investigation of high-quality synchronizationof coupled oscillatorsJonathan N. Blakelya)and Daniel J. GauthierDepartment of Physics and Center for Nonlinear and Complex Systems, Duke University,Box 90305, Durham, North Carolina 27708Gregg Johnson, Thomas L. Carroll, and Louis M. PecoraCode 6343, U. S. Naval Research Laboratory, Washington, DC 20375共Received 30 November 1999; accepted for publication 2 May 2000兲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. 关S1054-1500共00兲01203-9兴The once surprising fact that the irregular oscillations oftwo chaotic oscillators can be synchronized is now wellestablished. 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 oscil-lations due to small noise rather than deterministicchaos? We address this question here through two ex-periments on coupled periodic electronic circuits. In eachexperiment, we observe the degree of synchronization be-tween 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 addi-tion, we apply to our experimental systems several pro-posed criteria for high-quality synchronization developedin studies of synchronized chaos. We find that none ofthese criteria accurately predict the behavior observed inthe experiments. These results may provide some guid-ance in the development of practical applications of syn-chronized chaos such as secure communication schemeswhere a criterion for high-quality synchronization isneeded.I. INTRODUCTIONIt is now well established that the dynamics of a nonlin-ear system can become highly irregular when small randomnoise is injected into the system. For example, Ali andMenzinger1recently showed that a globally stable limit cycleoscillator subject to small amounts of noise can display anexplosive divergence of trajectories away from the limitcycle. The origin of this disproportionate response to smallperturbations is the fact that the limit cycle is composed ofsegments of varying local stability, most of which are stablebut some of which are highly unstable as shown schemati-cally in Fig. 1. In regions of pronounced local instability, aperturbation may undergo transient growth before decayingasymptotically. A similar behavior is displayed by non-normal linear systems as shown schematically in Fig. 2共a兲.2Non-normal systems are characterized by nonorthogonaleigenvectors. A small perturbation to such a system ex-pressed as a linear superposition of such vectors may havelarge coefficients but a small norm due to cancellation asdepicted in Fig. 2共b兲. As shown in Fig. 2共c兲, when the systemevolves in time, the coefficients of the superposition maydecay at different exponential rates so the cancellation is lost,causing the norm to increase even though the individual ei-genvector components are decaying asymptotically. Again,the result is a transient amplification of a perturbation by astable dynamical system. These two examples do not exhaustthe possible scenarios in which highly irregular oscillationsare generated by large noise amplification in a dynamicalsystem. An interesting question is whether the irregular os-cillations occurring in two identical noise-amplifying dy-namical systems can be synchronized.The primary objective of this paper is to present theresults of an experimental investigation of synchronization ofnoise-amplifying dynamical systems consisting of nonlinearelectronic periodic oscillators. In our experiments, the dy-namical behavior of the coupled oscillators is described by aset of nonlinear differential equations given bya兲Electronic mail: [email protected] VOLUME 10, NUMBER 3 SEPTEMBER 20007381054-1500/2000/10(3)/738/7/$17.00 © 2000 American Institute of PhysicsDownloaded 18 Nov 2002 to 152.3.183.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jspx˙m⫽ F关xm兴, 共1a兲x˙s⫽ F关xs兴⫹␥K共xm⫺ xs兲, 共1b兲where xm(xs) denotes the position in the N dimensionalphase space of the master 共slave兲 oscillator, F is a vectorfield describing the dynamics of an individual oscillator, K isan N⫻ N coupling matrix, and␥is the scalar couplingstrength. Note that the coupling is one-way so that the evo-lution of the master oscillator is unaffected by that of theslave. The synchronized state xs(t)⫽ xm(t) defines an N di-mensional invariant manifold 共called the synchronizationmanifold兲 residing in the 2N dimensional phase space of thecoupled system. We introduce new coordinates x储⫽ (xm⫹ xs)/2 and x⬜⫽ (xm⫺ xs)/2 to separate the dynamics alongand transverse to the synchronization manifold. In this coor-dinate system, synchronization may be defined as x⬜(t)⫽ 0.In each of two experiments, we couple two periodic oscilla-tors subject to small input noise and observe the degree ofsynchronization as a function of coupling strength. We usevarious norms of x⬜(t) to quantify the degree of synchroni-zation between the oscillators.In the first experiment, each oscillator is a one-dimensional 共1D兲 periodically driven system. The phasespace trajectory of a single oscillator brings the system veryclose to