Synchronization of coupled bistable chaoticsystems: experimental study

BY ALEXANDER N. PISARCHIK1,*, RIDER JAIMES-REATEGUI

2AND

J. HUGO GARCIA-LOPEZ2

1Centro de Investigaciones en Optica, Loma del Bosque 115,Lomas del Campestre, Leon 37150, Guanajuato, Mexico

2Universidad de Guadalajara, Centro Universitario de los Lagos,Lagos de Moreno 47460, Jalisco, Mexico

We carried out an experimental study of the synchronization of two unidirectionallycoupled Rossler-like electronic circuits with two coexisting chaotic attractors. Differentstages of synchronization are identified on the route from asynchronous motion tocomplete synchronization, as the coupling parameter is increased: intermittentasynchronous jumps between coexisting attractors; intermittent anticipating phasesynchronization; and generalized synchronization in the form of subharmonic entrainmentterminated by complete synchronization. All these regimes are analysed with time-series,power spectra and phase-space plots of the drive and response oscillators. Theexperimental study implicitly confirms the results of numerical simulations.

Keywords: synchronization; chaos; multistability; electronic circuits

On

*A

1. Introduction

Synchronization of coupled chaotic systems has been intensively investigated (e.g.Pecora & Caroll 1990; Pikovsky et al. 2001; Boccaletti et al. 2002 and referencestherein). At first, the notion of synchronization was understood to be thecoinciding of chaotic trajectories of coupled systems. Actually, this type ofsynchronization, which is observed in coupled identical systems, refers to completesynchronization (Murali & Lakshmanan 1994; Roy & Thornburg 1994; Sugawaraet al. 1994). Later, it was realized that complete synchronization is only aparticular type of synchronization. This type of synchronization has been widelyused for secure communication with chaos (Hayes et al. 1993; Argyris et al. 2005).In general, synchronization implies a certain relation between functionals of twoprocesses due to their interactions and can be observed not only in identicaloscillators. To describe synchronization of non-identical systems, Rulkov et al.(1995) introduced a concept of generalized synchronization, which means theexistence of some functional dependence between trajectories of coupledsubsystems. This type of synchronization also includes the case of subharmonicentrainment of periodic or chaotic oscillations with the fundamental frequency of

Phil. Trans. R. Soc. A (2008) 366, 459–473

doi:10.1098/rsta.2007.2103

Published online 6 August 2007

e contribution of 15 to a Theme Issue ‘Experimental chaos I’.

uthor for correspondence (apisarch@foton.cio.mx).

459 This journal is q 2007 The Royal Society

Rckey = a100K_LIN

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(a)

Figure 1. Electronic schemes of (a) drive, (b) response and (c) coupler circuits.

A. N. Pisarchik et al.460

a driving system (Kocarev & Parlitz 1996). Rosenblum et al. (1996) identifiedanother type of chaotic synchronization, phase synchronization, which implies aphase difference between chaotic oscillations that are locked within 2p. Later,other types of synchronization of coupled chaotic oscillators, such as lagsynchronization and anticipating synchronization, were also identified byRosenblum et al. (1997) and Voss (2000). All these types of synchronizationhave been detected in various experiments. Anticipating synchronization has

Phil. Trans. R. Soc. A (2008)

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Figure 1. (Continued.)

461Synchronization of chaotic systems

recently been observed by Ciszak et al. (2003) in semiconductor lasers, but only inthe presence of a time delay in coupling.

The majority of works on chaotic synchronization have been performed inmonostable systems. However, it is known that many nonlinear systems haveconcurrent and coexisting attractors for the same parameter values. Thedynamics of such multistable systems is defined by initial conditions. Recently,we have demonstrated numerically that the route from asynchronous motion tocomplete synchronization in two unidirectionally coupled bistable chaotic

Phil. Trans. R. Soc. A (2008)

xout

1.0kΩ

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Figure 1. (Continued.)

A. N. Pisarchik et al.462

systems is characterized by intermittent phase synchronization with anticipation(Pisarchik et al. 2006). This paper is devoted to the experimental study ofsynchronization in coupled bistable systems with coexisting chaotic attractors.As an example, we explore Rossler-like electronic circuits similar to those usedpreviously by Carroll & Pecora (1995) in their first experiments on chaoticsynchronization. We demonstrate different stages of chaotic synchronizationdepending on the coupling strength and discuss possible mechanisms underlyingthe synchronization phenomena.

2. Experimental set-up

We built the two identical electronic circuits shown in figure 1. One of them isthe drive circuit (figure 1a) and another one is the response circuit (figure 1b).We also constructed the electronic circuit shown in figure 1c to provideunidirectional coupling between the drive and response oscillators. Theparameters of all electronic components are indicated in figure 1a–c. Thesecircuits are an analogue version of the following model equations:

drive _x Z x

Kax1 Kbx2 Kx3

x 1 gx 2 0

gðx1Þ 0 Kx3

0B@

1CA; ð2:1Þ

response _y Z x

Kay1 Kb½y2 C3ðx2K y2Þ Ky3

y1 g½y2 C3ðx2K y2Þ 0

gðy1Þ 0 Ky3

0B@

1CAand ð2:2Þ

Phil. Trans. R. Soc. A (2008)

20 40 60 80 100

1

2

3

4

x 1 (V

)

Rc (kΩ)

Figure 2. Experimental bifurcation diagram of peak voltage x1 with resistance Rc as a control

463Synchronization of chaotic systems

gðx1; y1ÞZ0; if x 1; y1%3

mðx1; y1K3Þ; if x 1; y1O3

( ); ð2:3Þ

where xZ(x1, x2, x3) and yZ( y1, y2, y3) are the state vectors; _xZdx=dt,

_yZdy=dt, xZ104 sK1 is the time factor; aZ0.05, bZ0.5, mZ15, 32[0, 1] is thecoupling strength; and g is a variable parameter. The piecewise linear functionequation (2.3) is determined by the diode D at the entrance of the operationalamplifier in the x3 loop. The amplifier is switched on when the voltage exceeds3 V. The variable parameter gZR=RcK0:02, where RZ10 kU and Rc is acontrol resistance which can be varied between 27 and 200 kU.

parameter.

3. Dynamics of one isolated circuit

When the drive and response circuits are not coupled (eZ0), each of themexhibits complex behaviour defined by control resistance Rc and the initialconditions. Let us, first, consider the dynamics of the single circuit shown infigure 1a. The experimental bifurcation diagram of output voltage x1 versuscontrol resistance Rc is shown in figure 2. We record this diagram several timesby increasing and decreasing the control parameter. This procedure is equivalentto changing the initial conditions. For a large Rc, the behaviour of the singleoscillator is similar to that of the system discovered by Rossler (1977), with amultiplicative first-order nonlinearity in the x3 direction. The classical Rossleroscillator has a single saddle-node point located in the origin (0, 0, 0)T.

Similarly to the classical Rossler oscillator, our piecewise linear electroniccircuit undergoes a cascade of period-doubling bifurcations followed by chaoswhen Rc is decreased (figure 2). Several periodic windows can also bedistinguished in the diagram. At relatively large Rc (RcO36 kU), only Rossler-type chaos is observed. The chaotic attractor represents a so-called Rosslerfunnel (or an imperfect homoclinic orbit) around a saddle point. In our system,the lack of differentiability gives rise to coexistence of multiple attractors,cascades of period-adding bifurcations, period multiplication, homoclinic orbits,

Phil. Trans. R. Soc. A (2008)

A. N. Pisarchik et al.464

jumps to chaos and destruction of all periodic orbits. These bifurcations, whichare referred to as grazing bifurcations, are associated with portions of atrajectory being tangent to surfaces where the system is discontinuous. Such acomplex behaviour appears only for small Rc when a second saddle point arises.The homoclinic orbits become perfect when a condition derived by Shil’nikov(1994) is fulfilled. Homoclinic dynamics of the system equation (2.1) has beenanalysed by Pisarchik & Jaimes-Reategui (2005).

At relatively low values of the control parameter Rc, the bifurcation diagramrepresents a superposition of two different coexisting chaotic attractors, whichmerge in a crisis point at RcZ36 kU. To study the synchronization of multistablesystems, we fixed the control parameter atRcZ32 kU, where our circuits exhibit thecoexistence of two different chaotic attractors. Then, we chose the initial conditions(by switching on and off the power supply) for the drive and response circuits, so thattheir chaotic states would be different without coupling (3Z0). The experimentaltime series for the three output voltages of the uncoupled circuits are shown infigure 3, and the corresponding phase trajectories are shown in figure 4. It is clearthat the trajectories occupy different regions of the phase space. The naturalfrequencies of the two coexisting chaotic attractors are f1Z0.82 and f2Z1.022 kHz.

4. Synchronization stages

If two oscillators are identical and coupled by one variable, this variable can beconsidered as an external chaotic driver which increases the system dimension byone. In a unidirectionally coupled system with coexisting attractors, the responsesystem has no effect on the dynamics of the drive system, and hence the drivesystem always remains in the same attractor.

For very weak coupling 3!0.5%, the driving signal has no influence on theresponse system and the states are defined only by the initial conditions. The twooscillators can be considered to be isolated and their trajectories occupy differentregions of the phase space (see figures 3 and 4). However, a stronger coupling canchange the global structure of the phase space of the response system owing toincreasing dimension.

Let the initial states of the drive and response systems be chaotic attractorswith natural frequencies fdZf1 and frZf2. Synchronization can be characterizedquantitatively as a difference between phases of corresponding variables ofcoupled systems (Boccaletti et al. 2002) that can be measured from experimentaltime series as DfZ2p t krKt kd

fd, where t

kr and t kd are the times of the kth maxima

of the coupled variables for the response and drive oscillators, respectively. Whilethe coupling strength is increasing, the attractors in the response system canundergo different metamorphoses depending on the coupling strength.

(a ) Asynchronous intermittent jumps

The response system becomes sensitive to the drive as soon as the couplingexceeds 0.5%, when random intermittent jumps between coexisting attractorsoccur. Examples of the experimental time series and the corresponding powerspectra of the drive and response oscillations for different coupling strengths areshown in figure 5. When the response oscillator switches to the attractor similarto the drive system (figure 5a), driving frequency f1Zfd appears in the power

Phil. Trans. R. Soc. A (2008)

0 2 4 6 8

–6

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)y

(V)

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Figure 3. Experimental time-series of three variables for two different chaotic regimes in uncoupledcircuits: (a) x(V) and (b) y(V). The time traces of different variables are shown by the arrows.

–4

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Figure 4. Phase-space trajectories of two different chaotic regimes shown by open and closed circles.

465Synchronization of chaotic systems

Phil. Trans. R. Soc. A (2008)

–10

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er

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)x 2,

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)

Figure 5. (Caption opposite.)

A. N. Pisarchik et al.466

Phil. Trans. R. Soc. A (2008)

Figure 5. (Opposite.) Time-series (a,c,e,g,i ) and the corresponding power spectra (b,d,f,h, j )demonstrating different synchronization stages for coupling parameters (a,b) 3Z0.006, inter-mittent jumps between coexisting attractors, (c,d ) 3Z0.24, intermittent 1 : 2 phase synchroniza-tion, (e, f ) 3Z0.4, 1 : 2 phase synchronization, (g,h) 3Z0.7, intermittent 1 : 2–1 : 1 phasesynchronization and (i, j ) 3Z0.75, 1 : 1 or complete synchronization. The thin and thick linescorrespond, respectively, to the drive and response oscillators.

0.06 0.08 0.10 0.12 0.14 0.16 0.18–20

0

20

40

60

80

∆f

(a)

0.075 0.080 0.085 0.0904.0

4.5

5.0

5.5

6.0(b)

t (s)t (s)

Figure 6. Phase difference between drive and response oscillations as a function of time for3Z0.006. (a) Df increases almost linearly with time when the two systems stay in differentattractors. (b) Enlarged section demonstrating random fluctuations of Df on plateaus.

467Synchronization of chaotic systems

spectrum of the response system (middle part in figure 5b). Although at thisstage no synchronization is detected, such an intermittent behaviour is aprecursor of synchronization in multistable systems. The driving signal is toosmall to increase the system dimension and can be considered as external noisewhich just induces random switches between the coexisting attractors.

Figure 6a shows the phase difference Df as a function of time. Although thisdependence has a tendency to increase linearly with time, one can distinguishshort horizontal plateaus on which Df fluctuates in the 2p range (figure 6b). Onthe plateaus, the drive and response systems oscillate in antiphase, as seen in themiddle part of figure 5a.

(b ) Intermittent phase synchronization

A further increase in coupling parameter results in phase synchronization,which occurs only within the windows where the response system stays in theattractor similar to the attractor of the drive system. This synchronizationregime appears when 3O0.01. The examples of the time series and phasedifference Df(t) are shown in figure 7a,b. The fluctuation of Df on the plateaus isno longer random because its distribution is not Gaussian, rather it is Lorenzian.The duration of the synchronization windows is larger (figure 7b). Now, theresponse system is sensitive not only to some individual peaks of the driveoscillations, which induce switches between the coexisting attractors, but also tothe phase of oscillations while the systems stay in the similar attractors.

It is remarkable that intermittent phase synchronization is always accom-panied by anticipating synchronization, i.e. Df!0 inside the synchronous

Phil. Trans. R. Soc. A (2008)

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Df

Df

Figure 7. Time-series (a,c) and phase difference (b,d ) for coupling parameters (a,b) 3Z0.03,intermittent anticipating phase synchronization and (c,d ) 3Z0.12, combined attractor. The thin andthick lines indicate the time traces corresponding to the drive and response oscillators, respectively.

A. N. Pisarchik et al.468

windows. Anticipating phase synchronization is clearly seen from the time-seriesin figure 7a, in which the time trace of the response system anticipates thedriving trace. The averaged anticipation time is decreasing with increasing 3 andeventually it approaches 0, as soon as phase synchronization becomes perfect, i.e.when Df/0.

(c ) Combined attractor

For stronger coupling parameters (3O0.1), the dimension of the responsesystem is no longer the same as the dimension of the drive system; instead, itis higher. Owing to the increasing dimension, the global structure of the phasespace of the response system is completely different, and hence synchroniza-tion is hardly possible. The time-series and the phase dependence at this stageare shown in figure 7c,d; Df drifts in the vicinity of 0 (figure 7d ). Sometimes,very large phase slips can occur, when Df rises up to a half of the averagedperiod of the drive oscillations (as seen in figure 7c) although windows ofperfect phase synchronization (DfZ0) can appear as well. The dynamicalbehaviour is very similar to that of coupled non-identical (even structurallydifferent) systems.

(d ) Generalized synchronization

While the coupling strength is further increasing, the attractors in theresponse system can undergo a homeomorphism. This phenomenon wasdiscovered first by Afraimovich et al. (1986) and was later generalized by

Phil. Trans. R. Soc. A (2008)

469Synchronization of chaotic systems

Rulkov et al. (1995) to a system of coupled non-identical oscillators. In thecase of multistability, identical systems in different attractors can beconsidered as different systems. However, generalized synchronization ispossible. Mathematically, this may be formulated as follows. Generalizedsynchronization in coupled multistable systems occurs for the attractor Ai3R

m

of the drive system and a new homeomorphic attractor Bj3Rn of the response

system (iZ1, ., p and jZ1, ., q being the attractor numbers and m and nbeing the dimensions of the drive and response systems, respectively), if anattracting synchronization set MZfðx;yÞ2Bj!R

n : yZHðxÞg exists that isgiven by some function H : Bj/Ai3R

n and that possesses an open basinDIM su ch tha t limt/N jjyðtÞKHðxðtÞÞjjZ0 cðxð0Þ;yð0ÞÞ2D. Th i sdefinition also includes the case of subharmonic entrainment of periodic orchaotic oscillations with fundamental frequency f(Ai). For example, theentrainment with ratio f ðBjÞ : f ðAiÞZ1 : q (qO1) results from q attractors inthe response system.

The increasing coupling parameter above 20% gives rise to a period-doublingbifurcation for the chaotic attractor similar to the driving attractor that canlead to intermittent jumps to this new attractor. As a result, slips of period-doubling oscillations arise in the time-series (figure 5c), and frequency fd/2appears in the power spectrum (figure 5d ). The period-doubling oscillations arephase synchronized with the drive in the 1 : 2 ratio. We refer to this type ofsynchronization as intermittent period-doubling phase synchronization. As 3 isincreased, the period-doubling windows are enlarging and, eventually, the systembecomes perfectly phase synchronized in the period-doubling regime, i.e. period-doubling phase synchronization takes place as shown in figure 5e. Such afrequency entrainment is the manifestation of generalized synchronization.Recently, Pisarchik & Barmenkov (2005) observed a similar locking effect of thenatural frequency to the period-doubling frequency of driving oscillations in afibre laser with pump modulation.

(e ) Complete synchronization

Although the coupling is increasing above 60%, the period-doubling regimeis transforming into a period-one regime, while phase synchronization is beingconserved. The transition from 1 : 2 frequency locking to 1 : 1 frequencylocking is shown in figure 5g,h with the time-series and power spectra. Thechaotic attractor of the response system is converted into a chaotic attractorsimilar to the drive system. The peaks of the natural frequencies in the powerspectrum are broadened. This indicates a final stage on the route to completesynchronization that eventually occurs for 3O0.75 owing to the decreasingdimensions of the response system. Any trajectory of the response systemconverges to the same trajectory as the drive system, i.e. yðtÞKxðtÞZ0 fort/N, which results in the reducing number of coexisting attractors up to asingle attractor, identical to the attractor of the drive system. This case isshown in figure 5i, j.

It is worth mentioning that synchronization of coupled systems is closelyrelated to the problem of nonlinear control, although the goals may sometimes bedifferent (Kapitaniak 1994). In unidirectionally coupled systems, the drivesystem acts as a controller for the response system to attract its trajectory to a

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A. N. Pisarchik et al.470

desired trajectory defined by the controller. The synchronization problem forcoupled multistable systems can be considered as a problem of controllingmultistability or inter-attractor control.

5. Discussion

The synchronization phenomena described above can be interpreted in terms ofperiodic orbit structures of the driver and responder subsystems (Barreto et al.2000). Many papers describe synchronization of coupled chaotic systems in theframework of an invariant synchronization manifold I, which can easily be foundin coupled systems with symmetry, such as in our case. The results describedabove confirm the existence of I within a plane of symmetry for a wide range ofcoupling. The variables of the drive and response systems evolve identically on I

and hence the systems display identical synchrony. However, owing tomultistability in the coupled subsystems, I can become extremely complicatedand be destroyed as the coupling parameter is decreased.

In figure 8 we show the evolution of the attractor as the coupling parameter 3is decreased. Since the coupled oscillators are identical, the synchronizationmanifold I is simply the line xZy. It is invariant and attracting at 3Z1 andremains so for 3O0.75 (figure 8a) until a bubbling bifurcation occurs at criticalvalue 3b1z0.74. For 3!3b1, I is no longer invariant. Upon decreasing 3, thetrajectory first makes finite size intermittent short-lived excursions away from I

(figure 8b), i.e. a bubbling transition occurs. Before the bubbling transition takesplace, the chaotic attractor is asymptotically stable. At the bubbling bifurcation,the orbit within I loses transverse stability and after the bifurcation theattractor is only an attractor in the weaker sense of being a Milnor (measure)attractor. Such an emergence of the intermittent bursts of chaotic trajectoriesaway from the previously constrained attractor is the generic qualitativemanifestation of the bubbling transition in a physical system with an invariantmanifold, and this is often caused by the presence of symmetries in the system.A similar behaviour has been observed previously in monostable coupled chaoticsystems (Ashwin et al. 1994; Venkataramani et al. 1996). In our case of thesystem with two coexisting chaotic attractors, the bubbling bifurcation at whichthe orbit in I losses transverse stability is accompanied by a period-doubling(pitchfork) bifurcation at which a new chaotic attractor a is created outside of I.The trajectory spends a long time in the vicinity of I, but makes occasionalexcursions to another coexisting attractor. This leads to the attractor smearing(fattening) that can be seen in figure 8b. As 3 is further decreased, the chaoticattractor I itself becomes transverse unstable via a blowout bifurcation (Ott &Sommerer 1994), while the period-doubling attractor a becomes stable. One cansee in figure 8c that this new 1:2 synchronization manifold has a morecomplicated structure; the oscillations display generalized synchrony with a 1:2frequency ratio.

As the coupling is further reduced, the next desynchronization cycle isinitiated via the next bubbling transition, which appears at critical value3b2z0.32. The period-doubling orbit loses its transverse stability, makingexcursions away from a (figure 8d ). Finally, the attractor a itself becomestransverse unstable in the next blowout bifurcation that leads to the attractor

Phil. Trans. R. Soc. A (2008)

0

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–10 0 1 0–10 0 1 0

70%

75%

y2

y2

y2

y2

40%

24%

0. 6%

12%

3%

x2x2

30%

Figure 8. Attractor evolution in phase space as coupling is decreased for diffusively coupledcircuits. The coupling parameter 3 (in %) is indicated near each plot: (a) 75%, (b) 70%, (c) 40%,(d ) 30%, (e) 24%, ( f ) 12%, (g) 3% and (h) 0.6%.

471Synchronization of chaotic systems

smearing, as seen in figure 8e. For smaller 3 (figure 8f,g), the state variables maystill be functionally related, resulting in generalized synchronization. However,due to the coexistence of attractors with different, not multiple, frequencies, this

Phil. Trans. R. Soc. A (2008)

A. N. Pisarchik et al.472

function can be extremely complicated, and the identification of bubbling andblowout bifurcations is problematic. At very low couplings, no synchronization isobserved (figure 8h).

6. Conclusions

In this work we have studied experimentally the route from asynchronousbehaviour to complete synchronization in unidirectionally coupled oscillatorswith coexisting chaotic attractors. We have built two electronic Rossler-likecircuits and shown that the coexistence of attractors results from nonlinearitydue to a piecewise linear dependence in one of the system variables. We havedemonstrated experimentally different synchronization stages on the route fromasynchronous motion to complete synchronization. This scenario includesintermittent anticipating phase synchronization and period-doubling phasesynchronization. The latter is a particular case of generalized synchronization.We have demonstrated the bubbling transitions as the emergence of intermittentbursts of chaotic trajectories away from the previously constrained attractor to anewly created attractor intermittently synchronized with the drive. Such abehaviour of multistable coupled systems contains combined features inherent toboth identical and non-identical coupled systems. Furthermore, the coupledmultistable systems also have specific features that are not observed inmonostable coupled systems, for example two-state on–off intermittent phasesynchronization, a shift of the fundamental frequency of one of the chaoticattractors and period-doubling phase synchronization.

Similar bifurcation scenarios on the route to complete synchronization, as thecoupling parameter was increased, have also been found in the cases ofcoexistence of one periodic and one chaotic attractors and two periodicattractors with different fundamental frequencies. Although the dynamics inthese cases were not so rich as in the case of two coexisting chaotic attractors, thesynchronization features mentioned above are common. We believe that theresults described in this paper are quite general for a wide class of coupledbistable and multistable systems. The experimental results implicitly confirm theresults of numerical simulations (Pisarchik et al. in press). Synchronization ofbistable chaotic systems may be of interest for communication applications if theinformation is encrypted into proper switches between coexisting attractors orinto a qualitative change of a synchronization regime.

This work was supported by Consejo Nacional de Ciencia y Tecnologıa de Mexico (projectno. 46973) and PROMEP 103.5/05/2498. A.N.P. acknowledges support from the Spanish Ministryof Education and Science, project no. SAB2004-0038.

References

Afraimovich,V.S.,Verichev,N.N.&Rabinovich,M. I. 1986Stochastic synchronizationof oscillation indissipative systems. Radiophys. Quantum Electron. 29, 795–803. (doi:10.1007/BF01034476)

Argyris, A. et al. 2005 Chaos-based communications at high bit rates using commercial fibre-opticlinks. Nature 438, 343–346. (doi:10.1038/nature04275)

Ashwin, P., Buescu, J. & Stewart, I. 1994 Bubbling of attractors and synchronization of chaoticoscillators. Phys. Lett. A 193, 126–139. (doi:10.1016/0375-9601(94)90947-4)

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473Synchronization of chaotic systems

Barreto, E., So, P., Gluckman, B. J. & Schiff, S. J. 2000 From generalized synchrony to topologicaldecoherence: emergent sets in coupled chaotic systems. Phys. Rev. Lett. 84, 1689–1692. (doi:10.1103/PhysRevLett.84.1689)

Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L. & Zhou, C. 2002 The synchronization ofchaotic systems. Phys. Rep. 366, 1–101. (doi:10.1016/S0370-1573(02)00137-0)

Carroll, T. & Pecora, L. 1995 Nonlinear dynamics in circuits. Singapore: World Scientific Publishing.Ciszak, M., Calvo, O., Masoller, C., Mirasso, C. R. & Toral, R. 2003 Anticipating the response of

excitable systems driven by random forcing. Phys. Rev. Lett. 90, 204102. (doi:10.1103/PhysRevLett.90.204102)

Hayes, S., Grebogi, C., Ott, E. & Mark, A. 1993 Communicating with chaos. Phys. Rev. Lett. 70,3031–3034. (doi:10.1103/PhysRevLett.70.3031)

Kapitaniak, T. 1994 Synchronization of chaos using continuous control. Phys. Rev. E 50,1642–1644. (doi:10.1103/PhysRevE.50.1642)

Kocarev, L. & Parlitz, U. 1996 Generalized synchronization, predictability, and equivalence ofunidirectionally coupled dynamical systems. Phys. Rev. Lett. 76, 1816–1819. (doi:10.1103/PhysRevLett.76.1816)

Murali, K. & Lakshmanan, M. 1994 Drive-response scenario of chaos synchronization in identicalnonlinear systems. Phys. Rev. E 49, 4882–4885. (doi:10.1103/PhysRevE.49.4882)

Ott, E. & Sommerer, J. A. 1994 Blowout bifurcations: the occurrence of riddled basins and on–offintermittency. Phys. Lett. A 188, 39–47. (doi:10.1016/0375-9601(94)90114-7)

Pecora, L. M. & Carroll, T. L. 1990 Synchronization in chaotic systems. Phys. Rev. Lett. 64,821–824. (doi:10.1103/PhysRevLett.64.821)

Pikovsky, A., Rosenblum, M. & Kurths, J. 2001 Synchronization: a universal concept in nonlinearscience. New York, NY: Cambridge University Press.

Pisarchik, A. N.&Barmenkov, Y.O. 2005 Locking of self-oscillation frequency by pumpmodulation inan erbium-doped fiber laser. Opt. Commun. 254, 128–137. (doi:10.1016/j.optcom.2005.05.028)

Pisarchik, A. N. & Jaimes-Reategui, R. 2005 Homoclinic orbits in a piecewise linear Rossler circuit.J. Phys.: Conf. Ser. 23, 122–127. (doi:10.1088/1742-6596/23/1/014)

Pisarchik, A. N., Jaimes-Reategui, R., Villalobos-Salazar, J. R., Garcıa-Lopez, J. H. & Boccaletti, S.2006 Synchronization of chaotic systems with coexisting attractors. Phys. Rev. Lett. 96, 244102.(doi:10.1103/PhysRevLett.96.244102)

Pisarchik, A. N., Jaimes-Reategui, R., Villalobos-Salazar, J. R. & Garcıa-Lopez, J. H. In press.Synchronization of multistable systems. Int. J. Bifurcat. Chaos.

Rosenblum, M. G., Pikovsky, A. S. & Kurths, J. 1996 Phase synchronization of chaotic oscillators.Phys. Rev. Lett. 76, 1804–1807. (doi:10.1103/PhysRevLett.76.1804)

Rosenblum, M. G., Pikovsky, A. S. & Kurths, J. 1997 From phase to lag synchronization incoupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4196. (doi:10.1103/PhysRevLett.78.4193)

Rossler, O. E. 1977 Chaos in abstract kinetics: two prototypes. B. Math. Biol. 39, 275–289.Roy, R. & Thornburg Jr, K. S. 1994 Experimental synchronization of chaotic lasers. Phys. Rev.

Lett. 73, 2009–2012. (doi:10.1103/PhysRevLett.72.2009)Rulkov, N. F., Sushchik, M. M., Tsimring, L. S. & Abarbanel, H. D. I. 1995 Generalized

synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–983.(doi:10.1103/PhysRevE.51.980)

Shil’nikov, L. P. 1994 Chua’s circuit: rigorous and future problems. Int. J. Bifurcat. Chaos 4,489–520. (doi:10.1142/S021812749400037X)

Sugawara, T., Tachikawa, M., Tsukamoto, T. & Shimizu, T. 1994 Observation of synchronizationin laser chaos. Phys. Rev. Lett. 72, 3502–3505. (doi:10.1103/PhysRevLett.72.3502)

Venkataramani, S. C., Hunt, B. R., Ott, E., Gauthier, D. J. & Bienfang, J. C. 1996 Transitions tobubbling of chaotic systems. Phys. Rev. Lett. 77, 5361–5364. (doi:10.1103/PhysRevLett.77.5361)

Voss, H. U. 2000 Anticipating chaotic synchronization. Phys. Rev. E 61, 5115–5119. (doi:10.1103/PhysRevE.61.5115)

Phil. Trans. R. Soc. A (2008)