Eﬀect of noise on generalized synchronization of chaos...

Eur. Phys. J. B 82, 69–82 (2011) DOI: 10.1140/epjb/e2011-11019-1

Effect of noise on generalized synchronization of chaos: theoryand experiment

O.I. Moskalenko, A.E. Hramov, A.A. Koronovskii and A.A. Ovchinnikov

Eur. Phys. J. B 82, 69–82 (2011)DOI: 10.1140/epjb/e2011-11019-1

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Effect of noise on generalized synchronization of chaos: theoryand experiment

O.I. Moskalenkoa, A.E. Hramov, A.A. Koronovskii, and A.A. Ovchinnikov

Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, 410012 Saratov, Russia

Received 29 December 2010 / Received in final form 4 April 2011Published online 22 June 2011 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2011

Abstract. The influence of noise on the generalized synchronization regime in the chaotic systems withdissipative coupling is considered. If attractors of the drive and response systems have an infinitely largebasin of attraction, generalized synchronization is shown to possess a great stability with respect tonoise. The reasons of the revealed particularity are explained by means of the modified system approach[A.E. Hramov, A.A. Koronovskii, Phys. Rev. E 71, 067201 (2005)] and confirmed by the results ofnumerical calculations and experimental studies. The main results are illustrated using the examples ofunidirectionally coupled chaotic oscillators and discrete maps as well as spatially extended dynamicalsystems. Different types of the model noise are analyzed. Possible applications of the revealed particularityare briefly discussed.

1 Introduction

Synchronization is one of the most relevant directions ofnonlinear dynamics attracting great attention of modernscientists [1,2]. The interest to it is connected both with alarge fundamental significance of its investigation [1] anda wide practical applications, e.g. for the transmission ofinformation [3–18], diagnostics of dynamics of some bio-logical systems [19–24], control of chaos in the microwavesystems [25–29], etc.

Several types of the synchronous chaotic system behav-ior are traditionally distinguished. They are phase [1,30],generalized [31,32], lag [33,34], complete [35,36], timescale [28,37,38] synchronization and others.

One of the most important problems connected withthe study of the chaotic systems is the influence ofnoise on their behavior including the synchronous regimearising [39–50]. Noise is known to influence on sys-tem dynamics in different ways. In particular, in caseof complete synchronization of coupled chaotic oscilla-tors, noise may induce intermittent loss of synchroniza-tion due to local instability of the synchronization mani-fold [51,52]. At the same time, both periodic and chaoticnon-coupled identical dynamical systems subjected to acommon noise may achieve complete synchronization at alarge enough intensity [42,48,53–55]. Such phenomenon iscalled noise-induced synchronization regime. In the caseof phase synchronization of coupled oscillators noise caninduce phase slips in phase-locked periodic and chaoticoscillators [56,57]. On the other hand, noise can playa constructive role at phase synchronization enhancing

a e-mail: [email protected]

the synchronous regime below the threshold of phasesynchronization [45,58]. Nevertheless, for almost all typesof chaotic synchronization (phase synchronization, com-plete synchronization, lag synchronization) noise appearsto obstruct the synchronous motion and increase the valueof the coupling strength between oscillators correspondingto the onset of synchronization.

At the same time, effect of noise on the generalized syn-chronization regime is investigated poorly enough. As anexception one can refer to the paper [50] where the effectof noise on generalized synchronization in two character-istically different chaotic oscillators have been considered.In this case the effect of noise can be system dependent,i.e. common noise can either induce/enhance or destroythe generalized synchronization regime.

Systems studied in [50] are close to an attractor crisisbifurcation [59]. In this case external noise of small inten-sity may result in creation of a new chaotic attractor witha qualitatively different topology that results in chang-ing of the system behavior in the presence of noise. Inpresent paper we dwell for the first time upon the behav-ior of the generalized synchronization regime in systemswhich attractors are far away from the boundary bifurca-tion crisis or their basins of attraction are infinitely large.We report for the first time theoretical and experimen-tal results of the influence of noise on the threshold ofthe generalized synchronization regime in identical sys-tems with mismatched parameters whose attractors sat-isfy the conditions mentioned above. As it would be shownbellow, in this case the generalized synchronization on-set is almost independent on the noise intensity, i.e. thesynchronous regime appears in the absence and presenceof noise practically for the same values of the coupling

70 The European Physical Journal B

parameter strength. At the same time, if the system at-tractors are far away from the boundary bifurcation crisis,with their basins of attraction being limited, the stabilityof the generalized synchronization regime with respect toexternal noise would be observed in the large, but limitedrange of the noise intensity. The same findings also remainto be correct for the systems with the infinitely large basinof attraction, although the causes of such type of behaviorare different.

Revealed peculiarity of the behavior of the boundaryof the GS regime in the presence of noise could be usedin many relevant circumstances, e.g. for the secure trans-mission of information through the communication chan-nel [18,60], in the medical, physiological [61,62] and otherpractical applications where the level of natural noise issufficient.

The structure of the paper is as follows. Section 2contains brief description of the generalized synchroniza-tion regime, its methods for detection and mechanisms ofits arising both in the cases of the absence and presenceof noise. The reasons of the stability of the generalizedsynchronization regime with respect to noise are also dis-cussed in this section. Section 3 presents results of nu-merical simulation of the influence of noise on the thresh-old of the synchronous regime arising in several systemswith discrete and continuous time as well as spatially ex-tended systems demonstrating spatio-temporal chaos. InSection 4 we describe the experimental setup for the ob-servation of the generalized synchronization regime in thepresence of noise in the electronic chaotic circuit and givethe results obtained by means of it. Final discussions andremarks are given in conclusions.

2 Generalized synchronization regime

The generalized synchronization regime (GS) in two uni-directionally coupled chaotic oscillators with continuous

x(t) = G(x(t),gd)u(t) = H(u(t),gr) + εP(x(t),u(t)), (1)

or discrete

xn+1 = G(xn,gd)un+1 = H(un,gr) + εP(xn,un), (2)

time means the presence of a functional relation

u = F[x] (3)

between the drive x (x(t) or xn) and response u (u(t)or un) system states [31,63], i.e. in the GS regime theresponse system behavior converges to the synchronizedstate independently on the choice of initial conditionsbelonging to the same basin of attraction. In equa-tions (1), (2) x and u are the state vectors of the drive andresponse systems, respectively; G and H define the vector

fields of interacting systems, gd and gr are the control pa-rameter vectors, P denotes the coupling term, and ε is thescalar coupling parameter. Typically, the analytical formof the relation F[·] in (3) can not be found in most cases.Depending on the character of this relation – smooth orfractal – GS can be divided into the strong and the weakones [63], respectively. It is also important to note that thedistinct dynamical systems (including the systems withthe different dimension of the phase space) may be usedas the drive and response oscillators to achieve the GSregime.

To detect the GS regime both in flow systems and dis-crete maps the auxiliary system method [64] is frequentlyused. According to this method the behavior of the re-sponse system u is considered together with the auxiliarysystem v (v(t) in the case of the flow systems and vn ifmaps are considered). The auxiliary system is equivalentto the response one by the control parameter values, butstarts with other initial conditions belonging to the samebasin of chaotic attractor (if there is the multistability inthe system). If GS takes place, the system states u and vbecome identical after the transient is finished due to theexistence of the relations u = F[x] and v = F[x]. Thus,the coincidence of the state vectors of the response andauxiliary systems v ≡ u is considered as a criterion of theGS regime presence.

It is also possible to compute the conditional Lyapunovexponents to detect the presence of GS [63]. In this caseLyapunov exponents are calculated for the response sys-tem, and since the behavior of this system depends on thedrive system these Lyapunov exponents are called con-ditional. Negativity of the largest conditional Lyapunovexponent is a criterion of the GS presence in unidirection-ally coupled dynamical systems [63].

Methods for the GS regime detection described abovecould be easily applied for the investigation of the influ-ence of noise on the GS regime onset, with all criteriaof the GS regime appearance remaining unchangeable. Inother words, the auxiliary system method and the con-ditional Lyapunov exponent calculation may be used todetect the existence of this type of synchronization bothin flow systems and discrete maps in the presence of noise.

At the same time, taking into account the fact thatthe definition of the GS regime and methods for its detec-tion are the same both for systems with continuous anddiscrete time1, further in this Section we consider the GSregime onset in flow systems. Several peculiarities con-nected with the GS regime onset in discrete maps will bediscussed in Section 3.1.

GS is known to take place in systems with the differenttypes of coupling, the dissipative and non-dissipative ones.In the case of dissipatively coupled identical flow dynam-ical systems with mismatched parameters equations (1)can be rewritten as

x(t) = H(x(t),gd)u(t) = H(u(t),gr) + εA(x(t) − u(t)), (4)

1 Moreover, flow systems may be reduced to discrete maps,with all types of the synchronous behavior being connectedwith each other [65].

O.I. Moskalenko et al.: Effect of noise on generalized synchronization of chaos: theory and experiment 71

where A = {δij} is the coupling matrix, δii = 0 or δii = 1,δij = 0 (i �= j). The mechanisms of the GS regime arisingin systems with the dissipative coupling can be revealedby the modified system approach firstly proposed in ourprevious work [66]. Due to such approach the dynamicsof the response system may be considered as the non-autonomous dynamics of the modified system

um(t) = H′(um(t),gr, ε) (5)

where H′(u(t)) = H(u(t)) − εAu(t), under the externalforce εAx(t):

um(t) = H′(um(t),gr, ε) + εAx(t). (6)

One can easily see that the term −εAx(t) brings the ad-ditional dissipation into the system (5). The phase flowcontraction is characterized by means of the vector fielddivergence. Obviously, the vector field divergences of themodified and the response systems are related with eachother as

div H′ = div H− ε

N∑

i=1

δii (7)

(where N is the dimension of the modified system phasespace), respectively. So, the dissipation in the modifiedsystem is greater than in the response one and it increaseswith the growth of the coupling strength ε.

The GS regime arising in (4) may be considered as aresult of two cooperative processes taking place simulta-neously. The first of them is the growth of the dissipationin the system (5) and the second one is an increase ofthe amplitude of the external signal. Both processes arecorrelated with each other by means of parameter ε andcan not be realized in the coupled oscillator system (4)independently. Nevertheless, it is clear, that an increaseof the parameter ε in the modified system (5) results inthe simplification of its behavior and the transition fromthe chaotic oscillations to the periodic ones [66]. More-over, if the additional dissipation is large enough the sta-ble fixed state may be realized in the modified system. Onthe contrary, the external chaotic force εAx(t) tends tocomplicate the behavior of the modified system and im-pose its own dynamics on it. The GS regime is known totake place when own chaotic dynamics of the autonomousmodified system is suppressed [66]. At the same time, theresponse system demonstrates chaotic oscillations due tothe external signal coming from the drive system.

So, the stability of the GS regime is defined primarilyby the properties of the modified system. Adding of noisedoes not change the characteristics of the modified sys-tem (5) and does not seem to affect the threshold of theGS regime onset. Therefore, the GS regime should exhibitthe stability with respect to noise in the wide range of thenoise intensities. At that, it should be noted that the char-acteristics of noise does not matter and the similar stabil-ity of the GS regime would be observed both for additiveand multiplicative noise with different characteristics.

To verify the correctness of the statement men-tioned above we use the conditional Lyapunov exponent

method. We consider the evolution of both the refer-ence state of the response oscillator u(t) and the per-turbed one v(t) = u(t) + Δ(t) being close to each other(i.e., |Δ(t)| � 1). The conditional Lyapunov exponentsλr

i (i = 1, . . . , N) are determined by the exponential in-crease/decrease of the small perturbation Δ(t). To takeinto account the noise influence we have added the noiseterms ζ, ξ ∈ R

N into equations (4) describing the dynam-ics of the drive and response systems:

x(t) = H(x(t),gd) + ζ(t)u(t) = H(u(t),gr) + εA(x(t) − u(t)) + ξ(t). (8)

In equation (8) the stochastic processes are supposed to bedifferent for the more complicated case to be considered.

In this case the dynamics of the auxiliary (perturbed)system would be given by

v(t) = H(v(t),gr) + εA(x(t) − v(t)) + ξ(t). (9)

Note, the concept of the generalized synchronization andthe auxiliary system approach requires the identity ofthe signals driving both the response and auxiliary sys-tems. This requirement means that noise must be alsoidentical for the response and auxiliary system. In otherwords, the control parameter values and the noise signalsin the response and auxiliary systems should be fully iden-tical whereas initial conditions for them should be chosendifferent.

The equation determining the evolution of the pertur-bation Δ(t) may be obtained as follows

Δ(t) = H(v(t),gr) − H(u(t),gr) − εAΔ(t). (10)

Taking into account that v(t) = u(t) + Δ(t) and|Δ(t)| � 1, one can write

H(v(t),gr) ≈ H(u(t),gr) + JH(u(t),gr)Δ(t) (11)

(where J is a Jakobian matrix), and, as a consequence

Δ(t) = (JH(u(t)) − εA)Δ(t) = JH′(u(t))Δ(t), (12)

equation (12) is the variational equation for the com-putation of the conditional Lyapunov exponents of theresponse system describing by equation (8) as well asequation (4). Therefore, one can conclude that the largestconditional Lyapunov exponents (determining the thresh-old of the GS regime onset) would behave in the similarway both in the absence and presence of noise. Therefore,the threshold of GS should not considerably depend onthe noise intensity, whereas the GS regime should exhibitthe stability to the noise influence. Note, also, that thevector state u(t) in equation (12) depends on the noisesignals, and, therefore, the largest conditional Lyapunovexponents obtained for the cases with and without noiseare, however, not equivalent. As a consequence, the greatintensities of noise may change the stability properties ofthe modified system that may result in the variation ofthe value of the threshold of the GS regime.


It should be noted that onset of the GS regime issimilar to the last one for the cases of complete (CS)(identical) and lag (LS) synchronization. Such types ofthe synchronous chaotic system behavior could be consid-ered as partial cases of GS and they correspond to thestronger forms of such regime [63]. It is clear that themodified system approach could be applied for revealingthe mechanisms resulting in the synchronous regime onseteven in the presence of noise. At the same time, it shouldbe noted that even for identical dynamical systems withequal values of the control parameters GS regime arises abit earlier than the CS one [63,67]. As it would be shownbellow in Section 3.2, external noise added to the driveand response could destroy the CS (or LS) regime but itdoes not destruct the GS regime itself. Therefore, the sta-bility of the CS and LS regimes is less strong than for theGS one.

3 Influence of noise on the GS regime onsetin sample chaotic systems: numericalcalculations

To illustrate the stability of the GS regime with respectto noise we consider numerically three different pairs ofunidirectionally dissipatively coupled chaotic dynamicalsystems being capable to demonstrate the GS regime.As such model systems we have selected (i) systemswith discrete time – two unidirectionally coupled logisticmaps, (ii) chaotic oscillators – two unidirectionally cou-pled Rossler systems; (iii) spatially extended dynamicalsystems – unidirectionally coupled one-dimensional com-plex Ginzburg-Landau equations.

3.1 Logistic maps

We start our consideration with the GS regime arising intwo unidirectionally coupled logistic maps with additivenoise term:

xn+1 = f(xn, λx),yn+1 = f(yn, λy) + ε(f(xn, λx) + Df(ξn, λx) − f(yn, λy)),

(13)where f(x, λ) = λx(1 − x), ε < 1. Here λx,y are the con-trol parameter values of the drive and response systems,respectively, ε characterizes the coupling strength betweensystems, ξn is the stochastic process which probabilitydensity is distributed uniformly on the interval [0; 1], D de-fines the intensity of added noise.

Although the systems with the discrete time are thespecific class of dynamical systems, they are closely in-terrelated with the flow systems [68], with types ofthe chaotic synchronous motion corresponding with eachother in maps and flows [65]. Nevertheless, there are alsodifferences between these types of chaotic dynamical sys-tems. One of them is the type of coupling between oscilla-tors. Typically, for the logistic maps the coupling term isintroduced in the same way as it has been done in equa-tion (13) instead of the linear difference of the vectors

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4

ε

D20.1 14.05 10.5 8.0 SNR, [dB]

Fig. 1. Dependence of the threshold of the GS regime onset intwo unidirectionally coupled logistic maps (13) on the intensityof noise (the SNR values corresponding to the noise intensitiesare also shown) for different values of the control parameters:λx = 3.75, λy = 3.75 (•), λx = 3.75, λy = 3.79 (�), λx = 3.75,λy = 3.9 (�). Critical values of the noise intensity Dc, up towhich the GS regime in system (13) is observed, are markedby arrows.

(like in Eq. (4)), since for the maps it is this kind of termsthat provides the dissipative type of coupling [49,63,66]).Additionally, here and later the noise signal is introducedin the coupling term to emulate a natural noise added inthe communication channel [60]. To detect the GS regimein such system we have computed conditional Lyapunovexponents with further refinement of the threshold valuesby the auxiliary system method described above.

The dependence of the GS regime onset on the noiseintensity D for different values of the control parametersλx,y is shown in Figure 1. On the horizontal axis the sig-nal to noise ratios (SNR, [dB]) corresponding to thesenoise intensities are also indicated2. One can easily seethat the threshold value of the coupling parameter ε isalmost independent on the intensity of noise D ∈ [0; Dc]where Dc shown in Figure 1 by arrows, depends on thecontrol parameter values, Dc = Dc(λx, λy). For the se-lected values of the control parameters Dc ∈ [0.38; 0.44](SNR ∈ [8.5; 7.2] dB, respectively), i.e. the GS regimein unidirectionally coupled logistic maps (13) is stable tonoise up to the power of noise comparable with the chaoticsignal one.

To explain the reasons of stability of the GS regimewith respect to external noise we use the modified sys-tem approach described in Section 2. At the same time,due to the fact that the theory of the stability of the GSregime to noise proposed in Section 2, is applicable to flowsystems, and the noise added in system (13) is multiplica-tive, there are several peculiarities to be discussed bellow.Therefore, we use the modified system approach with re-gard to the system with discrete time and consider the

2 Here and later in the paper the SNR value has been com-

puted in traditional way, i.e. SNR = 10 lgPsign

Pnoise, where Psign

is a power of chaotic signal, Pnoise is a power of noise affectedthe chaotic system [69]. The power of the signal x(t) (indepen-dently of the fact whether it is deterministic or stochastic) onthe time interval [0; T ] has been computed by its time realiza-

tion, i.e. P =∫ T

0x2(t)dt.


Fig. 2. Bifurcation diagrams of the modified logistic map (15)in the absence (a) and presence (b) of noise (the noise is in-troduced in system (15) in the same way as in Eq. (13)). Thecontrol parameter λy = 3.79 in both cases, the noise intensityD = 0.1 in (b). The coupling parameter values εGS = 0.32corresponding to the GS regime (obtained by means of condi-tional Lyapunov exponent computation) are marked by arrowin both cases.

modified logistic map:

zn+1 = fm(zn, λy) =(1 − ε)f(zn, λy). (14)

One can see that the modified system (14) may be rewrit-ten in the form

zn+1 = azn(1 − zn), (15)

where a = λy(1 − ε). It is clearly seen that the term−εf(zn, λy) brings additional dissipation in system (14).The local phase volume contraction is characterized bymeans of the modulus of the derivative

∣∣∣∣dzn+1

dzn

∣∣∣∣ = (1 − ε)∣∣f ′

zn(zn, λy)

∣∣ (16)

where the modulus of multiplier |f ′zn

(zn, λy)| = λy |1−2zn|characterizes the phase volume contraction in the au-tonomous response system. The case of |dzn+1/dzn| = 1corresponds to the non-dissipative dynamics whereas|dzn+1/dzn| = 0 relates to the infinitely large dissipation.One can see that, as in the case of flow systems, the dis-sipation in the modified system is greater than in the re-sponse one and it increases with the growth of the couplingstrength ε, (0 < ε < 1). Bifurcation diagram characteriz-ing its behavior with the increase of ε-parameter is shownin Figure 2a. The value of parameter ε corresponding tothe onset of the GS regime (obtained by means of con-ditional Lyapunov exponent computation, see Sect. 2) ismarked by arrow. One can see that for a coupling pa-rameter strengths corresponding to the onset of the GSregime in system (13), in full agreement with the argu-ments discussed in [66], the modified system (15) demon-strates the periodic oscillations. External noise does not

almost change the characteristics of the modified systemand, therefore, it does not affect the threshold of the GSregime arising. Bifurcation diagram of the modified lo-gistic map in the presence of noise of intensity D = 0.1 isshown in Figure 2b. The noise is introduced in system (15)in the same way as in equation (13), i.e.

zn+1 = fm(zn, λy) + εDf(ξn, λx) (17)

to provide the same noise influence as in the coupled logis-tic maps. The level of noise is quite sufficient in compari-son with the signal amplitude what is clearly seen from thekind of bifurcation diagram. At the same time, it is easyto see that noise does not shift the bifurcation points inthis case but only leads to a noisiness of the system regime.Therefore, in the considered case one can say that, despiteof the large amplitude, the external noise does not affectthe threshold of the GS regime onset. The further increaseof the noise intensity D > Dc results in the runaway ofthe representation point to infinity.

The reasons of the jump of the representation pointto infinity can be explained in the following way. Logisticmap in autonomous regime

xn+1 = f(xn, λ), (18)

is known to have a finite basin of attraction, i.e., de-pending on the choice of the initial conditions, for thevalues of the control parameter λ mentioned above itdemonstrates either chaotic regime or the jump of rep-resentation point to infinity [70]. To provide the chaoticregime in system (18) we have to specify initial condi-tion in range x0 ∈ [0; 1], with the representation pointremaining in this range during the evolution of the sys-tem for an infinitely long time, at that the maximal valueof f(xn) = fmax = λ/4 would be achieved if xn = 1/2. Atthe same time, it is clear that external noise could makeit go out the range mentioned above.

The similar effect takes place for systems (13). Onecan estimate the intensity of noise Dc corresponding tothe jump of representation point of the response systemto infinity. For this purpose we consider the behavior of thedrive and response systems of equation (13). First equa-tion corresponds to the drive system and is not affected bythe influence of external noisy or chaotic signal. Thereforethe most probable value of f(xn) = 〈fx〉 where 〈fx〉 is astatistical mean of f(xn) (due to the properties of the au-tonomous logistic map). The maximal values of f(yn) andf(ξn) would be equal to fmax because of the uniform char-acter of the probability distribution of the random valueξn and the arguments discussed above. Due to the factthat a random variable ξn could not be negative the jumpof the representation point from the range [0; 1] could beperformed only through a right boundary of such range.Therefore the maximal value of yn+1 is 1. Substituting allquantities into the second equation of (13) and assumingε = εc (εc corresponds to the threshold value of the GSregime onset without noise) we can estimate the approxi-mate values of the noise intensity Dc up to which the jumpof representation point to infinity does not take place. Our


calculations show that

Dc ≈ 4 − 4εc〈fx〉 − (1 − εc)λy

εcλx, (19)

i.e. Dc ≈ 0.5 for the control parameter valuesλx = λy = 3.75, Dc ≈ 0.48 for λx = 3.75, λy = 3.79and Dc ≈ 0.42 for λx = 3.75, λy = 3.9, that agreeswell with the results of direct numerical calculations.Therefore, the GS regime for logistic maps, having alimited basin of attraction, exhibit the stability withrespect to noise in the large, but limited range of thenoise intensity.

In the considered case the GS regime destruction isconnected with the jump of representation point to infin-ity, which could be considered as an attraction of it to thesecond coexisting attractor being at the infinity [71]. Note,if the coexisting attractor was characterized by the limitedbasin of attraction, depending on the type of the regimebeing realized in the response system (and, correspond-ingly, to the second attractor), the increase or decrease ofthe threshold value of the GS regime would be observedwith the growth of the noise intensity [50].

The another important question is the stability of GSwith respect to the external noise in the case when sta-tistically independent noise sources affect the drive andresponse systems

xn+1 = f(xn, λx) + εDf(ζn, λx),yn+1 = f(yn, λy) + ε(f(xn, λx)

+Df(ξn, λx) − f(yn, λy)), (20)

where ζn is a stochastic process with the probability den-sity distributed uniformly in [0; 1]-range. Applying thearguments similar to the last one described for the sys-tem (13) to system (20), we can estimate the intensity ofnoise Dd

c corresponding to the jump of the representationpoint of the drive system to infinity. It is clear that due tothe absence of the dissipative term in the drive system thejump of representation point in it would take place for aless values of the noise intensity than for the response one.In this case the GS regime is stable to the noise influenceuntil D < Dd

c , where

Ddc ≈ 1 − λx/4

εcλx/4≈ 0.2. (21)

The further increase of the noise intensity D > Ddc re-

sults in the chaotic regime destruction in the drive system.Therefore, the values D > Dd

c are unapplicable for (20),since the jump of the representation point towards infinityin the drive system is observed.

Numerical calculations confirm the results obtainedanalytically. In Figure 3 dependencies of the critical valuesof the coupling parameter strength corresponding to theGS regime arising for different values of the control pa-rameters are shown (the SNR values are also indicated inthe second horizontal axis). As in the case of the absenceof noise in the drive system external noise does not affectthe threshold value of the GS regime onset. The very sim-ilar result is obtained in the case when both the drive and

0.2

0.3

0.4

ε

0 0.05 0.10 0.15 D26.0 19.8 16.1 SNR, [dB]

Fig. 3. Dependence of the threshold of the GS regime on-set in two unidirectionally coupled logistic maps (20) in thepresence of noise both in the drive and response on its in-tensity (the SNR values corresponding to the noise intensitiesare also shown) for different values of the control parameters:λx = 3.75, λy = 3.75 (•), λx = 3.75, λy = 3.79 (�), λx = 3.75,λy = 3.9 (�).

response systems are under the influence of the commonnoise source, i.e., ξn ≡ ζn.

It should be noted that the GS regime is also robustin the limited range against the perturbations in the con-trol parameters by noise. Therefore, one can conclude thatfor unidirectionally dissipatively coupled systems with dis-crete time the GS regime would exhibit the stability withrespect to noise.

3.2 Rossler systems

As a second example we consider two unidirectionally cou-pled flow Rossler oscillators:

x1 = −ωxx2 − x3 + D1ζ,

x2 = ωxx1 + ax2,

x3 = p + x3(x1 − c),u1 = −ωuu2 − u3 + ε(x1 + D2ξ − u1),u2 = ωuu1 + au2,

u3 = p + u3(u1 − c), (22)

where x(t) = (x1, x2, x3)T and u(t) = (u1, u2, u3)T arethe vector-states of the drive and response systems, re-spectively, a = 0.15, p = 0.2, c = 10, ωx and ωu = 0.95are the control parameter values, ε is a coupling param-eter. The parameters ωx,u define the natural frequenciesof the drive and response system oscillations. The termsD1ζ, D2ξ simulate the external noise influenced the driveand response systems. Here ξ and ζ are statistically in-dependent stochastic Gaussian processes described by thefollowing probability distribution

p(ξ) =1√2πσ

exp(− (ξ − ξ0)2

2σ2

), (23)

where ξ0 = 0 and σ = 1.0 are the mean value and variance.Parameters D1,2 define the intensities of the noise addedin the drive and response systems, respectively.

To integrate the stochastic equations (22) we haveused the four order Runge-Kutta method adapted for the


0

0.05

0.1

0.15

0.2

ε

1 2 4 8 16 32 64 128 256 D17.5 11.5 5.4 -0.5 -6.5 -12.5 -18.6 -24.6 -30.6 SNR, [dB]

Fig. 4. Dependence of the boundary value corresponding tothe GS regime arising in two unidirectionally coupled Rosslersystems with additive stochastic term (22) on the noise inten-sity D (the SNR values corresponding to the noise intensitiesare also shown) for different values of the drive system param-eter ωx: ωx = 0.99 (•), ωx = 0.95 (�), ωx = 0.91 (�). Thecritical value of the noise intensity Dc up to which the bound-ary value of the GS regime does not almost depend on the noiseintensity is marked by arrow.

stochastic differential equations [72] with time discretiza-tion step Δt = 0.001. The modified Runge-Kutta methodis applicable for delta-correlated Gaussian white noiseused frequently in our manuscript. At the same time, forthe integration of the stochastic differential equations inthe case of the other types of noise we have used one-stepEuler method. For the GS regime detection the auxiliarysystem method described in Section 2 has been used.

At first, we consider the behavior of chaotic sys-tems (22) in the presence of noise influenced only on theresponse system, i.e. D1 = 0, D2 = D. Figure 4 showsthe dependence of the threshold of the GS regime onseton the noise amplitude D (the SNR value) for three dif-ferent values of the control parameter ωx and fixed valuesof the other control parameters. To possess all necessaryknowledge about influence of noise on the system understudy we have chosen values of the parameter ωx in thedifferent ranges of the parameter mismatch where the dif-ferent mechanisms of the synchronous regime arising havebeen shown to take place [67]. Parameter ωx = 0.99 cor-responds to the case of the relatively large values of thefrequency detuning whereas ωx = 0.95 (interacting sys-tems are identical) and ωx = 0.91 relate to the small ones.It is easy to see that independently on the value of the con-trol parameter ωx the threshold of the GS regime onsetdoes not almost depend on the noise amplitude D ∈ [0; 40](SNR > −14.5 dB). Even for a great values of the noise in-tensity GS arises practically for the same values of the cou-pling parameter strength ε as for a noiseless case. Typicalsignals s(t) = x1(t)+Dξ(t) affecting the response and aux-iliary systems both in the absence and presence of noiseas well as the phase portraits of the response system and(u1, v1)-planes characterizing the response and auxiliarysystem behavior before (b, c, g, h) and after (d, e, i, j) theGS regime onset are shown in Figure 5. Pictures (a−e)correspond to the noiseless case whereas (f–j) refer to thepresence of noise of great intensity D = 40 affecting theresponse system (in the last case the signal is similar tothe stochastic one, with its amplitude being in approx-

Fig. 5. Signals s(t) affecting the response and auxiliary sys-tems (a, f), phase portraits (b, d, g, i) and (u1, v1)-planescharacterizing the response and auxiliary system behavior(c, e, h, j) before (ε = 0.05) and after (ε = 0.114) the GSregime onset in unidirectionally coupled Rossler systems withωd = 0.99, respectively. Pictures (a–e) correspond to the noise-less case (D = 0) whereas (f–j) refer to the noise one (D = 40).

imately 10 times more in comparison with the noiselesscase, compare Figs. 5a, 5f). One can easily see that char-acteristics of the response systems are changed slightlywith the noise intensity increasing (compare pictures b, dand g, i, respectively) and the boundary value of the GSregime remains practically the same. The causes deter-mining the stability of the GS regime with respect to theexternal noise influence are the same as in the alreadyconsidered case of the logistic maps (13) and could alsobe explained by the modified system approach. One cansay that for unidirectionally coupled Rossler systems thenoise of great intensity does not change the characteristicsof the modified system

z1 = −ωuz2 − z3 − εz1,z2 = ωuz1 + az2,z3 = p + z3(z1 − c),

(24)

where z = (z1, z2, z3)T is the vector state of the modifiedsystem and, as a consequence, of the response one. By theanalogy with the logistic maps the bifurcation diagramsfor the modified Rossler system are shown in Figure 6.Figure 6a corresponds to the noiseless case (D = 0)whereas in Figures 6b, 6c the modified Rossler systemswith additive noise of the different intensities (D = 10and D = 40, respectively) (the noise is introduced insystem (24) in the same way as in Eq. (22)) are shown.Independently on the noise intensity for the selected valuesof the control parameters the cycle-1 periodic oscillationsare observed in the modified system (24) (see also [66]).

The external noise does not shift the bifurcation pointsand, therefore, does not affect the boundary value of theGS regime. Therefore, we can conclude that the mecha-nisms determining the GS regime stability are the same asfor the system with discrete time (13). At the same time,


-14

-10

-6

-2

0 0.05 0.1 0.15 ε

xmin

-14

-10

-6

-2

0 0.05 0.1 0.15 ε

εGS

εGS

xmin

xmin

(a)

(b)

(c)

-14

-10

-6

-2

0 0.05 0.1 0.15 ε

εGS

Fig. 6. Bifurcation diagrams of the modified Rossler sys-tem (24) in the absence (a) and presence (b, c) of noise (thenoise is introduced in system (24) in the same way as inEq. (22)). The control parameter ωx = 0.99 in all consideredcases, the noise intensity D = 10 in (b) and D = 40 in (c). Thecoupling parameter values εGS = 0.112 corresponding to theGS regime (obtained by means of auxiliary system method, seeSect. 2) are marked by arrow in all cases.

since the basin of attraction in the Rossler system is un-bounded, the effect of the GS regime destruction describedabove in Section 3.1 could not be observed.

One more interesting question to be discussed is therelationship between the onset of the GS and CS regimes.According to the consideration made on the base of themodified system approach, GS and CS have the samemechanisms. At the same time, as we have mentioned inSection 2, the stability of the GS regime is stronger thanthe CS one. To confirm this statement we have analyzedthe CS regime arising in unidirectionally coupled identicalRossler systems (22) with ωd = ωr = 0.95 and comparedobtained results with the last one for the GS. Our calcu-lations show that in the absence of noise CS arises in thiscase for ε = 0.19, whereas GS takes place for ε ≥ 0.184.Adding noise of small intensity D = 0.1 results in theappearance of on-off intermittency [73] between the driveand response systems, at that the threshold value of theCS regime grows up, but the GS regime is still observed(see Figs. 4 and 7).

For the very large values of the noise intensity whenthe power of noise is much more than the Rossler sys-

Fig. 7. (x1; u1)- and (u1; v1)-planes characterizing the driveand response (a, c) and response and auxiliary (b, d) Rosslersystem behavior in the case of the absence (a, b) and the pres-ence (c, d) of noise in the response system (the noise intensityD = 0.1, the coupling strength ε = 0.19). The difference be-tween the drive and response system states in the presenceof noise (D = 0.1, ε = 0.19) shown in (e), illustrates the pres-ence of on-off intermittency. Parameter of the drive systemωd = 0.95.

tem signal one (D � 400, SNR � −34.5) the detectedsynchronous regime may be treated as the noise-inducedsynchronization, being the manifestation of the GS regimein the case when stochastic signal instead of the determin-istic one affects the response and auxiliary systems [49]. Inother words, the deterministic signal from the drive sys-tem practically does not play role and may be neglected incomparison with the stochastic one. At that, the boundaryvalue of the synchronous regime onset should not dependon the control parameter of the drive system ωx (see, Fig. 4for a large D) and is determined mainly by the characteris-tics of the noise signal. Therefore, for the noise intensitiesD � Dc = 45 (SNR < −15.5 dB) (shown in Fig. 4 by ar-row) the threshold value of the synchronous regime maystart increasing or decreasing depending on the value ofthe control parameter detuning.

It should be noted that the weak dependence of thethreshold value of the GS regime onset in the wide rangeof the noise intensity D takes place if the amplitude D1

of the additive noise term in equations (22) is not equalto zero. We have chosen it to be equal to D1 = εD. These


17.5 11.4 5.5 -0.6 -6.5 -12.6 -18.7 -24.9 -30.6 SNR, [dB]

0

0.05

0.1

0.15

0.2

ε

1 2 4 8 16 32 64 128 256 D

Fig. 8. Dependence of the boundary value of the GS regimearising in two unidirectionally coupled Rossler systems in thecase when statistically independent noise sources of intensity Daffect the drive and response on the noise intensity D (the SNRvalues corresponding to the noise intensities are also shown) fordifferent values of the drive system parameter ωx: ωx = 0.99(•), ωx = 0.95 (�), ωx = 0.91 (�). The critical value of thenoise intensity Dc up to which the boundary value of the GSregime does not almost depend on the noise intensity is markedby arrow.

dependencies for a different values of the drive systemparameter ωx are shown in Figure 8. Such behavior ofinteracting systems in the presence of noise is fully definedby mechanisms described above in this subsection.

Therefore, one can say that in both considered cases(maps and flows) in the wide range of the noise intensitythe external noise does not practically affect the thresh-old of the GS regime arising. Hence, we can say aboutstability of the GS regime with respect to external noisein dynamical systems with a few number of degrees offreedom.

3.3 Ginzburg-Landau equations

As a third example we consider the GS regime arisingin spatially extended self-oscillating media described bythe complex Ginzburg-Landau equations (CGLE). Thesystem under study is represented by a pair of unidirec-tionally dissipatively coupled complex Ginzburg-Landauequations (CGLE’s) being under influence of distributedin space source of the white noise. Equations describingsuch system may be written as

∂u

∂t= u − (1 − iαd)|u|2u + (1 + iβd)

∂2u

∂x2

+εDξ(x, t), x ∈ [0, L], (25)

∂v

∂t= v − (1 − iαr)|v|2v + (1 + iβr)

∂2v

∂x2

+ε(Dζ(x, t) + u − v), x ∈ [0, L]. (26)

Equation (25) describes the drive system and equa-tion (26) corresponds to the response one. It is knownthat in two unidirectional CGLE’s the GS regime may take

place [74]. In our investigation the parameters of the drivesystem are chosen as αd = 1.5, βd = 1.5. To study thegeneralized synchronization of the nonidentical systemswe have chosen the different values of control parametersαr ∈ [3; 5] and βr ∈ [3; 5] for the response system (26).The choice of such values of the control parameters resultsin the autonomous systems being in the spatiotemporalchaotic regime. Parameter ε determines the strength ofthe unidirectionally dissipative coupling between the re-sponse and drive systems, with the interaction of thembeing in each point of space. The terms Dξ(x, t), ζ(x, t)simulate complex model noise with Gaussian distributionof the random values ξ(x, t), ζ(x, t) with zero mean value:

〈ζ(x, t)〉 = 0,

〈ζ(x, t)ζ(x′, t′)〉 = δ(x − x′)δ(t − t′), (27)

D defines the noise intensity.Equations (25)–(26) have been solved with pe-

riodic boundary conditions u(x, t) = u(x + L, t) andv(x, t) = v(x + L, t), with all numerical calculations beingperformed for a fixed system length L = 40π and randominitial conditions. To evaluate (25)–(26) the standard nu-merical scheme for integration of the stochastic partialdifferential equations [75] has been used, the value of thegrid spacing is Δx = L/1024, the time step of the schemeis Δt = 0.0002.

To detect the presence of the GS regime we haveused the auxiliary system method described in Section 2.At that, we have assumed that auxiliary system va(t),also satisfying (26), has been under influence of the noisesource of the intensity D equal to the last one for the re-sponse system. As an criterion of the GS regime arising wehave chosen the following one. The GS regime takes placewhen the mean standard deviation of the response v andauxiliary va system states satisfies the following condition:

1T

∫

T

∫ L

0

|v(x, t) − va(x, t)|2 < δ, (28)

where δ = 10−5.Figure 9 shows the dependence of the boundary value

of the GS regime onset ε on the noise intensity D (SNRvalue) for several values of the control parameters of theresponse system. One can easily see that the noise ofintensity D ∈ [0; 64] (SNR ≥ −41 dB) does not almostaffect the threshold of the GS regime onset in spatially ex-tended systems described by the Ginzburg-Landau equa-tions. The time-space diagrams characterizing the behav-ior of unidirectionally coupled spatially extended mediabefore and after the GS regime onset both in the absenceand presence of noise are shown in Figure 10. Pictures(a–e) correspond to the case of the absence of noise bothin the drive and response systems whereas in the pictures(f−j) the white noise of intensity D = 0.4 affects both thedrive and response. Pictures (a, f) characterize the drivesystem behavior, whereas the other ones refer to the re-sponse system one before (b, g) and after (d, i) the GSregime onset. Figures 10c, 10e, 10h, 10j shows the spa-tiotemporal distributions of the module of the difference


(a) (f)

(h)(g)(c)(b)

(j)(i)(e)(d)

Fig. 10. The spatio-temporal diagrams characterizing behavior of the drive (a, f) and response (b, d, g, i) systems (25)–(26) aswell as the dependencies of the module of the difference between the states of the response and auxiliary systems |v − va| (c, e,h, j) for cases of absence (c, h) (ε = 0.2) and presence (e, j) (ε = 0.8) of the GS on time t and space x. The control parametervalues for the response system have been selected as αr = βr = 3. The time moments marked by arrows correspond to thecoupling switching-on between the drive and response systems. Pictures (a–e) correspond to the noiseless case (D = 0) whereas(f–j) refer to the noise one (D = 0.4).

ε

D~ 0

0.4

0.8

1.2

0.25 1 4 16 64 2567.19 SNR, [dB]-28.93-4.85 -16.89 -40.97 -53.01

Fig. 9. Dependence of the boundary value of the GS regimeonset in the coupled CGLE’s on the noise intensity D (the SNRvalues corresponding to the noise intensities are also shown)for different values of the control parameters of the responsesystem: αr = 3, βr = 3 (•), αr = 4, βr = 4 (�), αr = 5, βr = 5(�). The critical value of the noise intensity Dc up to whichthe boundary value of the GS regime does not almost dependon the noise intensity is marked by arrow.

between the states of the response and auxiliary systems|v − va| for cases of the absence (c, h) and the presence(e, j) of the GS regime. One can easily see, that in the sec-ond cases (e, j) the difference of the states of the response

and auxiliary systems in every point of space tends to bezero after coupling begins, which means the presence ofthe GS between the drive and response CGLE’s. It shouldbe noted that the length of the transient process precededthe GS regime onset is occurred to be rather more in thecase of the presence of noise whereas the threshold valueof the GS regime onset is the same as in the noiselesscase. Moreover, one can easily see that spatio-temporaldiagrams characterizing the response system behavior aresimilar to each other both in the presence and absence ofnoise (compare pictures (b, d) with (g, i), respectively).

The stability of the GS regime in Ginzburg-Landauequations with respect to noise is determined by thesame mechanisms, as in the cases of the systems witha few number of degrees of freedom considered in theprevious Sections 3.1 and 3.2. As well as for the logis-tic maps and Rossler systems, the modified system ap-proach may be used for the explanation of the observedphenomenon. Indeed, the noise of a large enough intensitydoes not almost affect the characteristics of the modified


Ginzburg-Landau equation

∂vm

∂t= vm − (1 − iαr)|vm|2vm

+(1 + iβr)∂2vm

∂x2− εvm, x ∈ [0, L] (29)

(and, as a consequence, of the response one), as well asin the case of Ginzburg-Landau equation with the addedconstant term [76]. Therefore, the noise does not changethe threshold value of the GS regime onset. At the sametime, as it has been discussed in Section 2, the boundaryvalue of the coupling parameter ε may start changing if thenoise intensity is a very great (D > 64, SNR < −41 dB).It is easy to see from Figure 9 that for such values of thenoise intensity the boundary value of the GS regime startsdecreasing. For the very large intensities D the couplingvalue εGS corresponding to the boundary of the GS regimetends to the constant value which does not depend on thethe control parameters α and β of the spatially extendedmedia. Such behavior of the boundary value of the GSregime onset, as in the case of unidirectionally coupledRossler systems considered in Section 3.2, is connectedwith the noise-induced synchronization regime realization.

Nevertheless, the noise of a large enough intensity doesnot almost alter the threshold value of the coupling pa-rameter strength between two unidirectionally coupledGinzburg-Landau equations. In this case one can sayabout stability of the GS regime with respect to noisein the coupled spatially extended self-oscillating media.

So, having considered three different examples ofmodel systems (discrete maps, flow systems, spatially-extended media) we can come to the conclusion that theGS regime demonstrates the significant stability with re-spect to noise in a wide range of the values of the externalnoise intensity.

4 Experimental study of the GS onsetin chaotic circuits in the presence of noise

To confirm the theoretical and numerical results given inthe previous sections we have also studied experimentallythe dynamics of the chaotic oscillator driven by the exter-nal chaotic signal in the presence of noise. In the experi-ment we have used the simple electronic circuit where allparameters including noise amplitude may be controlledprecisely.

The experimental setup is shown in Figure 11. As abasic element of the scheme we have used an electroniccircuit with nonlinear converter and linear feedback loopsimilar to the one described in [57,77] (it is shown inFig. 11 by dashed rectangle). Since the generator is capa-ble to demonstrate both periodic and chaotic oscillationsdepending on the choice of the parameter α of nonlinearconverter, it has been selected in such a way for the gener-ated signal to be chaotic (quantitative values of all controlparameters of the circuit are presented in the captions ofFigs. 11 and 12). Chaotic generator has been connectedto DAC/ADC board L-Card L-783 installed into personal

R4

R1

R2 R3

R5

R6

OP1 OP2

OP3

D1

D2

L r'CC

R

α

PC-basedDrive Chaotic Generator

NoiseGenerator Rc

Fig. 11. Block diagram of experimental setup. Chaotic gen-erator layout is shown by dashed rectangle. Here C = 330 nF,C′ = 150 nF, R = 630 Ω, r = 56 Ω, L = 3.3 mH, OP1,2

– TL082, OP3 – LF356N, D1,2 – 1N4148, R1 = 2.7 kΩ,R2 = R4 = 7.4 kΩ, R3 = 100 Ω, R5 = 186 kΩ, R6 = 4.7 kΩ,RC′, rLC – low-pass filters, α is parameter of nonlinear con-verter, Rc is a coupling resistance.

computer (PC) whereby we have recorded the dynamicsof potential on the capacitors C and C′. As a drive signalwe have used the last one generated by the circuit de-scribed above, digitized by ADC with further reconstruc-tion by DAC. The drive signal has been introduced intothe circuit via dissipative unidirectional coupling of vari-able dissipation value (see Fig. 11). The noise signal hasbeen produced with Agilent 33220 function generator, dig-itized and additively introduced into the coupling device(as it has been shown in Fig. 11). Characteristics of thenoise are close to the Gaussian one. Oscillations of the re-sponse system have been also digitized with ADC boardand transferred to personal computer for further numeri-cal processing.

As we have mentioned above, one of the easiest waysto detect the presence of the GS regime is the use of anauxiliary system, i.e. an additional response circuit, whichis a replica of the main one. But creation of the auxiliarysystem with parameters completely equal to the responsesystem ones is one of the most conceptual problems inthe experimental study of the GS regime. To solve thisproblem we have used an approach analogous to the onediscussed in [78]. As it has been specified above, the sig-nal from the drive system with additive noise has beenpreliminary recorded on PC. Therefore, it is evident thatin this case the response system could be subjected to theinfluence of identical drive signal (with additive noise) anynumber of times. For the realization of an auxiliary systemmethod it is quite sufficient to affect the response systemby the drive signal twice, alternating the period of the in-fluence with the time interval of autonomous dynamics (toprovide the different initial conditions), and then compareobtained data numerically.

The experiment has been performed for three maincases: (i) chaotic attractors both in the drive and responsesystems have identical band structure; (ii) chaotic regimewith band attractor has influenced on the regime withdouble scroll attractor; (iii) chaotic attractors both in thedrive and response systems have a double scroll struc-ture. Typical phase portraits of considered regimes areshown in Figure 12 (band attractor (a) and double scrollattractor (b)). The corresponding values of the control pa-


Fig. 12. Typical phase portraits of chaotic regimes observedin experiment: (a) band attractor (α ∼ 0.15), (b) double scrollattractor (α ∼ 0.25).

0.05

0.20

0.35

0.50

0 0.1 0.2 0.3 0.4 D

ε

10 6.99 5.23 3.98 SNR, [dB]

Fig. 13. Coupling strength value corresponding to the GSregime onset as a function of noise intensity (the SNR valuescorresponding to the noise intensities are also shown) in thecases when chaotic attractors both in drive and response sys-tems have the band structure (�); drive system in the bandchaotic regime influences on the response system in chaoticregime with the double scroll attractor (•); chaotic attractorsboth in drive and response systems have a double scroll struc-ture (�).

rameter α are indicated in the caption. Each case has beenstudied in the presence of Gaussian noise of different in-tensity. For experimental data the noise intensity has beencalculated as a ratio D = PN/PCS of a power of the noisesignal PN to the power of chaotic signal PCS .

Figure 13 shows the dependence of the coupling

strength value ε = 1Rc

√LC corresponding to GS regime on-

set on the noise intensity for three cases mentioned above.One can see that in the range of noise intensity [0; 0.5] thethreshold value remains nearly constant. Typical signalsfrom the drive system with and without additive noise af-fecting the response one as well as the phase portraits ofthe response system and (U, V )-planes characterizing theresponse and auxiliary system behavior both in the ab-sence and presence of the GS regime in chaotic circuitsin the case (i) are shown in Figure 14. One can easilysee that characteristics of the response circuit have notbeen changed noticeably with the appearance of noise.Analogous situation takes place in unidirectionally cou-pled chaotic circuits with initially double-scroll chaotic at-tractors in the one and both of them. One can say that inall considered cases the modified system (i.e. considered

(a) (f)

(h)(g)(c)(b)

(j)(i)(e)(d)

Fig. 14. Signals from the drive chaotic circuit without (a)and with (f) additive noise affecting the response circuit, phaseportraits of the response system (b, d, g, i) and (U, V )-planescharacterizing the response and auxiliary system behavior (c,e, h, j) before (ε = 0.22) and after (ε = 0.34) the GS regimeonset. The control parameters of the chaotic circuit has beenchosen in such a way that both the drive and response systemsin autonomous regime are characterized by the band attractors.Pictures (a–e) correspond to the noiseless case (D = 0) whereas(f–j) refer to the noise one (D = 0.4).

generator with additional dissipation) demonstrates thecycle-1 periodic oscillations. Further increase of the noiseintensity (when it becomes greater than the intensity ofthe deterministic signal) may result in the monotonousgrowth of the GS boundary value.

So, the experimental results satisfy the stability ofthe GS regime with respect to noise. They are also in agood agreement with the data obtained theoretically andnumerically.

5 Conclusions

In conclusion, we have analyzed both theoretically andexperimentally the influence of noise on the GS regimein different unidirectionally dissipatively coupled identicalchaotic systems with mismatched parameters with a smallnumber of degrees of freedom as well as spatially extendedmedia. The dependencies of the GS regime boundary onthe noise intensity in the cases when the drive and re-sponse systems are enforced both by common noise andby the statistically independent noise sources are also con-sidered. We have shown that if attractors of the drive andresponse systems have an infinitely large basin of attrac-tion, independently on the type of system and kind of thenoise distribution the GS regime possess a great stabilitywith respect to noise, i.e. the threshold of the synchronousregime arising does not almost depend on the intensity ofnoise. Such behavior of the boundary of the GS regime hasbeen explained by means of the modified system approach,


i.e., the joint action of dissipation and driving force is re-sponsible for the reported robustness of the GS regimeagainst noise.

Though the results described in the Manuscript referto the white noise we expect that they could be valid fordifferent noise forms. Similar results have been obtainedfor different types of noise, including colored noise.

It should be noted that the revealed peculiarity of theGS regime could be used in a number of practical appli-cations, i.e. for the transmission of information throughthe communication channels where the level of noise issufficient [18,60].

We thank the Referees of our Manuscript for the usefulcomments allowing us to improve the quality of the paper.This work has been supported by Federal special-purposeprogramme “Scientific and educational personnel of innovationRussia”, Russian Foundation for Basic Research (project 11-02-00047), and President’s Program for the Support of LeadingRussian Scientific Schools (project Nsh-3407.2010.2). A.K. alsothanks “Dynasty” Foundation.

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