attractor bubbling in coupled hyperchaotic...

International Journal of Bifurcation and Chaos, Vol. 10, No. 4 (2000) 835–847c© World Scientific Publishing Company

ATTRACTOR BUBBLING INCOUPLED HYPERCHAOTIC OSCILLATORS

JONATHAN N. BLAKELY and DANIEL J. GAUTHIERDepartment of Physics and Center for Nonlinear and Complex Systems,

Duke University, Box 90305, Durham, North Carolina 27708, USA

Received May 17, 1999; Revised August 25, 1999

We investigate experimentally attractor bubbling in a system of two coupled hyperchaotic elec-tronic circuits. The degree of synchronization over a range of coupling strengths for two differentcoupling schemes is measured to identify bubbling. The circuits display regimes of both at-tractor bubbling and high-quality synchronization. For the coupling scheme where high-qualitysynchronization is observed, the transition to bubbling is “soft” and its scaling with couplingstrength near the transition point does not fit into the known categories of transition types. Wealso compare the observed behavior to several proposed criteria for estimating the regime ofhigh-quality synchronization. It is found that none of these methods is completely satisfactoryfor predicting accurately the regimes of attractor bubbling and high-quality synchronization.

1. Introduction

One hallmark of chaos is that the trajectories oftwo identical chaotic systems starting from veryclose initial conditions diverge exponentially. It isnow known that this divergence can be overcomeby an appropriate coupling between the systems,thereby synchronizing their evolution [Fujisaka &Yamada, 1983; Pecora & Carroll, 1990]. In the pastdecade, the study of synchronized low-dimensionalchaotic oscillators has inspired several possible ap-plications including secure communication [Pecoraet al., 1997], dynamics-based computation [Sinha& Ditto, 1998], and schemes for controlling com-plex spatiotemporal dynamics. More recently, ithas become clear that such applications require abetter understanding of the dynamics of coupledhigher dimensional systems displaying hyperchaos(dynamics characterized by multiple positive Lya-punov exponents). For example, schemes for se-cure communication should be based on synchro-nized hyperchaotic oscillators since low-dimensionalchaos is not complex enough to securely mask in-formation [Perez & Cerdeira, 1995; Short, 1997].

One fundamental issue in synchronizing hy-perchaotic systems is the number of distinct cou-pling signals necessary to achieve synchronization.Early on, it was conjectured that the number ofcoupling signals must be greater than or equalto the number of positive Lyapunov exponents[Pyragas, 1993]. This conjecture has since provento be untrue. Theoretical studies of several hyper-chaotic systems show that it is possible to obtainasymptotic stability of the synchronized state us-ing only a single coupling signal [Peng et al., 1996;Tamaševičius & Cenys, 1997; Wang & Wang, 1998;Brucoli et al., 1998]. Synchronized hyperchaos us-ing such a scheme has been demonstrated experi-mentally in electronic circuits [Tamaševičius et al.,1996, 1997b; Johnson et al., 1998], erbium-dopedfiber ring lasers [Van Wiggeren & Roy, 1998], anddelayed feedback diode lasers [Goedgebuer et al.,1998].

These experiments demonstrate that hyper-chaos can be synchronized, but they do not re-port whether synchronization can be obtained overa wide range of coupling schemes and couplingstrengths. It is important to determine the full

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836 J. N. Blakely & D. J. Gauthier

range of synchronization behavior to thoroughlytest our theoretical understanding of this phe-nomenon. Large discrepancies between theory andexperiment were found in studies of coupled low-dimensional systems [Ashwin et al., 1994; Heagyet al., 1995; Gauthier & Bienfang, 1996; Rulkov& Sushchik, 1997]. The source of these discrepan-cies is attractor bubbling, a new type of dynami-cal behavior where a small amount of noise or mis-match between oscillators [Pikovsky & Grassberger,1991] gives rise to large intermittent bursts ofdesynchronization.

Attractor bubbling has yet to be investigatedexperimentally in systems of coupled hyperchaoticoscillators, although Ahlers et al. [1998] observedbubbling in a hyperchaotic numerical model of cou-pled diode lasers. None of the experiments citedabove report the close comparison of theory andexperiment over a range of coupling schemes andcoupling strengths necessary to characterize attrac-tor bubbling. Thus, important questions remainunanswered. Is the bubbling in hyperchaotic oscil-lators more severe than in low-dimensional systemsgiven the greater complexity of hyperchaos? Canexperiments on high-dimensional systems give ad-ditional guidance on the formulation of a methodfor predicting attractor bubbling?

In this paper, we address these questionsthrough an experimental investigation of attractorbubbling in a system of coupled hyperchaotic elec-tronic oscillators. We observe the degree of syn-chronization over a range of coupling strengths andcoupling schemes in order to determine the preva-lence of attractor bubbling. Our results show largeregimes of both attractor bubbling and high-quality(bubble-free) synchronization. We also develop aprecise mathematical model of our experimentalsystem in order to compare our observations to sev-eral proposed methods for estimating the range ofsynchronization. We find that none of these meth-ods is completely satisfactory for predicting accu-rately the regimes of attractor bubbling and high-quality synchronization.

This paper is structured as follows. InSec. 2, we briefly review synchronization andattractor bubbling in low-dimensional systems.Next, we describe the experiments performed toinvestigate attractor bubbling in a system of twocoupled hyperchaotic oscillators in Sec. 3 andpresent the results in Sec. 4. In Sec. 5, we applyseveral different criteria used to predict synchro-nization to our system and compare them to the

experimental results. Finally, in Sec. 6, we sum-marize our conclusions and discuss the implicationsof our results for using synchronized hyperchaos insecure communication applications.

2. Synchronization andAttractor Bubbling

In this section, we review the fundamental conceptsunderlying synchronization of chaotic oscillators,using a geometric description of this phenomenon.This sets the stage for discussing the previous re-search on attractor bubbling in systems of coupledlow-dimensional oscillators. For concreteness, welimit our discussion to the case of two one-way cou-pled oscillators.

The dynamics of a coupled system of two oscil-lators can be expressed succinctly as

dxmdt

= F[xm] , (1a)

dxsdt

= F[xs]− cK(xm − xs) , (1b)

where xm (xs) denotes the position in the n-dimensional phase space of the master (slave) os-cillator, F is the vector field governing the flow of asingle oscillator, K is a n× n coupling matrix, andc is the scalar coupling strength. Synchronizationof the oscillators is defined by xs(t) = xm(t) andhence the evolution takes place on an n-dimensionalhyperplane, called the synchronization manifold,within the full 2n-dimensional phase space. We in-troduce new coordinates specifying the dynamicswithin and transverse to the synchronization man-ifold as

x‖ = (xm + xs)/2 , (2a)

x⊥ = (xm − xs)/2 , (2b)

respectively. In this basis, the synchronization man-ifold is defined by x⊥(t) = 0.

According to Fujisaka and Yamada [1983], thesynchronization manifold is asymptotically stable ifand only if the largest transverse Lyapunov expo-nent is negative. These exponents are the averagerates of exponential expansion or contraction in di-rections transverse to the synchronization manifold.They are determined from the solution of the vari-ational equation

dδx⊥dt

= {DF[xm(t)]− cK}δx⊥ , (3)

Attractor Bubbling in Hyperchaotic Oscillators 837

obtained by linearizing Eq. (1) about x⊥ = 0, whereδx⊥ is a small perturbation away from the synchro-nization manifold, and DF[xm(t)] is the Jacobianof the vector field F evaluated on the attractor ofthe master oscillator.

In the absence of noise and mismatch of theoscillator parameters, the stability of the synchro-nization manifold determines whether or not syn-chronization will occur. Fujisaka and Yamada ap-parently thought that their result remains validwhen the oscillators are subject to small noise or arenot perfectly matched. They described how to de-termine experimentally the largest Lyapunov expo-nent of an oscillator by observing the transition tosynchronization as the coupling strength is varied.In just such an experiment, Schuster et al. [1986]determined the largest Lyapunov exponent of anoscillator by coupling it to a near-identical oscilla-tor and observing the coupling strength at whichthe average value of |x⊥| became small. Clearly,this method relies on the coincidence of the cou-pling strength at which synchronization becomesunstable and the coupling strength at which syn-chronization is lost experimentally. Many otherresearchers have assumed that the asymptotic sta-bility of the synchronization manifold is a goodpredictor of synchronization in real physical sys-tems despite the presence of noise and parametermismatch. For example, a demonstration that thelargest Lyapunov exponent is negative is presentedas proof that synchronization of chaotic oscillatorswill occur (see e.g. [Pyragas, 1993; Peng et al., 1996;Zonghua & Shigang, 1997]).

However, several researchers have found thatthis synchronization criterion is inconsistent withexperimental results [Ashwin et al., 1994; Heagyet al., 1995; Gauthier & Bienfang, 1996; Rulkov& Sushchik, 1997]. In some synchronization experi-ments where all transverse Lyapunov exponents arenegative, long intervals of synchronization are inter-rupted irregularly by large (comparable to the sizeof the attractor), brief desynchronization events.This behavior has been called attractor bubbling[Ashwin et al., 1994]. Ashwin et al. [1994, 1996]pointed out that the synchronization manifold maycontain unstable sets, such as UPO’s, characterizedby positive transverse Lyapunov exponents, eventhough the transverse exponents characterizing thechaotic attractor are all negative. Near such sets,the manifold is locally repelling so a small pertur-bation arising from experimental noise will resultin a brief excursion away from the synchronization

manifold. These desynchronization events recur be-cause the chaotic evolution of the system brings itinto the neighborhood of the repelling set an infinitenumber of times. This explanation implies that thetransition from bubble-free synchronization to bub-bling as the coupling strength is varied occurs whena set first becomes transversely unstable. Buildingon these ideas, Venkataramani et al. [1996a, 1996b]identified generic types of transitions from synchro-nization to bubbling by considering the behavior ofcoupled 2-D maps and electronic oscillators.

Several researchers have attempted to charac-terize attractor bubbling in experiments on cou-pled low-dimensional chaotic circuits. Heagy et al.[1995] located experimentally several unstable peri-odic orbits (UPO) embedded in the attractor of achaotic circuit and then used the UPO’s to drive aslave oscillator. They determined the critical cou-pling strength required for synchronizing the slaveoscillator to each UPO and found that it variedfrom orbit to orbit. In addition, some UPO’s re-quired a coupling strength larger than that neededto make the largest transverse Lyapunov exponentnegative. In a separate experiment, Gauthier andBienfang [1996] observed that the loss of synchro-nization of a system of coupled chaotic circuits co-incided with the first loss of transverse stabilityby a UPO as the coupling strength was decreased.In another study of coupled circuits, Rulkov andSushchik [1997] found “disruptive” regions on thesynchronization manifold that were characterizedby large positive local transverse exponents. Thesedisruptive regions were correlated with desynchro-nization events and associated with UPO’s embed-ded in the attractor. Recently, bubbling-like behav-ior has been observed in coupled periodic oscillatorswhere UPO’s play no role [Gauthier, 1998; Blakelyet al., 1999]

Attractor bubbling clouds the question ofwhether two oscillators are synchronized in an ex-periment. This is because, in the presence of bub-bling, the oscillators are synchronized on average(the time averaged value of |x⊥(t)| is very small) butlarge desynchronization events still occur intermit-tently (the maximum value of |x⊥(t)| is large). Toemphasize this distinction, Gauthier and Bienfang[1996] referred to synchronization as high-quality if|x⊥(t)| < ε for all time t > 0, where ε is a lengthscale small in comparison to the characteristic sizeof the chaotic attractor. In this definition, the met-ric, the value of ε, and the norm used for deter-mining |x⊥(t)| are arbitrary. However, in many


circumstances there may be a physically relevantchoice of metric, ε, and norm.

In light of the number experiments in whichattractor bubbling has been observed, it appearsthat bubbling is a typical behavior of coupled low-dimensional chaotic oscillators. Hyperchaotic dy-namics is significantly more complex and hence onemight expect a higher prevalence of attractor bub-bling as the complexity increases. In the next sec-tion, we describe our experimental system for inves-tigating attractor bubbling in coupled hyperchaoticoscillators.

3. Experimental Apparatus

Our experimental system consists of a pair of well-matched coupled hyperchaotic electronic circuitsbased on the design of Tamaševičius et al. [1997a].Electronic circuits are ideal for studying attractorbubbling because several coupling schemes can beimplemented and the coupling strength varied inorder to determine the prevalence of bubbling, andaccurate models can be developed for comparing ex-perimental observations to theoretical predictions.

A single circuit, shown schematically in Fig. 1,consists of inductors L1 = 250 mH and L2 = 97 mH,capacitors C1 = 198 nF and C2 = 60 nF, a 1N914diode, and a negative resistor −R implemented us-ing an Analog Devices OP-42 op amp. The dynam-ical evolution of the circuit is described by the setof equations

dv1dt

= av1 − i1 − g(v1 − v2) , (4a)

dv2dt

= −bi2 + bg(v1 − v2) , (4b)

di1dt

= v1 − hi1 , (4c)

di2dt

= dv2 − ei2 , (4d)

where v1 and v2 represent the voltage drop acrossthe capacitors. The variables i1 and i2 representthe current flowing through the inductors scaled bythe characteristic resistance ρ =

√L1/C1 so that

they have dimensions of voltage. The other param-eters are given by a = ρ/R, b = C1/C2, d = L1/L2and time is normalized to τ =

√L1C1 = 0.2 ms.

Also, h = r1/ρ and e = dr2/ρ, where r1 = 96 Ωand r2 = 57 Ω are the DC resistances of L1 and L2,respectively. The current flowing through the diode

Fig. 1. Schematic diagram of a single hyperchaotic oscillatorused in the experiments.

is represented by g(v) = ρId[exp(αv) − 1], whereId = 1.6 nA and α = 22.6 V

−1. We measurethe current–voltage curve of the diode and fit itwith the Schockley model I = Id[exp(αV ) − 1] toobtain the value of the parameters α and Id. Thevalue of the reverse saturation current Id is ad-justed to improve the agreement between the ex-perimentally observed and numerically generatedphase space projections of the attractor at variousvalues of a. With the chosen parameters, the modeland the circuit display several periodic windows ofsimilar structure, where the observed locations ofthese windows agree with the predictions of themodel to within 1%.

The circuit displays fixed point, periodic,chaotic, or hyperchaotic dynamics depending onthe value of the parameter a (set by the nega-tive resistor). To characterize the various dynamicregimes, we determine the Lyapunov exponents ofthe circuit from experimental time series using themethod of Eckmann et al. [1986]. We also com-pute the Lyapunov exponents of the circuit modeland find reasonable agreement between the theo-retical and experimental spectra, suggesting thatthe model closely resembles the experimental sys-tem. Figure 2 shows the two largest Lyapunov ex-ponents in the neighborhood of the transition tohyperchaos. Hyperchaos is especially apparent be-yond a = 0.6 where there are two large positiveexponents. We perform our synchronization exper-iments with a = 0.693, well into the hyperchaoticregime. Figure 3 shows experimental phase spaceprojections of the hyperchaotic attractor of the cir-cuit for this value of a.

We attempt to synchronize the behavior of thetwo circuits using either of the two coupling schemesillustrated by a block diagram in Fig. 4. For eachof these coupling schemes, there exists a range ofcoupling strength over which the largest transverse


Fig. 2. The two largest Lyapunov exponents as a function of the bifurcation parameter a for a single hyperchaotic oscilla-tor. The solid lines show the theoretically determined exponents and the solid circles show the exponents determined fromexperimental time series. The oscillator is hyperchaotic when both exponents are positive.

Fig. 3. Phase space projections of the hyperchaotic attractor of a single oscillator with a = 0.693.


Fig. 4. Block diagram of coupling between master and slavecircuits. The current Ic, proportional to the output of thedifferential amplifier, is injected into the slave circuit therebycoupling it to the master circuit. The coupling scheme isv2-to-v1 (v1) if the switch is up (down).

Lyapunov exponent is negative. In “v1 coupling”(K11 = 1, Kij = 0 otherwise), the voltages acrossthe capacitors C1 of the master and slave are mea-sured by high-impedance voltage followers (TexasInstruments TL072). The scaled difference of these

two signals is provided by an analog multiplier withdifferential inputs (Analog Devices AD633). Theresulting voltage is further scaled by an invertingamplifier (OP-42) and converted to a current bya voltage-to-current converter based on an instru-mentation amplifier (Analog Devices AD620) andoperational amplifier (OP-42). This current is in-jected directly into the node above the capacitor C1of the slave circuit thereby coupling it to the master.Similarly, in “v2-to-v1 coupling” (K21 = 1, Kij = 0otherwise), the voltages across the capacitors C2 ofthe master and slave circuits are measured by volt-age followers, subtracted and processed by an ana-log multiplier and an inverting amplifier, convertedto a current by an instrumentation amplifier, andinjected into the node above the C1 capacitor ofthe slave circuit. Note that this coupling schemeconsists of measuring one variable (v2) and feedingback to a different variable (v1).

Analog electronics are also used to obtain a sig-nal proportional to the square of the separation ofthe master and slave in phase space |x⊥(t)|2 where,in the notation of Sec. 2,

x⊥ = xm − xs =[v1m − v1s, v2m − v2s, i1m − i1s, i2m − i2s

]T. (5)

As described in the previous paragraph, volt-age followers are used to measure the voltagesacross the capacitors in each circuit. The currentsflowing through the inductors are determined by us-ing an instrumentation amplifier (AD620) to mea-sure the voltage across a 1.96 Ω resistor in serieswith each inductor. (In the model above, the re-sistors used to measure the currents are includedin the values stated for r1 and r2.) Analog mul-tipliers with differential inputs provide the squaresof the difference of these signals, which are thensummed using an operational amplifier (OP42).The resulting signal is recorded and scaled by acomputer using an analog-to-digital converter (Na-tional Instruments model AT-MIO-16H-9) to obtain|x⊥(t)|2.

In our experiments, we quantify the degree ofsynchronization over a range of coupling strengthsfor each of the two coupling schemes describedabove. At each coupling strength, we record a timeseries of |x⊥(t)|2 (where | • | denotes the Euclideannorm) of a duration on the order of 106τ . Fromthe time series, we determine the maximum ob-served value |x⊥(t)|2MAX and the root mean squareaverage |x⊥(t)|2RMS, quantifying the occurrence oflarge separations between the master and slave

dynamics and the duration of these separations, re-spectively. For example, a large value of |x⊥(t)|MAXand a small value of |x⊥(t)|RMS indicate the pres-ence of brief large desynchronization events in thetime series, the hallmark of attractor bubbling.

4. Results

We first consider “v2-to-v1 coupling” (K21 = 1,Kij = 0 otherwise) of the oscillators. Fig-ure 5(a) shows the experimentally observed degreeof synchronization, quantified by both the maxi-mum observed value |x⊥(t)|2MAX and the root meansquare average |x⊥(t)|2RMS as functions of the cou-pling strength c. Figure 5(b) shows the largesttransverse Lyapunov exponent determined numer-ically from Eq. (3) using the method described byJackson [1990]. The exponent is negative for cou-pling strengths in the range from c = 0.66 toc = 1.62, indicating that the synchronization man-ifold is asymptotically stable. Experimentally, weobserve that |x⊥(t)|2MAX is still comparable to thesize of the attractor in this range, even though|x⊥(t)|2RMS is very small, indicating the occurrence


Fig. 5. (a) Experimentally observed degree of synchronization for v2-to-v1 coupling. The squares indicate |x⊥|2RMS and thecircles |x⊥|2MAX. We observe desynchronization events at all coupling strengths as indicated by |x⊥| � 0. (b) The largesttransverse Lyapunov exponents as functions of coupling strength c calculated for both a chaotic trajectory (solid line) and anunstable periodic orbit embedded in the chaotic attractor (dashed line).

of attractor bubbling. With this coupling scheme,no high-quality synchronization is observed.

Next, we consider “v1 coupling” (K11 = 1,Kij = 0 otherwise) of the oscillators. Fig-ure 6 shows the experimentally observed de-gree of synchronization and the numerically de-termined largest transverse Lyapunov exponent.From Fig. 6(b), it is seen that the synchroniza-tion manifold is asymptotically stable for all cou-pling strengths larger than 0.26 since the exponentcrosses zero at this point and remains negative forlarger coupling strengths. Experimentally, we ob-serve a small region of attractor bubbling occur-ring for the range of coupling strengths between0.25 and 0.32. No desynchronization events greaterthan the noise level (|x⊥(t)|2MAX < 1.0 V2 or 0.5%of the maximum possible value of |x⊥(t)|2 on the at-tractor) are observed for coupling strengths greaterthan 0.32. Thus, there is a large range of cou-pling strengths where high-quality synchronizationcan be achieved despite the hyperchaotic nature ofthe system. Figure 7 shows projections of the fulleight-dimesional phase space of the coupled systemonto the v1m − v1s plane at three different coupling

strengths demonstrating the full range of observedbehavior: Fig. 7(a) c = 0, no synchronization oc-curs; Fig. 7(b) c = 0.28, synchronization is de-graded by attractor bubbling; Fig. 7(c) c = 0.36,high-quality synchronization is observed. Our ob-servations are consistent with those of Tamaševičiuset al. [1997b] who found that the transition to high-quality synchronization for this coupling scheme oc-curred for a coupling strength slightly higher thanthat expected based on the negativity of the trans-verse Lyapunov exponents.

Additional information concerning the bub-bling transition can be obtained by observing thescaling of |x⊥(t)|2MAX in the vicinity of the tran-sition. Venkataramani et al. [1996a] predictthat the transition can be “hard” (the bursts ap-pear abruptly with large amplitude as the couplingstrength is varied) or “soft” (the maximum burstamplitude increases continuously from zero), andthat the symmetry of the coupling has a funda-mental effect on the transition. From Fig. 6(a),it is seen that the transition is soft. For sucha transition and the one-way (asymmetric) cou-pling scheme used in our experiment, they predict


Fig. 6. (a) Experimentally observed degree of synchronization for v1 coupling. The squares indicate |x⊥|2RMS and the circles|x⊥|2MAX. We observe attractor bubbling in the range of coupling strengths 0.25 < c < 0.32.The straight line indicates|x⊥(t)|2MAX ∝ (cb − c) in the vicinity of the bubbling transition. (b) Largest transverse Lyapunov exponents as a function ofcoupling strength calculated for both a chaotic trajectory (solid line) and an unstable periodic orbit embedded in the chaoticattractor (dashed line). The vertical lines are a guide to the eye to facilitate the comparison between theory and experiment.The region in which the real parts of the eigenvalues of J are negative is indicated by the arrows on the right side of the figure.

Fig. 7. Projection of the full eight-dimensional phase space of the v1 coupled system onto the v1m–v1s plane at threedifferent coupling strengths. (a) c = 0, |x⊥(t)|2MAX = 254 volts2, |x⊥(t)|2RMS = 25 volts2, no synchronization; (b) c = 0.28,|x⊥(t)|2MAX = 16.1 volts2, |x⊥(t)|2RMS = 1.4 × 10−2 volts2, synchronization is degraded by attractor bubbling; (c) c = 0.36,|x⊥(t)|2MAX = 0.2 volts2, |x⊥(t)|2RMS = 1.1× 10−2 volts2, high-quality synchronization.

that |x⊥(t)|2MAX ∼ (cb − c)2, where cb is the cou-pling strength at which the transition occurs. Onthe contrary, we observe that |x⊥(t)|2MAX ∼ (cb−c),as illustrated by the solid straight line shown in the

figure. The fit between our data and the straightline is good in that the deviation from the straightline is comparable to the observed datapoint-to-datapoint variation, which is a reasonable estimate


of our experimental error. Our observation indi-cates the existence of a new type of bubbling transi-tion, or that the lowest-order nonlinear contributionto the transverse dynamics has an even symmetryeven though the coupling has an odd symmetry.

5. Synchronization Criteria

The experiments described in the previous section,taken together with previous research on attrac-tor bubbling, clearly demonstrate that the negativ-ity of the largest transverse Lyapunov exponent isnot a good predictor of high-quality synchroniza-tion in an experiment. Several alternative crite-ria for synchronization have been proposed, whichmake widely differing predictions for the range ofcoupling strength over which high-quality synchro-nization may be observed for our system. In thissection, we review these proposed criteria and ap-ply them to our experiment.

In the investigation of the stability of coupledoscillators, Ashwin et al. [1994] discovered the ex-istence of attractor bubbling and found that it oc-curs when any unstable periodic set embedded inthe chaotic attractor, such as an UPO, is character-ized by a positive transverse Lyapunov exponent.Thus, a criterion for high-quality, bubble-free syn-chronization is that the largest transverse exponentscharacterizing each of the unstable sets embeddedin the attractor must be negative. Since Lyapunovexponents are topological invariants, this criterionis independent of the choice of metric. Unfortu-nately, it is impossible to apply since there are typ-ically an infinite number of unstable sets embeddedin a chaotic attractor whose stability must be de-termined. There is some indication that the crite-rion might be applied in an approximate sense be-cause it has been suggested that a low-period UPOtypically yields the largest exponent [Hunt & Ott,1996]. However, it is now known that this conjec-ture does not hold for an attractor near a crisis[Zoldi & Greenside, 1998; Hunt & Ott, 1998; Yanget al., 1999].

To test this criterion, we have identified 11of the low-period UPO’s embedded in the hyper-chaotic attractor using the method described in[Parker & Chua, 1989]. Our test is only approx-imate in the sense that we do not determine thestability of all UPO’s. Note that x‖ = x⊥ = 0 is anunstable steady state of the system, but its neigh-borhood is never visited by the chaotic trajectory

and hence plays no role in attractor bubbling. For“v2-to-v1” coupling, the most transversely unstableset is a period-4 UPO (period of 17.07τ). The vari-ation of the largest transverse Lyapunov exponentfor this orbit as a function of coupling strength isshown in Fig. 5(b). This exponent is positive forall c. Thus, this criterion predicts attractor bub-bling over the entire range where the largest trans-verse exponent of the chaotic attractor is negative(0.66 < c < 1.62). These predictions are consistentwith our observations.

For v1 coupling, the most transversely unsta-ble set is a period-1 UPO (period of 5.26τ) whoselargest transverse exponent as a function of cou-pling strength is shown in Fig. 6(b). Attractor bub-bling is expected for coupling strengths in the range0.25 < c < 0.37 whereas it is observed experimen-tally in the smaller range 0.25 < c < 0.32. Thus,this criterion overestimates the range of attractorbubbling, which may be due to the fact that many ofthe other UPO’s become transversely stable aroundc = 0.32 or that the period-1 orbit is visited in-frequently. Numerical simulations suggest that theorbit is visited very infrequently if at all. For ourexperiments, this criterion is somewhat successfulat predicting attractor bubbling but its applicationsuffers from uncertainty knowing whether enoughunstable sets have been identified and whetherthe attractor visits the neighborhood of eachset.

Based on the same idea that the stability of theunstable sets govern the region of attractor bub-bling, Brown and Rulkov [1997] suggest an alterna-tive method for determining the stability of thesesets that is easier to implement. They derived a suf-ficient, but not necessary, condition for the asymp-totic stability using Gronwald’s theorem. Briefly,they decompose the matrix J = DF[xm(t)] − γKinto a time-independent part A ≡ 〈DF〉 − γK anda time-dependent part B(xm; t) ≡ DF[xm(t)] −〈DF〉, where 〈•〉 denotes a time average over thedriving trajectory. A trajectory is transversely sta-ble when

−Re[Λ1] > 〈‖P−1[B(xm; t)]P‖〉 , (6)

where Λ1 is the largest eigenvalue of A, P is amatrix of eigenvectors of A, and ‖ • ‖ denotes anorm whose choice is arbitrary. As for the pre-vious criterion, this condition must be evaluatedalong all unstable sets embedded in the attractorto determine the range of coupling strengths overwhich attractor bubbling can be avoided. Again,


this task is not trivial because of the infinite num-ber of unstable sets. In addition, the predictions ofthis criterion depend on both the choice of a normand the metric and hence it is not topologicallyinvariant.

We applied this criterion to our experimentby evaluating the stability of the 11 lowest periodUPO’s. For both coupling schemes and all couplingstrengths, the right-hand side of Eq. 6 is always ex-tremely large and the condition is never satisfied.Hence, this criterion correctly predicts that high-quality synchronization does not occur for v2-to-v1coupling. On the other hand, it predicts that high-quality synchronization cannot be achieved withv1 coupling, whereas our experiments show a largeregime of bubble-free synchronization. Apparentlythis sufficient, but not necessary, criterion is overlyconservative in this case.

To avoid the impracticality of applying the cri-teria described above, two others have been sug-gested. Pecora et al. [1995] attempt to guaranteehigh-quality synchronization by requiring all eigen-values of the matrix J have negative real parts at allpoints along the chaotic driving trajectory xm(t).The justification for this condition has some mo-tivation from the control literature [Brogan, 1974],even though the eigenvalues of a time-varying linearsystem cannot be used to determine the asymptoticbehavior of the solutions in general [Hale, 1969]. Asfor the previous case, this criterion depends on thechoice of metric.

Applying this criterion to our experiment bydetermining numerically the eigenvalues of J alonga chaotic trajectory of duration 106τ , we find thatthere are always points where the real part of aneigenvalue is positive for v2-to-v1 coupling. Thus,no high-quality synchronization is predicted for thiscoupling scheme, consistent with our experimen-tal observations. For v1 coupling, all eigenval-ues have negative real parts for c > 0.61. Thus,the predicted range of high-quality synchronizationis smaller than that observed in our experiments.This criterion provides a reasonable estimate of theregime of high-quality synchronization without theimpracticality of determining the transverse stabil-ity of an infinite number of unstable sets. We note,however, that this criterion has failed in other ex-periments [Blakely et al., 2000].

A second criterion was suggested by Gauthierand Bienfang [1996]. They argued that the ef-fect of noise in attractor bubbling is to repeatedlyset the system into a transient behavior where a

perturbation grows rapidly for a brief time beforedecaying. Using this reasoning, they proposed amethod for testing whether perturbations undergotransient growth. This criterion is based on the timederivative of the Lyapunov function L ≡ |δx⊥(t)|2.A sufficient condition that all perturbations decayto the manifold without transient growth is

dLdt

= 2δx⊥(t) · Jδx⊥(t) < 0 (7)

for all times. As in the two previous cases, thiscriterion depends on the choice of metric and norm.

For our experiment, Eq. (7) is simple enough tobe evaluated analytically. For the v2-to-v1 coupledoscillators,

dLdt

= (a− g′)δx21 − hδx23 − e(δx4 −

(d− b)2e

δx2

)2− [(b+ 1)g′ − c]δx1δx2

+

((d− b)2

4e− bg′

)δx22 , (8)

where δx⊥ = [δx1, δx2, δx3, δx4]T , and g′[v] =

αρId exp(αv). We see that dL/dt can be positiveregardless of the value of the coupling strength c byconsidering the case when δx2 = δx3 = δx4 = 0,and δx1 6= 0. In this situation, only the first termis nonzero and can be positive since g′ is near zerofor some part of the trajectory. Thus, the criterionpredicts no high-quality synchronization, consistentwith our experimental observations.

For the v1 coupled oscillators,

dLdt

= (a− g′ − c)δx21 − hδx23

− e(δx4 −

(d− b)2e

δx2

)2− (b+ 1)g′δx1δx2

+

((d− b)2

4e− bg′

)δx22 . (9)

We see that dL/dt can be positive regardless ofthe value of the coupling strength c by consider-ing the case when δx1 = δx3 = 0, and δx4 =(d − b)δx2/2e 6= 0. In this situation, only the lastterm remains and can be positive since g′ is nearzero for some part of the trajectory. Thus, no high-quality synchronization should be expected. In fact,a large range of high-quality synchronization is ob-served in the experiment and hence this criterion ismuch too conservative.


6. Discussion

In summary, we investigate attractor bubbling inan experimental system of coupled hyperchaotic os-cillators. Our results show clearly that attractorbubbling is present over all coupling strengths forone coupling scheme and hence high-quality syn-chronization is not possible for this scheme. On theother hand, a large regime of high-quality synchro-nization is observed for a second coupling scheme.In this case, we measure the scaling of the squareof the burst amplitude as a function of couplingstrength near the bubbling transition point. Thescaling does not fit into the known categories, po-tentially indicating a new type of transition. Weattempt to predict the regime of high-quality syn-chronization using several proposed synchronizationcriteria. The criterion proposed by Ashwin et al.[1994] appears to provide a reasonable estimate ofthe range of high-quality synchronization but it canonly be implemented approximately because it re-quires analysis of an infinite number of unstablesets. The criterion proposed by Pecora et al. [1995]provides the next best estimate and is much simplerto apply. However, it has been shown to fail in otherexperiments [Blakely et al., 2000]. The criteria pro-posed by Brown and Rulkov [1997] and Gauthierand Bienfang [1996] are both too conservative forour experimental system.

Our results have clear implications for securecommunications using synchronized chaos. Com-munication systems based on low-dimensional chaoshave proven to be susceptible to eavesdroppingmethods based on nonlinear prediction techniques[Short, 1997] or even simple return maps [Perez &Cerdeira, 1995]. To improve security, it has beensuggested that synchronized hyperchaos providesmore complex signal masking [Peng et al., 1996].However, faithful recovery of a masked message re-quires high-quality synchronization. We find thatattractor bubbling does occur over a wide range ofcoupling strengths for some coupling schemes butit is not highly prevalent for other schemes. There-fore, attractor bubbling may not hinder the designof communication systems based on hyperchaoticoscillators.

Unfortunately, our comparison of the proposedsynchronization criteria indicates that there is nosimple and generally reliable way to determine howto couple two oscillators to achieve high-quality syn-chronization. Additional research is needed to ad-dress this issue.

Acknowledgments

We gratefully acknowledge the financial support ofthe U. S. ARO through Grants DAAD19-99-1-0199and DAAG55-97-1-0308.

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Appendix A

Determining the Lyapunov Exponents

We determine the Lyapunov exponents of our os-cillators from experimental time series using themethod introduced by Eckmann et al. [1986] andextended by Brown et al. [1990]. Briefly, thismethod consists of the following steps: (1) for eachdata point xi, search the time series to obtain a ballof points in a small neighborhood of xi; (2) approx-imate the evolution of each neighborhood one timestep forward with a linear map; (3) determine theeigenvalues of a product of the matrices using the


QR decomposition to obtain the Lyapunov expo-nents. Note that all four variables are directly ac-cessible in our experiment and hence no time-delayphase space reconstruction is necessary. We mea-sure all four variables and represent the range ofvoltages from −2.82 volts to 2.82 volts in integerformat using 65536 integers. Each time series con-sisted of 200,000 points with a sample interval of0.21 µsec. At each point in the time series we searchfor 32 neighbors. Following Brown et al. [1990], weuse a Hanning window to reduce end effects.

We use a similar procedure to determine the ex-ponents of the circuit model. Rather than search-ing a time series for near neighbors, we generaterandom nearby points in phase space and evolvethem forward using the equations of the model.This allows us to obtain arbitrarily small and well-populated neighborhoods. The values reported inSec. 3 are obtained using 16 point neighborhoodswithin a distance of 10−5 V. The length of the modeltime series is 200,000 points.

attractor bubbling in coupled hyperchaotic...

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