VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996

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Intermittent Loss of Synchronization in Coupled Chaotic Oscillators:Toward a New Criterion for High-Quality Synchronization

Daniel J. Gauthier and Joshua C. Bienfang*Department of Physics and Center for Nonlinear and Complex Systems,

Duke University, Box 90305, Durham, North Carolina 27708(Received 24 October 1995)

We observe incomplete synchronization of coupled chaotic oscillators over a wide rangcoupling strengths and coupling schemes for which high-quality synchronization is expeLong intervals of high-quality synchronization are interrupted at irregular times by large,desynchronization events that can be attributed to “attractor bubbling,” clearly demonstratingthe standard synchronization criterion is not always useful in experiments. We suggest a smethod for rapidly selecting the coupling schemes that are most likely to produce high-qsynchronization. [S0031-9007(96)01028-9]

PACS numbers: 05.45.+b, 84.30.Ng

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Spontaneous synchronization of dynamical systesuch as that appearing in clocks [1] and fireflies [for example, has been the subject of curiosity ascholarly inquiry for many years. Recently, severesearch groups [3] have synchronizedchaoticsystems; asurprising result considering that initially close trajectorof chaotic systems diverge exponentially. One motivatfor researching chaos synchronization techniques isexplore their practical application to various problemscommunication [4], optics [5], and nonlinear dynammodel verification [6]. Also, a detailed understandingthe synchronization process may lead to new schemecontrolling complex spatiotemporal dynamics that ocduring cardiac fibrillation [7] or in diode laser arrays [8for example.

Recent reports indicate that our understanding ofsynchronization process is not complete: two wematched chaotic systems do not necessarily synchrounder conditions when high-quality synchronizationexpected [9–12]. Rather, long intervals of high-quasynchronization are interrupted irregularly by lar(comparable to the size of the attractor), brief desynchnization events that may be undesirable or even harmin some applications. It is proposed [9–12] that tbehavior, calledattractor bubbling, is associated withinvariant sets embedded within the synchronizatmanifold that are unstable to perturbations causednoise or slight parameter mismatch [13].

The primary objectives of this Letter are to demonstrthat the popular and widely used criterion for synchronition of coupled chaotic oscillators entirely fails to predthe regime of high-quality (burst-free) synchronizationa simple experimental system, and to compare our obvations with recent theories [9–12]. A secondary objtive is to suggest a new, simple method for estimatingrange of high-quality synchronization.

In our investigation, we consider one-way couplingtwo chaotic electrical circuits. The dynamical evoluti

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of a single circuit [14], shown schematically in Fig. 1,governed by the set of dimensionless equations

ÙV1j ­V1j

R12 gfV1j 2 V2jg , (1a)

ÙV2j ­ gfV1j 2 V2jg 2 Ij , j ­ m, s , (1b)

ÙIj ­ V2j 2 R4Ij , (1c)

where V1j and V2j represent the voltage droacross the capacitors (normalized to the diovoltage Vd ­ 0.58 V), Ij represents the currenflowing through the inductor (normalized toId ­VdyR ­ 0.25 mA for R ­

pLyC ­ 2, 345 V), gfV g ­

VyR2 1 IrfexpsaV d 2 exps2aV dg represents the current (normalized to Id) flowing through the parallel combination of the resistor and diodes, and tiis normalized to t ­

pLC ­ 2.35 3 1025 sec.

The circuit displays “double scroll” behavior foIr ­ 2.25 3 1025, a ­ 11.6, R1 ­ 1.2, R2 ­ 3.44,R3 ­ 0.043, Rdc ­ 0.15 (the dc resistance of the indutor), andR4 ­ R3 1 Rdc ­ 0.193, where all resistancehave been normalized toR.

ny

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r--eFIG. 1. Chaotic electronic oscillator consisting of a negaresistorR1 ­ 2814V, capacitorsC ­ 10 nF, an inductorL ­55 mH (dc resistance353V), a resistorR3 ­ 100V, and apassive nonlinear element (resistorR2 ­ 8, 067V, diodes type1N914, dashed box).

© 1996 The American Physical Society 1751

VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996

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The dynamics of the coupled system can be expressuccinctly as

Ùxm ­ Ffxmg , (2a)

Ùxs ­ Ffxsg 2 cKsxm 2 xsd , (2b)

where xm sxsd denotes the position inn-dimensionalphase space of the master (slave) oscillators,F representsthe flow of the oscillators,K is ann 3 n coupling matrix,c is the scalar coupling strength, andxT

j ­ sV1j , V2j, Ijd.We match all components to within 1%, construct tcircuits, and select two from the group whose bifurction diagrams are most similar (there are only slight dferences in the bifurcation diagrams for all ten). Tfacilitate our discussion of the synchronization procewe introduce new coordinatesxk ­ sxm 1 xsdy2 andx' ­ sxm 2 xsdy2 that specify the dynamics within antransverse to the synchronization manifold, respectivel

Synchronization of the oscillators occurs whenxsstd ­xmstd ; sstd which is equivalent tox'std ­ 0; thesystem resides on ann-dimensional synchronizationmanifold within the2n-dimensional space. In practicethe occurrence of high-quality synchronization is indcated by jx'stdj , ´, where ´ is a length scale smal(typically 1%) in comparison to the typical dimensionthe chaotic attractor. We note that the synchronizatcondition has been generalized [15] to include the pobility that the variables of the slave oscillator are equaa function of the variables of the master oscillator.

The widely used criterion for synchronization of copled chaotic oscillators was proposed by Fujisaka aYamada [16] over a decade ago. They investigatestability of the synchronized statex' ­ 0 by determin-ing the transverse Liapunov exponentsl

1' $ l

2' $ · · · $

ln' characterizing the dynamics transverse to the sync

nization manifold. The exponents are determined frthe solution to the variational equation

d Ùx' ­ hDFfsstdg 2 cKjdx' , (3)

obtained by linearizing Eq. (2) aboutx' ­ 0, whereDFfsstdg denotes the Jacobian ofF evaluated onsstd.They propose that high-quality synchronization occursvalues of the coupling strengthc wherel

1' , 0.

In stark contrast to the expected results, weserve incomplete synchronization for all coupling schemover a wide range of coupling strengths wherel

1' , 0.

For example, consider “V2 coupling” (K22 ­ 1, Kij ­ 0otherwise) of the oscillators forc ­ 4.6. A numericalanalysis of Eq. (3) reveals thatl

1' , 0 whenc . c22

crit .0.64. As seen in Fig. 2, we observe brief, large-scaletermittent desynchronization events in the experimentobserved temporal evolution ofjx'stdj. This behaviorpersists indefinitely.

To demonstrate that the standard synchronizacriterion fails entirely for theV2-coupled oscillators, wemeasure the average distance from the synchronizamanifold jx'stdjrms, which is sensitive to the globa

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FIG. 2. Experimentally observed intermittent loss of synchnization in V2-coupled chaotic oscillators forc ­ 4.6. Longintervals of high-quality synchronizationfjx'stdj ø 0g are in-terrupted by brief, large-scale (comparable to the sizethe synchronization manifold) desynchronization events. Tcharacteristic time scale of the system corresponds to,6t ­0.141 msec.

transverse stability of the synchronized state [17], andmaximum observed value of the distance from the mafold jx'stdjmax, which is sensitive to the local stabilitof the state [10]. Figure 3(a) shows the experimentameasured values ofjx'stdjrms (solid line) andjx'stdjmax

(dashed line) as a function of the coupling strengCompare these measurements to the predicted valuel

1' (solid line) in Fig. 3(b). It is seen thatjx'stdjrms de-

creases rapidly as the coupling strength increases and

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FIG. 3. (a) Experimentally observed degree of synchronition and (b) theoretically predicted stability of the synchronizstate for V2-coupled chaotic oscillators. We observe desychronization events forc . c22

crit, as indicated byjx'stdjmax ¿jx'stdjrms ø 0. High-quality synchronization is never observefor this coupling scheme even though the standard syncnization criterion predicts its occurrence forl1

' , 0. Recenttheories predict attractor bubbling forh' . 0, in agreementwith our observations.

VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996

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it is near zero forc . c22crit, where l

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servation indicates that our model of the electrical circaccurately describes its dynamics [17]. Persistentsynchronization events occur forc . c22

crit, wherejx'stdjmax remains large; high-quality, bubble-free sychronization [jx'stdjmax comparable to the noise level]never observed. Surprisingly, similar results are foufor most other coupling schemes.

We find that high-quality synchronization can onbe obtained for coupling schemes whereK11 fi 0, al-though the range is less than that expected basedthe standard synchronization criterion. For exampconsider “V1-coupled” oscillators (K11 ­ 1, Kij ­ 0otherwise) where high-quality synchronization is epected for c . c11

crit . 0.305 based on a numericaanalysis of Eq. (3). Figure 4(a) shows the experimtally observed variation ofjx'stdjrms (solid line) andjx'stdjmax (dashed line) with a coupling strength whicshould be compared to the predicted values ofl

1'

(solid line) shown in Fig. 4(b). Again, it is seen thjx'stdjrms decreases rapidly as the coupling strenincreases and that it is near zero forc . c11

crit wherel

1' , 0. Note that high-quality synchronization

only obtained for coupling strengths much greater thexpected.

Our results indicate that the criterion proposedFujisaka and Yamada [16], and widely used in theoretstudies of synchronization [3,4,6], is not a sufficient contion for high-quality synchronization of chaotic oscillato

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FIG. 4. (a) Experimentally observed degree of synchrontion and (b) theoretically predicted stability of the synchnized state forV1-coupled chaotic oscillators. High-qualisynchronizationfjx'stdjmax ø 0g is observed only forh' , 0.The range of high-quality synchronization predicted by ourproximate method isc . 1yR1.

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in an experimental setting. Recent research [9–12] sgests that the criterion fails when the transverse Liapuexponents characterizing invariant sets embedded wthe synchronization manifold are greater than zero unconditions whenl

1' , 0. A desynchronization even

corresponds to the growth of a perturbation (due to noor parameter variation) during the interval when ttrajectory is in a neighborhood of these invariant sets.

To test this hypothesis, we determine the mtransversely unstable invariant set, characterized bymaximum transverse Liapunov exponenth', since itmediates the transition from attractor bubbling to higquality synchronization. A numerical analysis of tlow-period unstable orbits [18] indicates that the unstasteady-statexk ­ 0 is the most unstable set. Figures 3(and 4(b) show the dependence ofh' on the couplingstrength (dashed line) forV2- andV1-coupled oscillatorsrespectively. ForV2-coupled oscillators, it is seen thh' . 0 for all c, consistent with our observation thdesynchronization events occur for allc. For V1-coupledoscillators, it is seen that the transition to high-quasynchronization (jx'jmax comparable to the noise leveoccurs near the point whereh' becomes less than zerThe transition is not sharp, which may be the resof the finite noise level and parameter variation inexperiment.

Based on our observations, it appears that the procriterion for high-quality synchronization of chaotic osclators ish' , 0. While this criterion is mathematicallprecise, it may be difficult to apply in practice becauthere are an infinite number of invariant sets whosebility must be determined [18]. Is there a different methfor estimating the range of high-quality synchronizatithat captures the essence of the mathematically precriterion without being overly complex? We believe threcent studies of the dynamics of linear systems chaterized by non-normal matrices offers some guidanTrefethen [19] shows that perturbations can grow signcantly in the transient phase of the dynamics of a linsystem even when the eigenvalues of the matrix goving the dynamics are all negative and distinct. Henthe eigenvalues do not necessarily say much aboutbehavior of the system in the transient phase, rathercharacterize the asymptotic, long-term behavior.

In a similar vein, our observations suggest thatLiapunov exponents characterizing the dynamics of a nlinear system do not necessarily say much about the tsient behavior. Noise and the unstable invariant setsrise to persistent transient behavior in which the effeof perturbations are magnified during brief intervals.simple method for testing whether perturbations can gin the transient phase is to investigate the time dertive of the Liapunov functionL ­ jdx'stdj2 computedfrom a mathematical model of the system. The fution L is equal to the square of the distance betweentrajectory and the synchronized statex' ­ 0 for small

1753

VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996

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distances [20]. A sufficient condition that all perturbtions decay to the manifold without transient growth is

dL

dt­ 2dx'std ? hsssDFfsstdg 2 cKddddx'stdj , 0 (4)

for all times. We suggest that condition (4) canused to quickly estimate the range of coupling strengthat result in high-quality, burst-free synchronizationcoupled chaotic oscillators. Note that the Liapunfunction depends on the choice of the metric (it is ninvariant), and hence it underestimates the range of hquality synchronization.

For theV2-coupled oscillators,dLydt ­ 2hR211 dx2

1 2

g0fV1mstd 2 V2mstdg sdx1 2 dx2d2 2 cdx22 2 R4dx2

3 j,where dx'std ­ sdx1, dx2, dx3d, and g0fV g ­ R21

2 1

aIr fexpsaV d 1 exps2aV dg. We see that dLydtcan be greater than zero regardless of the valuethe coupling strengthc. Hence, attractor bubblingmay be present for allc, in agreement with ourexperimental observations. For theV1-coupledoscillators, dL ydt ­ 2hsR21

1 2 cddx21 2 g0fV1mstd 2

V2mstdg sdx1 2 dx2d2 2 R4dx23 j can be greater than zer

for all times whenc , R211 ­ 0.83, also in reasonable

agreement with our observations. These results sugthat our method is useful for estimating the rangehigh-quality synchronization without the need of complnumerical calculations.

We gratefully acknowledge fruitful discussions of thwork with P. Ashwin, J. Heagy, E. Ott, L. Pecora, aD. Schaeffer. This work was supported by the U.S. ArmResearch Office under Grant No. DAAH04-95-1-052the U.S. Air Force Phillips Laboratory under ContraNo. F29601-95-K—0058, and the National Science Fodation under Grant No. PHY-9357234.

*Current address: Department of Physics, UniversityNew Mexico, Alburquerque, NM 87131.

[1] A. B. Pippard, The Physics of Vibration(CambridgeUniversity Press, Cambridge, England, 1989) Sect. 12.

[2] S. H. Strogatz,Nonlinear Dynamics and Chaos(Addison-Wesley, Reading, MA, 1994) Sect. 4.5.

[3] V. S. Afraimovich, N. N. Verichev, and M. I. RabinovichIzv. Vyssh. Uchebn. Zaved., Radiofiz.29, 1050 (1986)[Radiophys. Quantum Electron.29, 795 (1986)]; L. M.Pecora and T. L. Carroll, Phys. Rev. Lett.64, 821 (1990);Phys. Rev. A 44, 2374 (1991); N. F. Rul’kov, A. R.Volkovskii, A. Rodriguez-Lozano, E. Del Rio, and M. GVerlarde, Int. J. Bif. Chaos2, 669 (1992); Y. C. Lai,M. Ding, and C. Grebogi, Phys. Rev. E47, 86

1754

s

th-

of

stf

,

-

f

(1993); T. C. Newell, P. M. Alsing, A. Gavrielides, anV. Kovanis, Phys. Rev. Lett.72, 1647 (1994); Phys. RevE 51, 2963 (1995).

[4] Lj. Kocarev, K. S. Halle, K. Eckert, L. O. Chua, anU. Parlitz, Int. J. Bif. Chaos2, 709 (1992); S. HayesC. Grebogi, and E. Ott, Phys. Rev. Lett.70, 3031 (1993);K. M. Cuomo and A. V. Oppenheim, Phys. Rev. Lett.71,65 (1993); K. Murali and M. Lakshmanan, Phys. RE 48, R1624 (1993); P. Colet and R. Roy, Opt. Le19, 2056 (1995); T. Kapitaniak, Phys. Rev. E50, 1642(1994); G. Perez and H. A. Cerdeira, Phys. Rev. L74, 1970 (1995); L. Kocarev and U. Parlitz, Phys. RLett. 74, 5028 (1995); J. H. Peng, E. J. Ding, M. Ding, aW. Yang, Phys. Rev. Lett.76, 904 (1996).

[5] R. Roy and K. S. Thornburg, Jr., Phys. Rev. Lett.72, 2009(1994); and T. Sugawara, M. Tachikawa, T. Tsukamand T. Shimizu, Phys. Rev. Lett.72, 3502 (1994).

[6] K. Pyragas, Phys. Lett. A170, 421 (1992); R. BrownN. F. Rul’kov, and E. R. Tracy, Phys. Rev. E49, 3784(1994); U. Parlitz, Phys. Rev. Lett.76, 1232 (1996).

[7] P. V. Bayly, E. E. Johnson, P. D. Wolf, H. S. GreensiW. M. Smith, and R. E. Ideker, J. Cardiovas. Electphysiol.4, 533 (1993), and references therein.

[8] H. G. Winful and L. Rahman, Phys. Rev. Lett.65, 1575(1990); R.-D. Li, and T. Erneux, Phys. Rev. A46, 4252(1992); D. Merbach, O. Hess, H. Herzel, and E. SchPhys. Rev. E52, 1571 (1995).

[9] P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett A.193,126 (1994); Nonlinearity,9, 703 (1996).

[10] J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev52, R1253 (1995).

[11] S. C. Venkataramani, B. R. Hunt, and E. Ott (topublished).

[12] N. F. Rulkov and M. M. Sushchik (to be published).[13] A. S. Pikovsky and P. Grassberger, J. Phys. A24, 4587

(1991); R. Brown, N. F. Rulkov, and N. B. Tufillaro, PhyLett. A 196, 201 (1994); T. Kapitaniak, J. Wojewoda, aJ. Brindley, Phys. Lett. A210, 283 (1996).

[14] Our circuit is a variation of chaotic oscillators describedM. P. Kennedy, IEEE Trans. Circuits Syst. Video Techn40, 657 (1993); A. Kittel, K. Pyragas, and R. RichtPhys. Rev. E50, 262 (1994).

[15] N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D.Abarbanel, Phys. Rev. E51, 980 (1995).

[16] H. Fujisaka and T. Yamada, Prog. Theor. Phys.69, 32(1983);70, 1240 (1983).

[17] H. G. Schuster, S. Martin, and W. Martienssen, Phys. RA 33, 3547 (1986).

[18] B. R. Hunt and E. Ott, Phys. Rev. Lett.76, 2254 (1996).[19] L. N. Trefethen, inNumerical Analysis,edited by D. F.

Griffiths and G. A. Watson (Longman, Birmingham, A1992), pp. 234–266.

[20] K. Ogata,Modern Control Engineering, 2nd Ed.(Prentice-Hall, Englewood Cliffs, NJ, 1990), Sect. 9-7.