Nonlinear Analysis: Real World Applications 11 (2010) 3059–3065

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Lag and anticipatory synchronization based parameter estimationscheme in modulated time-delayed systemsDibakar Ghosh a,∗, A. Roy Chowdhury ba Department of Mathematics, Dinabandhu Andrews College, Garia, Kolkata - 84, Indiab High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata - 32, India

a r t i c l e i n f o

Article history:Received 6 July 2009Accepted 30 October 2009

Keywords:Adaptive demodulatorSynchronizationParameter estimationModulated delay time

a b s t r a c t

In this article, we have shown that using delay dynamical systems as base, one canuse the modulation of the various parameters, to transmit multiple signals through asingle chaotic time series. It is shown that under certain conditions the original signalsand parameters of the driving signals can be recovered with the help of an adaptivedemodulator. An important aspect of the present method is that the communication ispossible between two time-delayed systems with parameter mismatch using lag andanticipatory synchronization. The whole idea is presented on the basis of Mackey–Glasssystem, with variable time delay.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Synchronization of unidirectionally coupled chaotic system has become an area of active research and it is used forcommunication [1]. Such communications are founded on the principles of chaoticmasking [2], whereby amessage is addeddirectly to the chaotic carrier with an amplitude much lower than the chaotic carrier or chaotic modulation [2] wherebythe chaotic carrier undergoes analog or digital modulation or chaotic switching [3] where a binary message is transmittedby switching between two chaotic attractors associated with two sets of parameters of the system. It is proved that lowdimensional chaotic systems do not ensure a sufficient level of security for communications, as the associated chaoticattractors can be reconstructed with some effort and the hidden message can be retrieved by an attacker [4]. One way toovercome this problem is by considering high dimensional system with more than one positive Lyapunov exponent. In thisregard, time-delayed system received a lot of attention, i.e. x = f (x(t), x(t − τ)), where τ is a delay time. With increase intime-delay τ , the system is more complex and number of positive Lyapunov exponents increases and the system eventuallytransits to hyperchaos. For this reason, a time-delayed system was considered as a suitable candidate for communicationsystems [5]. In a recent study, it was discussed that time-delayed system is still vulnerable for communication because timedelay τ can be exposed by several measures, e.g. autocorrelation [6], filling factor [7], etc. If the delay time τ is known, thetime-delayed system become quite a simple and themessage encoded by the chaotic signal can be extracted by the commonattack methods [4]. If we consider a time-delayed system in which the delay time is not constant but modulated in time,then the communication systemwill bemore secure. This is the practical significance for the study of chaos synchronizationin time-delayed systems with modulated delay time. In general, security of chaotic communication system depends on thecomplexity degree of master’s dynamics and transmitted signal.An important issue in time series analysis of chaotic system is one or multiple parameter estimation from a single

or more than one time series. Parameter estimation method may be broadly classified into online and offline estimation

∗ Corresponding author.E-mail addresses: drghosh_chaos@yahoo.com (D. Ghosh), asesh_r@yahoo.com (A. Roy Chowdhury).

1468-1218/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2009.10.025

3060 D. Ghosh, A. Roy Chowdhury / Nonlinear Analysis: Real World Applications 11 (2010) 3059–3065

methods. The online estimation method, e.g. adaptive control approach [8], though simple to implement, cannot be used toestimate all the parameters. The offline method, e.g. active–passive-decomposition method [9], autosynchronization [10],error minimization strategy [9], statistical method [11] and iterative method [12], has been demonstrated to be able toestimate all the system’s parameters. Recently, Konnur [13] proposed an online estimation of all model parameters of agiven chaotic–hyperchaotic system using least-squares approach. This method is more analytical and effective but requiremore than one scalar time series for estimating all parameters of a system. Chen and Kurths [14] also proposed an observerbased approach for chaos synchronization and parameter estimation from a scalar output signal. This signal could be either avariable from themaster or a scalar nonlinear combination of all variables from themaster. Adaptive chaos synchronizationbased parameter estimation is recently developed by Huang et al. [15]. In the past few years, various methods have beendeveloped for the estimation of multiple parameters of a chaotic system and used to secure communication [15]. Mostof the methods required more than one time series to estimate all the parameters of a chaotic–hyperchaotic system.In practice, such requirement increases implementation cost and sometimes it is not possible in engineering. Therefore,multiparameter estimation and multiparameter modulation for communication using only single scalar chaotic time seriesis much important.In our previous study [16], adaptive scheme for multiparameter estimation in modulated time-delayed system from a

single chaotic time series is described. In another study [17], nonlinear observer based multiparameter estimation schemeis derived. In these twoworks [16,17], we have used complete synchronization for multiparameter estimation between twoidentical chaotic time-delayed systems. But, practically complete synchronization between two identical systems is almostimpossible for the presence of the signal transmission and evolution time of response system itself. Nowadays, most securecommunications are based on complete synchronization, which limits its application in engineering. In this respect, securecommunication using lag and anticipatory synchronization can reduce the complexities of the transmitters and receivers.In this article, we have proposed a method for parameter estimation using lag and anticipatory synchronization.

2. General formula

2.1. Description of method

We begin by briefly describing the setup of the parameter modulation. Let the driving system as

x = f (x, xτ , ξ1, ξ2) (1)

where x(t) is the system state at time t , xτ = x(t− τ), τ is the delay time and ξ1, ξ2 are system parameters. The informationsignals are used tomodulate one ormore parameters so that the output of system (1) is chaotic.We assume that there existsan unidirectional coupling scheme and the response system as

y = f (x, xτ , y, yτ , η1, η2). (2)

Systems (1) and (2) are asymptotically synchronized if y→ x, when ξ1 = η1 and ξ2 = η2. The recovery of information andestimation of parameters of the driving system (1) are possible with the help of an appropriate adaptive demodulator.

2.2. Synchronized systems

Weconsider synchronization between twounidirectional nonlinearly coupled time-delayed systems,where the couplingterm is a time delay. The coupled system can be considered as

x(t) = −ax(t)+ r1f (xτ1) (3)

y(t) = −ay(t)+ r2f (yτ1)+ kf (xτ2) (4)

where k is the coupling strength. Systems (3) and (4) allow for the synchronization manifold xτ2−τ1 = y under the conditionk = r1 − r2. In Eq. (3), f (xτ1) is a nonlinear function of xτ1 characterizing the system, e.g. f (xτ1) =

xτ11+xcτ1

for Mackey–Glass

model [18], f (xτ1) = π [A−µ sin2(xτ1−x0)] for the Vallee model [19], f (xτ1) = sin

2(xτ1−x0) for the sine squaremodel [20]and f (xτ1) = sin(xτ1) for the Ikeda model [21].We denote the error by ∆ = xτ2−τ1 − y (which corresponds to anticipatory synchronization when τ2 < τ1, complete

synchronization for τ2 = τ1 and lag synchronization when τ2 > τ1). We can derive the error dynamics as

∆ = −a∆+ r2f ′(xτ2)∆τ1 (5)

where f ′ represents the derivative of f with respect to time and bounded. We take feedback delay and coupling delay as amodulated time delay, respectively, in the form of

τ1 = τ10 + α1 sin(ω1t) (6)

τ2 = τ20 + α2esin(ω2t) (7)

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where τ10 and τ20 are the zero-frequency components, α1 and α2 are the amplitudes and (ω1/2π), (ω2/2π) are the twofrequencies of the modulation, respectively.It is obvious that ∆ = 0 is the only trial solution of Eq. (5). The synchronization manifold is locally stable if the origin

of Eq. (5) is asymptotically stable. To study the stability of the synchronization manifold xτ2−τ1 = y, we use Krasovskii–Lyapunov approach [22].According to [5,23], the sufficient condition for the trial solution∆ = 0 of the time-delayed system ∆(t) = −r(t)∆(t)+

s(t)∆(t − τ) is r(t) > |s(t)|. For variable time delay, the sufficient condition [5] for synchronization is

r(t) >|s(t)|√1− τ ′1

.

For the time-delayed system (5), the analytical condition for synchronization is

r(t) >|r2| sup |f ′(xτ2)|√

1− τ ′1. (8)

Note that for constant time delay cases the inequality (8) is reduced to the well known sufficient stability condition [5].

2.3. Single parameter modulation

We take an information signal and use it to modulate various parameters of a chaotic system. The resulting transmittedsignal consists of the information hidden in the signal from the chaotic system. At the receiver end, themessage is recoveredby demodulation of parameterwith the help of adaptive learningmechanism. First, we try to vary the parameter ‘a’ in Eq. (3).We take the driver system (3) as

x(t) = −a(t)x(t)+ r1f (xτ1)a(t) = 0

}(9)

and the response system (4) as

y(t) = −a′(t)y(t)+ r2f (yτ1)+ kf (xτ2)a′(t) = fa

}(10)

where fa is a function for the learning to be determined. Next, wewant to find out the function fawith the help of Krasovskii–Lyapunov functional approach [22,23].We denote the error signals as ∆ = xτ2−τ1 − y and ∆a = aτ2−τ1 − a

′. Then, from Eqs. (9) and (10), we can derive thedynamics of the error as

∆ = −aτ2−τ1∆− y∆a + r2f′(xτ2)∆τ1

∆a = −fa.

}(11)

We define a positive definite Krasovskii–Lyapunov functional of the form [5]

V (t) =12

(∆2 +

1γ∆2a

)+ h(t)

∫ 0

−τ1(t)∆2(t + θ)dθ

where h(t) > 0 and γ is learning parameter. Then

V (t) = ∆∆+1γ∆a∆a + h(t)

∫ 0

−τ1(t)∆2(t + θ)dθ + h(t)[∆2 −∆2τ1 +∆

2τ1τ ′1]

if h(t) ≤ 0 for arbitrary t , then

V (t) ≤ −aτ2−τ1∆2+ r2f ′(xτ2)∆∆τ1 +∆a

(∆a

γ−∆y

)+ h(t)[∆2 −∆2τ1 +∆

2τ1τ ′1].

If we choose∆a

γ= ∆y, i.e. fa = −γ y∆ = −γ y(xτ2−τ1 − y)

thenV (t) ≤ −(aτ2−τ1 − h(t))∆

2+ r2f ′(xτ2)∆∆τ1 − (1− τ

′

1)h(t)∆2τ1

= −∆2

[aτ2−τ1 − h(t)− r2f

′(xτ2)∆τ1

∆+ (1− τ ′1)h(t)

∆2τ1

∆2

]= −∆2F(X, h(t))

where F(X, h(t)) = aτ2−τ1 − h(t)− r2f′(xτ2)X + (1− τ

′

1)h(t)X2 with X =

∆τ1∆.

3062 D. Ghosh, A. Roy Chowdhury / Nonlinear Analysis: Real World Applications 11 (2010) 3059–3065

In order to show that V (t) < 0 for all∆ and∆τ1 , i.e. for allX ’s, it is sufficient to show that Fmin > 0. The absoluteminimum

of F occurs at X =r2f ′(xτ2 )2h(t)(1−τ ′1)

with Fmin = aτ2−τ1 − h(t)−{r2f ′(xτ2 )}

2

4h(t)(1−τ ′1). Thus, the sufficient condition for synchronization is

aτ2−τ1 > h(t)+{r2f ′(xτ2)}

2

4h(t)(1− τ ′1)= ψ(h(t)).

Again, ψ(h(t)) is a function of h(t). The minimum value of ψ(h(t)) occurs at h(t) =|r2f ′(xτ2 )|√1−τ ′1

with ψmin =|r2| |f ′(xτ2 )|√

1−τ ′1.

For system (11), the stability condition for the synchronization manifold xτ2−τ1 = y can be written as

aτ2−τ1 >|r2| sup |f ′(xτ2)|√

1− τ ′1. (12)

Then, the resulting response system becomes

y(t) = −a′(t)y(t)+ r2f (yτ1)+ kf (xτ2)a′(t) = −γ y(xτ2−τ1 − y).

}(13)

From the above discussion, we can obtain that xτ2−τ1 = y is the synchronization manifold between systems (9) and (13)with the existence condition r1 = r2 + k and stability condition (12).

2.4. Two parameters modulation

We now study the modulation for two parameters for communication. We modulate the parameters a and r1 by twoencrypted information signals in the driver (transmitter) system. The driver and response systems consist the state variablesx, a, r1 and y, a′, r ′1, respectively. We consider the coupled system as

x(t) = −a(t)x(t)+ r1(t)f (xτ1)a(t) = 0r1(t) = 0

}(14)

and

y(t) = −a′(t)y(t)+ r ′1(t)f (yτ1)+ k(xτ2−τ1 − y)a′(t) = far ′1(t) = fr1 .

(15)

If we can calculate the functions fa and fr1 , so that synchronization between Eqs. (14) and (15) occurs, then the demodula-tion of the parameters will be realized.We now derive the functions fa and fr1 with the help of analysis of its error dynamics.We denote the error signals as ∆ = xτ2−τ1 − y, ∆a = aτ2−τ1 − a

′ and ∆r1 = r1τ2−τ1 − r′

1. Then, from Eqs. (14) and (15),we can derive the dynamics of error as

∆ = −(aτ2−τ1 + k)∆−∆ay+ f (xτ2)∆r1 + r′

1f′(xτ2)∆τ1

∆a = −fa∆r1 = −fr1 .

To design the functions fa and fr1 , we choose a positive definite Lyapunov functional of the form

V (t) =12

(∆2 +

1γ1∆2a +

1γ2∆2r1

)+ h(t)

∫ 0

−τ1(t)∆2(t + θ)dθ

where γ1 and γ2 are learning parameters and γ1 > 0 and γ2 > 0. Then

V (t) ≤ −(aτ2−τ1 + k)∆2+∆a

(∆a

γ1−∆y

)+∆r1

[∆r1

γ2+ f (xτ2)∆

]+ h(t)

[∆2 −∆2τ1 +∆

2τ1τ ′1]+ r ′1f

′(xτ2)∆∆τ1 .

If we choose the parameter error dynamics as

∆a = γ1y∆ = γ1y(xτ2−τ1 − y)

∆r1 = −γ2f (xτ2)∆ = −γ2f (xτ2)(xτ2−τ1 − y)

then

V (t) ≤ −(aτ2−τ1 + k)∆2+ h(t)[∆2 −∆2τ1 +∆

2τ1τ ′1] + r

′

1f′(xτ2)∆∆τ1 .

D. Ghosh, A. Roy Chowdhury / Nonlinear Analysis: Real World Applications 11 (2010) 3059–3065 3063

According to Ref. [5], V (t) < 0 if

aτ2−τ1 + k >|r ′1| sup |f

′(xτ2)|√1− τ ′1

. (16)

Finally, we obtain the response system as

y(t) = −a′(t)y(t)+ r ′1(t)f (yτ1)+ k(xτ2−τ1 − y)a′(t) = −γ1y(xτ2−τ1 − y)r ′1(t) = −γ2f (xτ2)(xτ2−τ1 − y)

. (17)

In Section 3, we test the correctness of theoretical analysis with the help of numerical simulation. For numerical simulation,we take time delay Mackey–Glass differential equation with variable time delay. For one parameter modulation, we takenonlinear direct feedback coupling, i.e. of the form f (xτ2) and for two parameters modulation, linear feedback coupling isused. It is found that numerical simulations fully support the analytical approach.

3. Numerical example

Complete synchronization in coupled chaotic systems occurs only at a certain point in the system’s parameter space, thusis a fairly restrictive concept and often difficult to achieve except under ideal conditions. For complete synchronization, thestates of two systems remaining identical is practically impossible for the presence of the signal transmission and evolutiontime of response system itself. So, knowledge of the lag or anticipatory synchronization is of highly practical importance.For this purpose, we take the drive system (9) and response system (13) as

x(t) = −a(t)x(t)+r1x(t − τ1)

1+ x10(t − τ1)a(t) = 0

(18)

and

y(t) = −a′(t)y(t)+r2y(t − τ1)

1+ y10(t − τ1)+

kx(t − τ2)1+ x10(t − τ2)

a′(t) = −γ y(xτ2−τ1 − y)

. (19)

In this case, we choose the parameter values τ10 = 10, τ20 = 25, α1 = 1.5, α2 = 3.0, ω1 = 0.0001, ω2 = 0.00011, γ1 =γ2 = 25.0, provided the general stability condition (12) is satisfied along the parameter condition r1 = r2 + k. For theseparameter values, τ1(t) varies between 8.5 and 11.5 and τ2(t) oscillates between the values 26.1 and 33.15. In this case,the coupling delay τ2(t) is greater than the feedback delay τ1(t) and the lag time τ(t) = τ2(t) − τ1(t) is positive and therelation between the drive x(t) and the response y(t) now becomes x(t − τ) = y(t). The message signal is selected asm(t) = | sin(0.0085t)|. The encrypted message signal is

a(t) = 0.4+m(t − τ)

and decrypted message can be obtained by

m(t) = a′(t)− 0.4.

The encrypted message a(t) and decrypted message a′(t) denoted by solid line and dotted line, respectively, are shown inFig. 1(a). In Fig. 1(b), the evolution of drive system x(t) (solid line) and response system y(t) (dotted line) are shown. Theircorresponding lag synchronization manifold xτ2−τ1 and y are shown in Fig. 1(c).For two information signalsm1(t) = | sin(0.008t)| andm2(t) = | cos(0.008t)|, we take the drive and response systems

as

x(t) = −a(t)x(t)+r1(t)x(t − τ1)1+ x10(t − τ1)

a(t) = 0r1(t) = 0

(20)

and

y(t) = −a′(t)y(t)+r ′1(t)y(t − τ1)1+ y10(t − τ1)

+ k(xτ2−τ1 − y)

a′(t) = −γ1y(xτ2−τ1 − y)

r ′1(t) = −γ2(xτ2−τ1 − y)x(t − τ2)

1+ x10(t − τ2)

. (21)

3064 D. Ghosh, A. Roy Chowdhury / Nonlinear Analysis: Real World Applications 11 (2010) 3059–3065

a b c

Fig. 1. (a) Encrypted signal a(t) and decrypted signal a′(t), (b) lag synchronization between x(t) and y(t) and (c) lag synchronization manifold forr1 = 2.0, r2 = 1.0, and k = 1.0.

a b c

Fig. 2. (a) Encrypted signal a(t) and decrypted signal a′(t), (b) encrypted and decrypted signalm2(t) and (c) corresponding lag synchronization manifold.

The corresponding formula for encrypted messages are

a(t) = 0.8+m1(t − τ), r1(t) = 3.0+m2(t − τ)

and decrypted message can be obtained by

m1(t) = a′(t)− 0.8, m2(t) = r ′1(t)− 3.0.

The variation of a(t) and a′ are shown in Fig. 2(a). In Fig. 2(b), the encrypted and decrypted messages of the signal m2(t)are shown. Lag synchronization manifold is shown in Fig. 2(c). For τ10 = 25, τ20 = 10 and other parameter are same aslag synchronization gives τ1(t) > τ2(t) and depicting the existence of anticipatory synchronization. In a similar way, thetransmission of one and two information signals are possible through anticipatory synchronization.

4. Conclusions

In conclusion, we have developed a method based on lag and anticipatory synchronization by which more than oneinformation is simultaneously transmitted through a chaotic time series. The advantage of this method is that it does notrequire always a complete synchronization, but can operate even when there is lag or anticipatory synchronization. Pastresults indicate that the method proposed here would be more easily applicable in experiment. The proposed methodis very much applicable in experiments, where only single output is available. The demodulation design that we havepresented is general and provides a design procedure that can be applied to a wide array of chaotic systems. Resultingsystems can be easily and inexpensively implemented using simple analog circuit,making this approach feasible for practicalcommunication applications.In comparison with previous methods [active–passive-decomposition method [9], autosynchronization [10], error min-

imization strategy [9], statistical method [11], iterative method [12], least-squares approach [13], observer based ap-proach [14], etc.], our method has the following advantages: (1) more than one parameter can be estimated through a single

D. Ghosh, A. Roy Chowdhury / Nonlinear Analysis: Real World Applications 11 (2010) 3059–3065 3065

chaotic time series, wheremultiple informationmay be simultaneously transmitted through just a single chaotic time series.This scheme is highly applicable in experiment, where only single output is available and it decreases the implementationcost of the communication. (2) Compared with the previously proposed chaos synchronization communication scheme, ourscheme increases the difficulty of directly finding out the information using the single transmitted signal. (3) One importantsignificance of the present method is that lag or anticipatory synchronization is also applicable for transmission of multipleinformation. Complete synchronization is only occurred between two identical systems, which is practically impossible dueto the presence of signal transmission or parameter mismatch.

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