Anticipating and projective–anticipating synchronization of coupled multidelay feedback systems

Physics Letters A 365 (2007) 407–411 www.elsevier.com/locate/pla

Anticipating and projective–anticipating synchronization of coupled multidelay feedback systems Thang Manh Hoang ∗ , Masahiro Nakagawa Department of Electrical Engineering, Faculty of Engineering, Nagaoka University of Technology, Kamitomioka 1603-1, Nagaoka, Niigata 940-2188, Japan Received 25 October 2006; received in revised form 8 January 2007; accepted 22 January 2007 Available online 3 February 2007 Communicated by A.R. Bishop

Abstract In this Letter, the model of coupled multidelay feedback systems is investigated with the schemes of anticipating and projective–anticipating synchronizations. Under these synchronization schemes, the slave anticipates the master’s trajectory. Moreover, with projective–anticipating synchronization there exists a scale factor in the amplitudes of the master’s and slave’s state variables. In the both cases, the driving signals are in the form of multiple nonlinear transformations of delayed state variable. The Krasovskii–Lyapunov theory is used to consider the sufficient condition for synchronization. The effectiveness of the proposed schemes is confirmed by the numerical simulation of specific examples with modified Ikeda and Mackey–Glass systems. © 2007 Elsevier B.V. All rights reserved. Keywords: Chaos synchronization; Anticipating synchronization; Projective synchronization; Time-delayed feedback system

1. Introduction Since first coupling model of chaotic systems [1] was proposed by Pecora and Carroll, chaos synchronization has attracted considerably increasing attention and recently becomes an active area of research. That is because of its potential applications, e.g. in secure communications [2], lasers [3,4], biological [5,6] and physiological [7] tasks, etc. Over the last decade, following the complete synchronization [1], several new types of chaos synchronization of coupled oscillators have appeared, i.e. generalized synchronization [8], projective synchronization [9], phase synchronization [10], lag synchronization [11], anticipating synchronization [12]. Roughly speaking, one chaotic system (plays a role of master) sends a driving signal to the other system(s) (plays a role of slave(s)) to establish a synchronous regime, as a result, their chaotic trajectories remain in step with each other during temporal evolution. More specifically, in the complete synchronization, the amplitude of the * Corresponding author.

E-mail address: [email protected] (T.M. Hoang). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.01.041

master’s state variable is equal to that of slave’s. In the generalized synchronization, the amplitude relation between state variables in the complete synchronization is extended in a way that the amplitude of the slave’s state variable correlates with that of master’s by a generic function. In the projective synchronization, there exists a scale factor in the amplitude of the master’s state variable and that of the slave’s. In the lag synchronization, the state variable of slave is retarded with the time length of τ (τ is nonnegative real) in compared to that of master. In contrast to the lag synchronization, the anticipating synchronization is interpreted that the slave anticipates the master’s motion. In the phase synchronization, the difference between the phase of the master’s state variable and that of the slave’s is constant during interaction while the amplitudes of their state variables evolves freely. Recently, the cooperative behavior of coupled multidelay feedback systems has been investigated [14–17] in such a way that the master synchronizes with the slave under the scheme of lag synchronization. However, most of synchronization models reported so far have driving signals in the form of linear or single nonlinearly transform of state variable. Differing from conventional synchronization models [12–16], the driving sig-

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nals of the synchronization systems presented in [17,18] are constituted by the sum of  multiple nonlinear transformations of delayed state variable, j kj f (x(t − τP +j )). In addition, the synchronous regime of coupled non-identical multidelay feedback systems [19] can be established, there, the equation  of driving signal consists of various function forms, i.e. j kj fj (x(t − τP +j )). Note that, functions fj (·) are from the master’s and/or slave’s equations. Practically, the complexity of driving signal offers a great significance to the application in secure communications due to the restriction of existing reconstruction methods in dealing with multidelay feedback systems (see [20]). Moreover, investigation of other synchronization schemes for such systems will allow to diversify the application in secure communications. In the present Letter, the anticipating synchronization of coupled multidelay feedback systems is investigated, in which the slave anticipates the master’s trajectory. Further, the scheme of anticipating synchronization of coupled multidelay feedback systems is extended to obtain the projective–anticipating synchronization in which the amplitudes of the master’s and slave’s state variables correlate by a scale factor. Moreover, the driving signal is in the form of sum of multiple nonlinear transformations of delayed state variable like that in [17,18]. The Krasovskii–Lyapunov theory is used to consider the sufficient condition for synchronization. The numerical simulations demonstrate with the examples of modified Mackey–Glass and Ikeda systems to confirm the effectiveness of the proposed schemes. 2. Scheme of anticipating synchronization Let us consider the synchronization model of following coupled multidelay feedback systems, its dynamical equations are defined as: Master: dx = −αx + dt

P 

mi f (xτi );

DS(t) =

y(t) = x(t + τd ),

(4)

where τd is an anticipation time by which the slave anticipates trajectory of the master. By applying the anticipation time τd to the state variable of (1), the synchronization error  = y − x(t + τd ) has dynamics described as d dy dx (t + τd ) = − dt dt dt   Q P   ni f (yτi ) + kj f (xτP +j ) = −αy + j =1

i=1



− −αx(t + τd ) +

P 



mi f (xτi −τd )

i=1

= −α +

P 

ni f (xτi −τd + τi )

i=1

+

Q 

kj f (xτP +j ) −

j =1

P 

mi f (xτi −τd ),

(5)

i=1

τP +j = τi − τd .

(6)

If the chosen value of delays is satisfied the relation as shown in (6), the components with coefficient kj will be affiliated to those with coefficient mi and (5) can be rewritten as  d ni f (xτi −τd + τi ) = −α + dt P

i=1

−

i=1

P ,Q 

(mi − kj )f (xτi −τd ),

(7)

i=1,j =1

kj f (xτP +j );

mi − kj = ni . (2)

j =1

Slave: dy = −αy + dt

relation:

(1)

Driving signal: Q 

Fig. 1. Chaos synchronization of multidelay feedback systems.

(8)

Also supposed that the adopted value of mi , kj and ni are fulfilled (8), f (·) is bounded and τi is small, the dynamics of synchronization error given in (7) is reduced to  d ni f  (xτi −τd )τi , = −α + dt P

P 

ni f (yτi ) + DS(t),

(3)

i=1

where α, mi , ni , kj , τi ∈ ; integers P , Q (Q  P ), f (·) is the differentiable generic nonlinear function. xτi and yτi stand for delayed state variables x(t − τi ) and y(t − τi ), respectively. As shown in (2), the driving signal is constituted by the sum of multiple nonlinear transformations of delayed state variable, and it is generated by driving signal generator (DSG) as illustrated in Fig. 1. The synchronization model is desired to create an anticipating synchronization manifold which is determined by following

(9)

i=1

where f  (·) stands for the derivative of f (·). Applying the Krasovskii–Lyapunov theory [21,22] to the case of multiple delays in (9), the sufficient condition for synchronization is expressed as α>

P 

  |ni |supf  (xτi −τd ),

(10)

i=1

where supf  (·) stands for the supreme limit of f  (·). The following example demonstrates with the modified Ikeda systems

T.M. Hoang, M. Nakagawa / Physics Letters A 365 (2007) 407–411

(a)

409

(b)

Fig. 2. (a) State variables x(t) and y(t). (b) Portrait of x(t + τd ) versus y(t).

to support for above general description. Assumed that the chosen value of τi , τP +j and τi is nonnegative and constant for simulation. Therefore, the condition to assure for τP +j in (6) being nonnegative is τi  τd (i = 1, . . . , P ) correspondingly.

tension of the anticipating synchronization. Let us consider the configuration with dynamical equations given in (1)–(3). The desired synchronization manifold is expressed as

Example 1. Let us consider the synchronization of two fourdelay Ikeda systems with the dynamical equations as follows: Master:

where a and b are nonzero real numbers, τd is taken a positive value. Hence, the synchronization error is

(11)

i=1

Driving signal: DS(t) =

Q=2 

kj sin xτP +j ;

(12)

j =1

Slave: P =4

 dy ni sin yτi + DS(t). = −αy + dt

(14)

 = ay − bx(t + τd ),

P =4

 dx mi sin xτi ; = −αx + dt

ay(t) = bx(t + τd ),

(13)

i=1

According to (6) and (8), the relation between the value of parameters and delays is chosen as: m1 = n1 , m2 − k1 = n2 , m3 = n3 , m4 − k2 = n4 , τ5 = τ2 − τd , τ6 = τ4 − τd . Anticipating synchronization manifold considered in this example is y(t) = x(t + τd ), and chosen τd = 6.0. The adopted values of parameters and delays for simulation are as: α = 2.5, m1 = −0.5, m2 = −13.5, m3 = −0.6, m4 = −14.6, n1 = −0.5, n2 = −0.9, n3 = −0.6, n4 = −0.2, k1 = −12.6, k2 = −14.4, τ1 = 1.5, τ2 = 7.2, τ3 = 2.6, τ4 = 8.4, τ5 = 1.2, τ6 = 2.4. It is clear to observe from Fig. 2(a) that the master’s state variable is delayed in comparison to the slave’s with the anticipation time length of τd = 6.0, in other words, the slave anticipates the master’s trajectory. As illustrated in Fig. 2(b), the attractor of x(t + τd ) versus y(t) shows the desired anticipating synchronization manifold.

(15)

and the dynamics of synchronization error is d dy dx (t + τd ) (16) =a −b . dt dt dt By applying the anticipation time τd to the state variable of (1) and substituting into (16), the dynamics of anticipating synchronization error can be written as   Q P   d ni f (yτi ) + kj f (xτP +j ) = a −αy + dt i=1 j =1   P  − b −αx(t + τd ) + (17) mi f (xτi −τd ) . i=1

Moreover, the relation of yτi can be deduced from (15) as below bxτi −τd + τi . a Thus, (17) becomes

yτi =

(18)

 P  bxτi −τd + τi d ni f = −α + a dt a i=1

+a

Q  j =1

kj f (xτP +j ) − b

P 

mi f (xτi −τd ).

(19)

i=1

The right-hand side of Eq. (18) can be represented as

3. Scheme of projective–anticipating synchronization

bxτi −τd + τi = xτi −τd + τ (app) , a i

In this section, the projective–anticipating synchronization of couped multidelay feedback systems is investigated as an ex-

is a time delay at which the synchronization erwhere τi ror satisfies (20). As a special case, a is equal to b and  is

(app)

(20)

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T.M. Hoang, M. Nakagawa / Physics Letters A 365 (2007) 407–411

(a)

(b)

Fig. 3. (a) State variables x(t) and y(t). (b) Portrait of x(t + τd ) versus y(t).

small; hence, the synchronization scheme is anticipating synchronization and the synchronization error vanishes eventually as presented in the previous section. By substituting the righthand side of (20) into (19), (19) is rewritten as  d ni f (xτi −τd + τ (app) ) = −α + a dt i i=1

+a

kj f (xτP +j ) − b

j =1

P 

mi f (xτi −τd ).

(21)

By the similar reasoning as given in the case of anticipating synchronization, the relation between the value of some delays satisfies (6). The dynamics given in (21) can be rewritten as P ,Q 

d ani f (xτi −τd + τ (app) ) = −α + dt i i=1,j =1  − (bmi − akj )f (xτi −τd ) ,

(22)

bmi − akj = ani .

(23)

Supposed that the value relation between some parameters as given by (23), the dynamics presented in (22) can be reduced to (24) if f (·) is bounded and τ (app) is small: i

P

(24)

i=1

According to the Krasovskii–Lyapunov theory [21,22], the sufficient stability condition for the trivial solution  = 0 is α>

P 



 |ani | supf  (xτi −τd ) ,

(26)

i=1

DS(t) =

i=1

 d ani f  (xτi −τd )τ (app) . = −α + dt i

P =5

 xτi dx mi ; = −αx + dt 1 + xτci Driving signal:

P

Q 

Master:

(25)

i=1

where supf  (·) is the supreme limit of f  (·). Similar to the case of anticipating synchronization presented above, the chosen value of delays is τi  τd to ensure that τP +j is positive. The specific example is simulated to verify the effectiveness of the proposed scheme. Example 2. In this example, the model of projective–anticipating synchronization of coupled five-delay Mackey–Glass systems is defined as below:

Q=3  j =1

kj

xτP +j 1 + xτcP +j

;

(27)

Slave: P =5

 dy yτi ni + DS(t). = −αy + dt 1 + yτci

(28)

i=1

As an exemplar case, the relation between parameters and delays is adopted as: τ6 = τ1 − τd , τ7 = τ3 − τd , τ8 = τ5 − τd , bm1 − ak1 = an1 , m2 = n2 , bm3 − ak2 = an3 , m4 = n4 , bm5 − ak3 = an5 . Moreover, the supreme limit of the function 2 c+1 1c f  (x) is equal to (c−1) 4c at x = ( c−1 ) (see [13]). The values of parameters and delays for simulation are set at: a = −2.5, b = 1.5, α = 16.3, c = 10, m1 = −16.2, m2 = −0.3, m3 = −14.5, m4 = −1.0, m5 = −18.6, n1 = −0.4, n2 = −0.3, n3 = −0.8, n4 = −1.0, n5 = −0.7, k1 = 10.12, k2 = 9.5, k3 = 11.86, τd = 4.6, τ1 = 4.8, τ2 = 3.8, τ3 = 6.2, τ4 = 5.5, τ5 = 4.6, τ6 = 0.6, τ7 = 2.0, τ8 = 0.4. It is clear to observe the scale factor as well as the delay of the master’s state variable in comparison to the slave’s from Fig. 3(a). Moreover, the master’s state variable is inversely proportional to the slave’s due that the negative and positive values are assigned to a and b, respectively. In Fig. 3(b), the attractor of synchronization manifold illustrates the correlation between the state variables of x(t + τd ) and y(t). 4. Concluding remarks In the present Letter, we have realized the schemes of anticipating and projective–anticipating synchronizations of coupled multidelay feedback systems. The simulation result for the case of anticipating synchronization shows that the slave anticipates the motion of the master. In the scheme of projective– anticipating synchronization, accompany with the property of

T.M. Hoang, M. Nakagawa / Physics Letters A 365 (2007) 407–411

anticipation, there exists a scale factor between the amplitude of the master’s and slave’s state variables. It is easy to observe from (23) that the scale factor ab can be changed by changing the value of mi , kj , or ni . Moreover, the driving signal is constituted by the sum of multiple nonlinear transformations of delayed state variable in these synchronization schemes, thus, the driving signal is much more complex in comparison to those in conventional schemes [12–16]. That is really important in consideration to apply these schemes in secure communications. Moreover, the generalized synchronization of coupled multidelay feedback systems can be determined by looking closely in the synchronization manifolds, sufficient conditions, and the relation between parameters. Specifically, let us consider the synchronization system as given in (1)–(3). The scheme of anticipating synchronization with the manifold of y(t) = x(t +τd ) is established if the sufficient condition in (10) is satisfied, or ⎧ P  ⎪ ⎨ α > i=1 |ni ||supf (xτi −τd )|, (29) τP +j = τi − τd (τi  τd for ∀i), ⎪ ⎩ m i − kj = n i . By changing the relation between delays, from τP +j = τi − τd to τP +j = τi + τd , the scheme of lag synchronization with the manifold y(t) = x(t − τd ) is obtained as presented in [17]. In addition, the scheme of projective–anticipating synchronization with the manifold of ay(t) = bx(t + τd ) (a and b are nonzero constants) is achieved by modifying the relation between coefficients given in (29) to ⎧ P  ⎪ ⎨ α > i=1 |ani ||supf (xτi −τd )|, (30) τP +j = τi − τd (τi  τd for ∀i), ⎪ ⎩ bmi − akj = ani . Similarly, the scheme of projective-lag synchronization with the manifold of ay(t) = bx(t − τd ) is resulted by changing the relation between delays given in (30) to ⎧ P  ⎪ ⎨ α > i=1 |ani ||supf (xτi +τd )|, (31) τP +j = τi + τd , ⎪ ⎩ bmi − akj = ani . Obviously, it can be seen as a special case in which the value of τd is set to zero, as a result, the lag and anticipating synchronizations become the scheme of complete synchronization, and the schemes of projective-lag and projective–anticipating synchronizations turn in to the projective synchronization. In summary, the transition between the lag and anticipating synchronization as well as between the projective-lag to

411

projective–anticipating synchronization can be done by adjusting the relation between delays. In addition, the change from the lag to projective-lag synchronization and from the anticipating to projective–anticipating synchronization is realized by modifying the relation between coefficients. Acknowledgements This study was supported in part by the 21st Century COE (Center of Excellence) Program “Global Renaissance by Green Energy Revolution” and the Grant-in-Aid for Scientific Research (15300070) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18] [19] [20] [21] [22]

L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. T. Yang, Int. J. Comput. Cogn. 2 (2004) 81. L. Fabiny, P. Colet, R. Roy, D. Lenstra, Phys. Rev. A 47 (1993) 4287. R. Roy, K.S. Thornburg, Phys. Rev. Lett. 72 (1994) 2009. R.C. Elson, A.I. Selverston, R. Huerta, N.F. Rulkov, M.I. Rabinovich, H.D.I. Abarbanel, Phys. Rev. Lett. 81 (1998) 5692. S.K. Han, C. Kurrer, Y. Kuramoto, Phys. Rev. Lett. 75 (1995) 3190. M.D. Prokhorov, V.I. Ponomarenko, V.I. Gridnev, M.B. Bodrov, A.B. Bespyatovl, Phys. Rev. E 68 (2003) 041913. N.F. Rulkov, M.M. Sushchik, L.S. Tsimring, H.D.I. Abarbanel, Phys. Rev. E 51 (1995) 980. R. Mainieri, J. Rehacek, Phys. Rev. Lett. 82 (1999) 3042. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 76 (1996) 1804. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78 (1997) 4193. H.U. Voss, Phys. Rev. E 61 (2000) 5115. K. Pyragas, Phys. Rev. E 58 (1998) 3067. E.M. Shahverdiev, R.A. Nuriev, R.H. Hashimov, K.A. Shore, Chaos Solitons Fractals 29 (2006) 854. E.M. Shahverdiev, R.A. Nuriev, L.H. Hashimova, E.M. Huseynova, R.H. Hashimov, K.A. Shore, Chaos Solitons Fractals (2006), doi:10.1016/ j.chaos.2006.06.026. E.M. Shahverdiev, R.H. Hashimov, R.A. Nuriev, L.H. Hashimova, E.M. Huseynova, K.A. Shore, Chaos Solitons Fractals 29 (2006) 838. T.M. Hoang, D.M. Minh, M. Nakagawa, J. Phys. Soc. Jpn. 74 (2005) 2374. T.M. Hoang, M. Nakagawa, J. Phys. Soc. Jpn. 75 (2006) 094801. T.M. Hoang, M. Nakagawa, Phys. Lett. A 363 (2007) 218. T.M. Hoang, M. Nakagawa, J. Phys. Soc. Jpn. 75 (2006) 064801. J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. N.N. Krasovskii, Stability of Motion, Stanford Univ. Press, Stanford, 1963.