Chaos, Solitons & Fractals 53 (2013) 1–9

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

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

Projective synchronization of time-varying delayed neuralnetwork with adaptive scaling factors

0960-0779/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.chaos.2013.04.007

⇑ Corresponding author.E-mail addresses: dibakar@isical.ac.in (D. Ghosh), santo@iscass.org (S.

Banerjee).

Dibakar Ghosh a,⇑, Santo Banerjee b

a Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, Indiab Institute for Mathematical Research, University Putra Malaysia, Malaysia

a r t i c l e i n f o

Article history:Received 7 May 2012Accepted 22 April 2013Available online 21 May 2013

a b s t r a c t

In this work, the projective synchronization between two continuous time delayed neuralsystems with time varying delay is investigated. A sufficient condition for synchronizationfor the coupled systems with modulated delay is presented analytically with the help of theKrasovskii–Lyapunov approach. The effect of adaptive scaling factors on synchronizationare also studied in details. Numerical simulations verify the effectiveness of the analyticresults.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The synchronization phenomenon has received greatattention in recent years, in particular for the potentialtechnological applications in all areas of engineering, sci-ence and management. Several theoretical works and lab-oratory experimentations have also demonstrated thepivotal role of this phenomenon in secure communications[1,2]. Different types of synchronization techniques havebeen proposed in the literatures, e.g., complete synchroni-zation (CS) [1,3], phase synchronization (PS) [4], lag syn-chronization (LS) [5], generalized synchronization (GS)[6], generalized projective synchronization (GPS) [7], mul-tiplexing synchronization (MS) [8], etc.

Gonzalez-Miranda [9] observed that when chaotic sys-tems exhibit invariance properties under a special type ofcontinuous transformation, amplification and displace-ment of the attractor is occurred. This degree of amplifica-tion or displacement obtained is smoothly dependent onthe initial condition. By this definition complete synchroni-zation is not a special case of projective synchronization.Later Mainieri and Rehacek [10] in 1999, this type of syn-chronization is called as projective synchronization, wherethey declared that the two partially linear three-dimen-

sional chaotic systems could be synchronized up to a scal-ing factor a. The scaling factor is a constant transformationbetween the synchronized variables of the master andslave systems. Projective synchronization is not in the cat-egory of GS because the slave system of projective syn-chronization is not asymptotically stable. The responsesystem attractor possesses the ‘‘same topological charac-teristic (such as Lyapunov exponents and fractal dimen-sions)’’ as the slave system attractor [11]. Recently,Hoang and Nakagawa [12] proposed a new type of syn-chronization in multi-delay feedback systems, which iscalled projective-anticipating synchronization. This syn-chronization is a combination of projective and anticipa-tory synchronization and the states of master and slaveare related by ay(t) = bx(t + s) (s > 0) where a and b arenonzero real numbers. Projective-lag synchronization aredefined as ay(t) = bx(t � s) (s > 0).

Projective synchronization is very important for its pro-portionality between the synchronized dynamical states.There has been a lot of interest on projective synchroniza-tion with time delayed systems due to its potential appli-cations in encryption. The inherent delay can generatehigh order chaos which is effective in terms of security.Also the scaling factor/function can transfer a binary digitalto M-nary, which is very useful for faster communications.With the development of research on complex systems,more and more researchers carried out the study aboutcomplex dynamical behaviors on networks. While most

2 D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9

studies focused on complete and phase synchronization invarious networks, little attention has been paid to projec-tive synchronization. Very recently there has been somedevelopments on the generalized analytical conditions[13,14] on projective synchronization of time delayedsystems.

In this paper we have investigated the projective syn-chronization phenomenon in two neural time delayed sys-tems with delay time modulation. The linear couplingyielding synchronization between two systems can be con-sidered as a combination of a positive-delayed feedback ofthe drive variables X on Y and of a negative feedback of thedriven variables Y onto itself. By exploiting the analogywith electronic circuits this could be thought as meta mod-el of hierarchical signal transfer between neural structures.The idea based on the Lyapunov stability, we theoreticallyanalyze both the existence and sufficient stability condi-tions of the projective-anticipating, projective, and projec-tive-lag synchronization of time-delayed chaotic systemson random networks. In this study, we extend the studyof projective synchronization in infinite-dimensional cha-otic system, while most of the studies [9,10] are done inlow dimensional systems. Projective synchronization isusually observed in partially linear system and the scalingfactor is dependent on initial conditions but here we pro-pose a projective synchronization scheme without the lim-itation of partial linearity and initial conditions. Finally wehave shown that our numerical calculations well supportthe analytic results.

The plan of the rest paper is as follows: In Section 2,some chaotic properties is investigated in two time delayneural system. Section 3 represents stability condition foranticipatory, complete and lag synchronization in delayfeedback coupled. The stability condition for projective-anticipatory, projective and projective-lag synchronizationin coupled neural systems is investigated in Section 4. InSection 5, adaptive scaling factor is considered. Finally con-clusions are made in Section 6.

2. Two-neural system and its chaotic properties

In general n-neuron systems with a single time-delaycan be written as

_xiðtÞ ¼ �uixiðtÞ þXn

j¼1

v ijf ðxjðtÞÞ þXn

j¼1

wijf ðxjðt � sÞÞ þ Ii;

i ¼ 1;2; . . . ;n ð1Þ

where f(�) is the neural activation function, ui > 0, vij and wij

are real numbers, s P 0 is the only time-delay and Ii areexternal inputs. The initial conditions are xi(t) = /i(t),t 2 [�s,0], with some given continuous functions/i : [�s,0] ? R. In this paper, without loss of generality,initial conditions are always chosen as constant functionson [�s,0].

In this paper, we consider a two-neuron network mod-eled by the following nonlinear delay differential equationas

_xðtÞ ¼ �UxðtÞ þ VfðxðtÞÞ þWfðxðt� sÞÞ þ I ð2Þ

where x = (x1,x2)T, f(x(t)) = (f(x1(t)), f(x2(t)))T, f(x(t � s)) =

(f(x1(t � s)), f(x2(t � s)))T and U ¼ u1 00 u2

� �,

V ¼ v11 v12

v21 v22

� �, W ¼ w11 w12

w21 w22

� �, I ¼ I1

I2

� �. Such a

model describes the computational performance of a Hop-field network [15], where, each neuron is represented by alinear circuit consisting of a resistor and a capacitor andeach neuron is connected to another via nonlinear activa-tion function f(�) by the synaptic weight wij (i – j).

If we choose a odd symmetry activation functionf(x) = tanh(x) and I = 0, it is easy to see that (0,0) is alwaysa fixed point of system (2) and if ðx�1; x�2Þ be a fixed pointthen ð�x�1;�x�2Þ is also a fixed point. So the fixed pointsare symmetry in the real plane about the origin. Lu [16]investigated numerically the eigenvalues and eigenvectorsat these three fixed points for different values of ui, vij andwij(i, j = 1,2). In this section we investigate the changes inthe qualitative behavior of the system’s attractor as theparameters w22 and s varied. We choose the parametervalues as u1 = 1.0 = u2, v11 = 2.0, v12 = �0.1, v21 = �5.0,v22 = 1.5, w11 = �1.5, w12 = �0.1, w21 = �0.2, w22 = �1.0. Ats = 1.0, the system (2) shows chaotic attractor in Fig. 1(a)with initial conditions x1(t) = �0.5, x2(t) = �0.6 (solid line)and x1(t) = 0.5, x2(t) = 0.6(dotted line) for t 2 [�1,0]. Bifur-cation diagram with changes parameter w22 is shown inFig. 1(b) with initial condition x1(t) = �0.5, x2(t) = �0.6 fort 2 [�1,0]. For 0.9 < s < 0.978, there is a stable limit cycleattractor [Fig. 1(c)]. A period two orbit is observed for0.978 6 s < 0.984. For 0.984 6 s 6 0.9865, there is periodfour orbits. For s > 0.9865, numerical simulation showschaotic attractors. An interesting phenomena is observedin this bifurcation diagram that for 1.005 6 s 6 1.18, thereare double-scroll-like chaotic attractor with initial condi-tions x1(t) = �0.5, x2(t) = �0.6 or x1(t) = 0.5, x2(t) = 0.6 fort 2 [�s,0]. There are several periodic windows betweenthis interval, such as, period three window exist nears = 1.01 and 1.101 6 s 6 1.104, period five window existin 1.0335 6 s 6 1.0375. In Fig. 1(d), the parameter regionbetween w22 and s is shown. The gray region representperiod one, black region represent period two and whiteregion for chaotic region.

In next we replace the constant time-delay s to modu-lated time-delay [17] as s(t) = s0 + asin(xt) where s0 is thezero-frequency component, a is amplitude and xis the fre-quency of the modulation. For s0 = 1.07, a = 0.03 andx = 0.1, s(t) varies between 1.04 and 1.1 where the system(2) shows double-scroll-like chaotic attractor (in Fig. 1(e)).

3. Coupled system and stability condition for chaossynchronization

We consider drive system as

_xðtÞ ¼ �UxðtÞ þ VfðxðtÞÞ þWfðxðt� sðtÞÞÞ ð3Þ

and response system as

_yðtÞ ¼ �UyðtÞ þ VfðyðtÞÞ þWfðyðt� sðtÞÞÞþ K½xðt� s1Þ � yðtÞ� ð4Þ

−1.5 −1 −0.50.9

1

1.1

1.2

1.3

1.4

1.5

w22

τ

−0.5 0 0.5

−3

−2

−1

0

1

2

3

x1(t)

x 2(t)

−1.5 −1 −0.5

0.5

0.6

0.7

0.8

0.9

w22

x 1(t)

0.9 1 1.1 1.2 1.3

−2

0

2

4

τ

x 2(t)

−0.5 0 0.5

−3

−2

−1

0

1

2

3

x1(t)

x 2(t)

(a) (b)

(c) (d)

(e)

Fig. 1. (a) Chaotic attractor of system (2) with initial conditions x1(t) = �0.5, x2(t) = �0.6 (solid line) and x1(t) = 0.5, x2(t) = 0.6 (dotted line) for t 2 [�1,0]. (b)Bifurcation diagram with respect to w22 for s = 1.0. (c) Bifurcation diagram with respect to s for w22 = �1.0. (d) Two parameters bifurcation diagram in therange w22 2 (�1.5,0.0) and s 2 (0.9,1.5). (e) Double scroll like chaotic attractor for s0 = 1.07, a = 0.03, x = 0.1.

D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9 3

where K = (k1,k2)T is coupling strength vector, s1 iscoupling delay. For s1 < 0, s1 = 0 and s1 > 0, anticipating,complete and lag synchronization occur between (3) and(4) respectively.

Let D1 = x1(t � s1) � y1, D2 = x2(t � s1) � y2 be the syn-chronization errors. Then error dynamics is

_D1 ¼ �r1D1 þ r2D2 þ s1D1ðt � sÞ þ s2D2ðt � sÞ ð5Þ

_D2 ¼ �r3D1 þ r4D2 þ s3D1ðt � sÞ þ s4D2ðt � sÞ ð6Þ

where r1 = u1 + k1 � v11f0(n1), r2 = v12f0(n2), r3 = v21f0(n1), r4 -4 = u2 + k2 � v22f0(n2), s1 = w11f0(n3), s2 = w12f0(n4), s3 = w21-

f0(n3), s4 = w22f0(n4), ni is a value between xi(t � s1) and yi

(i ¼ 1;2), n3 is a value between x1(t � s1 � s) andy1(t � s), n4 is a value between x2(t � s1 � s) andy2(t � s). D1 = 0 = D2 is the fixed point of the error system

(5) and (6). From Krasovskii theory [18], we introduce apositive definite functional as

VðtÞ ¼ 12ðD2

1 þ D22Þ þ h1ðtÞ

Z 0

�sðtÞD2

1ðt þ h1Þdh1

þ h2ðtÞZ 0

�sðtÞD2

2ðt þ h2Þdh2 ð7Þ

where h1(t) > 0, h2(t) > 0 are two functions of time[8,17,19]. Taking differentiation with respect to t of (7)

_VðtÞ ¼ ðD1_D1 þ D2

_D2Þ þ _h1ðtÞZ 0

�sðtÞD2

1ðt þ h1Þdh1

þ _h2ðtÞZ 0

�sðtÞD2

2ðt þ h2Þdh2 þ h1ðtÞ½D21 � D2

1ðt � sÞ

þ D21ðt � sÞs0� þ h2ðtÞ½D2

2 � D22ðt � sÞ þ D2

2ðt � sÞs0�

where s0 ¼ dsdt.

0 50 100 150 200−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

t

x 1(t) −

y1(t)

0 20 40 60 80 100 120−2

−1.5

−1

−0.5

0

0.5

1

t

x 1(t), y

1(t)

0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

1

t

x 1(t),

y 1(t)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x1(t+τ1)

y 1(t)

(a) (b)

(c) (d)

Fig. 2. (a) Complete synchronization error between (3) and (4) for k = 0.4, (b) time series of x1(t) (solid line) and y1(t) (dotted line) showing lagsynchronization, (c) time series of x1(t) (solid line) and y1(t) (dotted line) showing anticipatory synchronization and (d) corresponding anticipatingsynchronization manifold.

4 D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9

If _h1ðtÞ 6 0 and _h2ðtÞ 6 0 for arbitrary t and using Eqs.(5) and (6),

_VðtÞ < � r1 � h1 �s2

1

4h1ð1� s0Þ �s2

2

4h2ð1� s0Þ

� �D2

1

� r4 � h2 �s2

3

4h1ð1� s0Þ �s2

4

4h2ð1� s0Þ

� �D2

2

þ r2 þ r3 þs1s3

2h1ð1� s0Þ þs2s4

4h2ð1� s0Þ

� �D1D2

Using the formula 2AB 6 A2 + B2 we can obtain

_VðtÞ 6 �F1ðh1;h2ÞD21 � F2ðh1; h2ÞD2

2

where

F1ðh1; h2Þ ¼ r1 � h1 �s1ðs1 þ s3Þ4h1ð1� s0Þ �

s2ðs2 þ s4Þ4h2ð1� s0Þ �

r2 þ r3

2

F2ðh1; h2Þ ¼ r4 � h2 �s3ðs1 þ s3Þ4h1ð1� s0Þ �

s4ðs2 þ s4Þ4h2ð1� s0Þ �

r2 þ r3

2

_VðtÞ < 0 if minimum value of F1(h1,h2) and F2(h1,h2) great-er than zero. From this one can obtain

r1 >r2 þ r3

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis1ðs1 þ s3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� s0p

and

r4 >r2 þ r3

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis4ðs2 þ s4Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� s0p

Finally the sufficient condition for synchronization are

k1 > v11f 0ðn1Þ � u1 þv12f 0ðn2Þ þ v21f 0ðn1Þ

2

þ f 0ðn3Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw11ðw11 þw21Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� s0p

and

k2 > v22f 0ðn2Þ � u2 þv12f 0ðn2Þ þ v21f 0ðn1Þ

2

þ f 0ðn4Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw22ðw12 þw22Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� s0p

The above result can be summarized below in form of atheorem as

Theorem I. If the derivative of the activation function f(�) isbounded i.e. there exist a real constant M <1 such that jf0(�)j < M,then the coupled system (3) and (4) are synchronized if

k¼ k1 ¼ k2 >Mðv12þv21Þ

2

þM max v11�u1

Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw11ðw11þw21Þ

pffiffiffiffiffiffiffiffiffiffiffiffi1�s0p ;v22�

u2

Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw22ðw12þw22Þ

pffiffiffiffiffiffiffiffiffiffiffiffi1�s0p

( )

From the above set of parameter values, systems (3) and(4) are synchronized if k > 0.0493. For s1 = 0, k = 0.4 com-plete synchronization shown in Fig. 2(a). For s1 = 1.0, lagsynchronization occurs in Fig. 2(b). Anticipating synchroni-zation occurs for s1 = �1.0, are shown in Fig. 2(c). Thecorresponding anticipating synchronization manifold isshown in Fig. 2(d).

0 50 100 150 200−2

−1

0

1

2

t

x 1(t), y

1(t)

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

x1(t)

y 1(t)

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

x1(t)

y 1(t)

0 50 100 150 200−2

−1

0

1

2

t

x 1(t), y

1(t)

(a) (b)

(c) (d)

Fig. 3. (a) Projective synchronization between x1(t) and y1(t) for a = 3.0, b = 2.0, k = 2.5, (b) corresponding projective synchronization manifold, (c) antiphaseprojective synchronization between the systems (8) and (9) for a = �3.0, b = 2.0 and (d) corresponding antiphase projective synchronization manifold.

0 50 100 150 200−3

−2

−1

0

1

2

t

x 1(t), y

1(t)

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

x1(t−τ1)

y 1(t)

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

x1(t−τ1)

y 1(t)

0 50 100 150 200−2

−1

0

1

2

3

t

x 1(t), y

1(t)

(a) (b)

(c) (d)

Fig. 4. (a) Projective lag synchronization between the systems (8) and (9) for a = 3.0, b = 2.0, k = 2.5, (b) corresponding manifold, (c) antiphase projective lagsynchronization for a = �3.0 and (d) corresponding antiphase projective lag synchronization manifold.

D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9 5

0 20 40 60 80 100 120 140 160 180 200−1.5

−1

−0.5

0

0.5

1

1.5(a)

t

x 1(t),

y 1(t)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−1.5

−1

−0.5

0

0.5

1

1.5(b)

x1(t+τ1)

y 1(t)

Fig. 5. (a) Projective anticipating synchronization between the systems (8) and (9) and (b) corresponding manifold.

6 D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9

4. Sufficient condition for projective synchronization

We consider coupled system for projective synchroniza-tion as

_xðtÞ ¼ �UxðtÞ þ VfðxðtÞÞ þWfðxðt� sðtÞÞÞ ð8Þ

and

_yðtÞ¼�UyðtÞþVfðyðtÞÞþWfðyðt�sðtÞÞÞþK½axðt�s1Þ�byðtÞ�ð9Þ

where a and b are desired scaling factors for projective syn-chronization. For a = b = 1, we get the anticipating, completeand lag synchronization as before and for different sign ofscaling factors, a new types of anti-phase synchronizationarises. We define projective synchronization errors as

D1 ¼ ax1ðt � s1Þ � by1 ð10Þ

D2 ¼ ax2ðt � s1Þ � by2 ð11Þ

then one have an error dynamics as

_D1¼�ðu1þk1ÞD1þv11½af ðx1ðt�s1ÞÞ�bf ðy1Þ�þv12½af ðx2ðt�s1ÞÞ�bf ðy2Þ�þw11½af ðx1ðt�s�s1ÞÞ�bf ðy1ðt�sÞÞ�þw12½af ðx2ðt�s�s1ÞÞ�bf ðy2ðt�sÞÞ�

ð12Þ

_D2¼�ðu2þk2ÞD1þv21½af ðx1ðt�s1ÞÞ�bf ðy1Þ�þv22½af ðx2ðt�s1ÞÞ�bf ðy2Þ�þw21½af ðx1ðt�s�s1ÞÞ�bf ðy1ðt�sÞÞ�þw22½af ðx2ðt�s�s1ÞÞ�bf ðy2ðt�sÞÞ�

ð13Þ

Eqs. (10) and (11) can be written as

x1ðt � s1Þ ¼by1 þ D1

a¼ y1 þ Dapp

1 ð14Þ

x2ðt � s1Þ ¼by2 þ D2

a¼ y2 þ Dapp

2 ð15Þ

where Dappi (i ¼ 1;2) represents time-delay at which

synchronization error satisfies (14) and (15) for a specialcase a = b and Di (i ¼ 1;2) are small. Eqs. (12) and (13)can be written as

_D1 ¼ �r01Dapp1 þ r02D

app2 þ s01D

app1 ðt � sÞ þ s02D

app2 ðt � sÞ ð16Þ

_D2 ¼ r03Dapp1 � r04D

app2 þ s03D

app1 ðt � sÞ þ s04D

app2 ðt � sÞ ð17Þ

where r01 ¼ r1, r02 ¼ ar2, r03 ¼ ar3, r04 ¼ r4, s01 ¼ as1, s02 ¼ as2,s03 ¼ as3, s04 ¼ as4. Here ri and si (i ¼ 1;2;3;4) are same asin previous section. Similarly as previous section, we ob-tain the sufficient condition for projective synchronizationas

k¼ k1 ¼ k2 >Maðv12þv21Þ

2

þM max v11�u1

Mþa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw11ðw11þw21Þ

pffiffiffiffiffiffiffiffiffiffiffiffi1�s0p ;v22�

u2

Mþa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw22ðw12þw22Þ

pffiffiffiffiffiffiffiffiffiffiffiffi1�s0p

( )

ð18Þ

For s1 = 0 and a = 3.0, b = 2.0, k = 2.5 which satisfied thestability condition (18), the time series of the drive systemx1(t) (solid line) and response system y1(t) (dotted line)shown in Fig. 3(a). The phase angles between the synchro-nized chaotic attractors is zero and identical pattern.Synchronization manifold in x1(t) � y1(t) plane is shown

0 100 200 300 400 5000.6

0.65

0.7

0.75

0.8

0.85

t

α,β

0 50 100 150 200−1

−0.5

0

0.5

1

t

x 1(t) −

y1(t)

0 100 200 300 400 5001

1.1

1.2

1.3

1.4

1.5

t

α,β

0 5 10 15 20−1

−0.5

0

0.5

1

t

x 1(t) −

y1(t)

(a) (b)

(c) (d)

Fig. 6. (a) Time variation of the scaling factors a and b for low adaptive gain ca = 0.1, cb = 0.2, k = 2.5, (b) projective synchronization error, (c) for highadaptive gain ca = 1.1, cb = 1.2, time variation of a and b and (d) corresponding error.

D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9 7

in Fig. 3(b). For opposite sign of scaling factors, we getanti-phase synchronization in Fig. 3(c) and correspondingmanifold in Fig. 3(d).

For s1 > 0 as before and a = 3.0, b = 2.0, k = 2.5, we getprojective-lag synchronization with identical phase pat-tern are shown in Fig. 4(a) and corresponding manifoldin Fig. 4(b). For a = �3.0, we get anti-phase lag synchroni-zation with anti-phase pattern and difference of phase an-gels of the synchronized trajectories is p are shown inFig. 4(c) and corresponding manifold in Fig. 4(d).

For s1 = �1.0, we get projective-anticipating synchroni-zation with identical phase pattern. Time series of x1(t) andy1(t) are shown in Fig. 5(a) and corresponding synchroniza-tion manifold in Fig. 5(b).

5. Projective synchronization using adaptive scalingfactors

In the previous section, we have chosen the scaling fac-tors a, b arbitrarily. Here we consider a method of projec-tive synchronization to generate the adaptive scalingfactors. Corresponding driving and response systems aregiven by

_xðtÞ ¼ �UxðtÞ þ VfðxðtÞÞ þWfðxðt� sðtÞÞÞ ð19Þ

_yðtÞ ¼ �UyðtÞ þ VfðyðtÞÞ þWfðyðt� sðtÞÞÞþ K½aðtÞxðt� s1Þ � bðtÞyðtÞ� ð20Þ

where the generating equations for a(t) and b(t) can bewritten as

_a ¼ ca½x1ðt � s1Þ � y1ðtÞ� ð21Þ

_b ¼ cb½x2ðt � s1Þ � y2ðtÞ� ð22Þ

where ca and cb are the adaptive gains. The synchroniza-tion time depends upon the numerical values of the adap-tive gains. Fig. 6(a) is the time variation of a (solid line) andb (dotted line), and converges to a (t) ? 0.87183 andb(t) ? 0.8699 as t ?1.

We define the synchronization errors as

D1 ¼ aðtÞx1ðt � s1Þ � bðtÞy1ðtÞ ð23Þ

D2 ¼ aðtÞx2ðt � s1Þ � bðtÞy2ðtÞ ð24Þ

Fig. 6(b) represents the error of exact projective synchroni-zation for k = 2.5, ca = 0.1, cb = 0.2, s1 = 0.0. For the lowadaptive gains, we can see that the synchronization occursafter t = 150. Fig. 6(c) is the time variation of a (solid line),b(dotted line) for ca = 1.1, cb = 1.2 where a, b converges to1.45891 and 1.4579 respectively. Fig. 6(d) represents thecorresponding exact projective synchronization error D1

occurs after t = 7.For s1 > 0, ca = 0.005, cb = 0.009, the scaling factors a(t)

and b(t) converges respectively to the values a(t) ?0.8074 and b(t) ? 0.7914. For large adaptive gainsca = 0.01, cb = 0.02 and the scaling factors converges toa(t) ? 0.8096 and b(t) ? 0.8033 (Fig. 7(a)) and corre-sponding time variation of x1(t) (solid line) and y1(t) (dot-ted line) are shown in Fig. 7(b).

For s1 < 0, ca = 0.01, cb = 0.02, the scaling factors con-verges to a(t) ? 0.82 (solid line) and b(t) ? 0.8135 (dotted

0 200 400 600 800 10000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9(a)

α,β

t0 20 40 60 80 100

−2

−1.5

−1

−0.5

0

0.5

1

1.5(b)

t

x 1(t), y

1(t)

0 200 400 600 800 10000.6

0.65

0.7

0.75

0.8

0.85

t

α,β

(c)

0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

1

1.5

t

x 1(t),

y 1(t)

(d)

Fig. 7. (a) For low adaptive gain, the time variation of the scaling factors a and b, (b) projective lag synchronization between x1(t) and y1(t), (c) variation of aand b for high adaptive gain and (d) projective anticipating synchronization between x1(t) and y1(t).

8 D. Ghosh, S. Banerjee / Chaos, Solitons & Fractals 53 (2013) 1–9

line) (Fig. 7(c)) and corresponding time variation of x1(t)(solid line) and y1(t) (dotted line) are shown in Fig. 7(d).

6. Conclusions

In conclusion, we have studied analytically and numer-ically the projective-lag, projective or projective-anticipa-tory synchronization in modulated time-delayed coupledsystems, related to neural networks. Transition among pro-jective-lag, projective or projective-anticipatory synchro-nization can be obtained by adjusting the coupling delay.Compared with the previous works [7,9–12], the proposedone has the following advantages: (1) The projective syn-chronization is usually observed in partially linear chaoticsystems and the scaling factor is depended on initial condi-tions. In this analysis, the synchronization in two neuraltime delayed systems is without the limitation of partiallylinearity and the scaling factor is depended on the controllaw but not on the initial conditions. (2) While most of thestudy on projective is done in chaotic or hyperchaotic sys-tems, in this paper we investigated projective-lag, projec-tive or projective-anticipatory synchronization in twocoupled delayed neural chaotic systems with modulateddelay time. (3) The choice of scaling factors in projective

synchronization is very important. An adaptive rule forthe scaling factors is introduced in this analysis.

Acknowledgement

The authors are very thankful to Prof. Roberto Livi andProf. J.M.G. Miranda for their significant comments andsuggestions.

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