International Journal of Bifurcation and Chaos, Vol. 11, No. 6 (2001) 1707–1722c© World Scientific Publishing Company

MASTER SLAVE SYNCHRONIZATION OF LUR’ESYSTEMS WITH TIME-DELAY

M. E. YALCIN, J. A. K. SUYKENS∗ and J. VANDEWALLEKatholieke Universiteit Leuven, Department of Electrical Engineering, ESAT-SISTA,

Kardinaal Mercierlaan 94, B-3001 Leuven (Heverlee), Belgium

Received July 21, 2000; Revised August 29, 2000

In this paper time-delay effects on the master–slave synchronization scheme are investigated.Sufficient conditions for master–slave synchronization of Lur’e systems are presented for aknown time-delay in the master and slave systems. A delay-dependent synchronization cri-terion is given based upon a new Lyapunov–Krasovskii function. The derived criterion is asufficient condition for global asymptotic stability of the error system, expressed by means of amatrix inequality. The feedback matrix follows from solving a nonlinear optimization problem.The method is illustrated for the synchronization of Chua’s circuits, 5-scroll attractors andhyperchaotic attractors.

1. Introduction

Since the work of Pecora and Carroll [1991], the re-search on chaotic synchronization has received con-siderable attention. It basically deals with sufficientconditions for synchronization of identical or non-identical systems [Wu & Chua, 1994]. Synchroniza-tion schemes have been investigated with local andglobal synchronization [Hasler, 1994], robust syn-chronization [Suykens et al., 1999], partial synchro-nization [Hasler et al., 1998] and generalized syn-chronization [Kocarev et al., 1996]. An overviewof synchronization methods has been recently pre-sented in [Chen & Dong, 1998]. Synchronizationhas opened the way to investigate an engineeringapplication of chaos, which is chaotic communica-tions. Since 1992, a number of chaotic commu-nication schemes have been proposed [Oppenheimet al., 1992; Hasler, 1994; Wu & Chua, 1994]. In thispaper, we deal with propagation delay in master–slave synchronization schemes. This problem hasbeen recently reported in [Chen & Liu, 2000] whichintroduces the possibility of applying chaotic syn-

chronization to optical communication. Chen andLiu [2000] called this problem a phase sensitivitydue to the distance between two remote chaoticsystems and has reported that the existence of atime-delay can destroy synchronization. Further-more, the synchronization and bifurcation phenom-ena of two chaotic circuits which are coupled by adelay line have been investigated e.g. by Koike et al.[1997]. Experimental confirmation of synchroniza-tion of two chaotic circuits with the existence ofdelay has been studied by Kawate et al. [2000].However, theoretical studies of this problem are stilllacking.

On the other hand, in the area of control the-ory time-delay systems have been investigated andit is well known that delays often result in insta-bilities. Therefore, stability analysis of time-delaysystems is an important subject in control the-ory [Hale, 1977; Mori et al., 1983; Kamen, 1982,1983; Tissir & Hmamed, 1996]. In the literature,stability criteria for time-delay systems are classi-fied into two main categories: delay-independentcriteria [Kamen, 1982, 1983; Chen & Latchman,

∗Author for correspondence.E-mail: johan.suykens@esat.kuleuven.ac.be

1707

1708 M. E. Yalcin et al.

1995] and delay-dependent criteria [Mori, 1985;Mori & Kamen, 1989; Chen, 1995]. Recently,time-delay systems have been intensively studied in[Mahmoud, 2000]. The Lyapunov–Krasovskii func-tion [Krasovskii & Brenner, 1963] is a candidatefunction for asymptotical stability analysis of lin-ear time-delay systems. Boyd et al. [1994] derivedlinear matrix inequalities (LMI’s) for multiple de-lay [Boyd et al., 1994] from a Lyapunov–Krasovskiifunction.

In this paper we consider master–slave schemeswith identical Lur’e systems. Lur’e systems are aclass of nonlinear systems which can be representedas a linear dynamical system, feedback intercon-nected to a nonlinearity that satisfies a sector con-dition. We suppose that the output of the mastersystem is received at the slave system with delay(τ), which is assumed to be a known value. A delay-dependent criterion for global asymptotic stabilityof the error system is given which is expressed as amatrix inequality and is derived from an extendedLyapunov–Krasovskii function. The theoretical re-sult is illustrated by a number of simulation ex-amples for Chua’s circuits, n-scroll attractors andhyperchaotic systems.

This paper is organized as follows. In Sec. 2we present the master–slave synchronizationscheme of Lur’e systems with time-delay. In Sec. 3the error system and sufficient conditions forglobal asymptotic stability based on a Lyapunov–Krasovskii function for the delay-dependent anddelay-independent cases are derived. In Sec. 4 weintroduce a new Lyapunov function for the master–slave synchronization scheme and a sufficient con-dition for global asymptotic stability is given. Fi-nally, in Sec. 5 examples are given for Lur’e systems:

Chua’s circuit, 5-scroll attractors and hyperchaoticattractors.

2. Time-Delay SynchronizationScheme

Consider the following master–slave synchroniza-tion scheme with static error feedback and time-delay τ

M :

{x(t) = Ax(t) +Bσ(Cx(t))

p(t) = Hx(t)

S :

{y(t) = Ay(t) +Bσ(Cy(t)) + u(t)

q(t) = Hy(t)

C :{u(t) = G(p(t− τ)− q(t− τ))

(1)

with master system M, slave system S and con-troller C (Fig. 1). The master and slave systems areLur’e systems with state vectors x, y ∈ Rn, outputof subsystems p, q ∈ Rl, respectively, and matricesH ∈ Rl×n, A ∈ Rn×n, B ∈ Rn×nh , C ∈ Rnh×n.σ(·) satisfies a sector condition [Vidyasagar, 1993;Khalil, 1993] with σi(·) i = 1, 2, . . . , nh belongingto sector [0, k], i.e. σi(ξ)[σi(ξ) − kξ] ≤ 0, ∀ ξ fori = 1, 2, . . . , nh. The scheme aims at synchroniz-ing the master system to the slave system by ap-plying full state error feedback to the slave systemwith control signal u ∈ Rn and feedback matrixG ∈ Rn×l. Master–slave synchronization has beenstudied for τ = 0 in [Wu & Chua, 1994; Curranet al., 1997]. The difference between this work andthe present paper is the time-delay in the outputs,i.e. p(t − τ) and q(t − τ) instead of p(t) and q(t),respectively.

Fig. 1. Synchronization scheme.

Master–Slave Synchronization with Time-Delay 1709

3. Error System of Time-DelaySynchronization Scheme

Defining the signal e(t) = x(t) − y(t) one obtainsthe error system E

E : e = Ae+Bη(Ce; y) + Fe(t− τ) (2)

with e = e(t), F = −GH and η(Ce; y) = σ(Ce +Cy) − σ(Cy). One assumes that the nonlinearityη(Ce, y) belongs to sector [0, k] [Curran & Chua,1997; Suykens & Vandewalle, 1996]

0 ≤ ηi(cTi e; y)

cTi e=σi(c

Ti e+ cTi y)− σi(cTi y)

cTi e≤ k ,

∀ e, y; i = 1, 2, . . . , nh (3)

The following inequality holds then [Boyd et al.,1994; Khalil, 1992; Vidyasagar, 1993]

ηi(cTi e; y)[η(cTi e; y)− kcTi e] ≤ 0 ,

∀ e, y; i = 1, 2 . . . , nh . (4)

Stability of the error system without time-delay(τ = 0) in the feedback has been derived forquadratic Lyapunov functions [Wu & Chua, 1994;Curran & Chua, 1997] and Lur’e–Postnikov func-tions [Curran et al., 1997]. The Lyapunov–Krasovskii function is a candidate Lyapunov func-tion for time-delay systems [Krasovskii & Brennel,1963; Hale, 1977; Mahmoud, 2000]:

V1(e) = eTPe+

∫ 0

−τe(t+ s)TQe(t+ s)ds ,

P = P T > 0 , Q = QT > 0 .

(5)

By taking Lyapunov–Krasovskii function Eq. (5), itis straightforward to find a sufficient condition forglobal asymptotic stability of the error system E .

Theorem 1. Let Λ = diag{λi} be a diagonal ma-trix with λi ≥ 0 for i = 1, 2, . . . , nh then a suffi-cient condition for global asymptotic stability of theerror system E , based on the Lyapunov–Krasovskiifunction, is given by the matrix inequality

Y =

ATP + PA+Q PB + kCTΛ PF

BTP + kΛTC −2Λ 0

FTP 0 −Q

< 0 .

(6)

Proof. By taking the time derivative of the

Lyapunov–Krasovskii function Eq. (5) and apply-ing the S-procedure [Boyd et al., 1994], using theinequalities from the nonlinearities, one obtains

V1(e) = eTPe+ eTP e+ e(t)TQe(t)

− e(t− τ)TQe(t− τ)

≤ eT (ATP + PA)e+ ηTBTPe+ eTPBη

+ e(t− τ)TFTPe+ eTPFe(t− τ)

+ e(t)TQe(t)− e(t− τ)TQe(t− τ)

−∑i

2λiηi(ηi − kcTi e)

= ξTY ξ < 0

where ξ = [e(t); η; e(t− τ)]. �

The matrix inequality (6) does not include in-formation on the delay. Therefore this result isa delay-independent stability criterion for synchro-nization. A necessary condition for Y < 0 is thatthe linear part of this system (A matrix) must bestable. An analysis for Chua’s circuit, 5-scroll at-tractors and hyperchaotic attractors shows that afeasible point could not be found for these exam-ples with Y negative definite. For Chua’s circuit,a Lur’e representation is given by Guzelis [1993],which has a stable A matrix, but even in this caseone could not find a feasible point. In the literatureit has been recommended that if delay-independentcriteria fail, delay-dependent conditions should beapplied [Tissir & Hmamed, 1996]. The followingLyapunov–Krasovskii function is a candidate func-tion for a delay-dependent condition [Mahmoud,2000]

V2(e) = eTPe+ r1

∫ t

t−τ

∫ t

t+θ[eT (s)ATAe(s)]dsdθ

+ r2

∫ t

t−τ

∫ t

t−τ+θ[eT (s)FTFe(s)]dsdθ (7)

where P = P T > 0 and r1 > 0, r2 > 0 are weightfactors. However, taking Eq. (7) as a candidatefunction and deriving an LMI from this would notbe possible. For this reason we propose a new candi-date Lyapunov function in the next section in orderto find a sufficient condition for global asymptoticstability of error system E .

1710 M. E. Yalcin et al.

4. Delay-Dependent SynchronizationCriterion for Lur’e Systems

Now, we propose a new Lyapunov–Krasovskii func-tion for delay-dependent stability criteria

V3(e) = eTPe+ r1

∫ t

t−τ

∫ t

t+θ[eT (s)ATAe(s)]dsdθ

+ r2

∫ t

t−τ

∫ t

t+θ−τ[eT (s)FTFe(s)]dsdθ

+ r3

∫ t

t−τ

∫ t

t+θ[ηT (Ce(s);

y(s))BTBη(Ce(s); y(s))]dsdθ (8)

where P = P T > 0 and r1 > 0, r2 > 0, r3 > 0.By taking this new Lyapunov function Eq. (8) wefind a new sufficient condition for global asymptoticstability of error system E .

Theorem 2. Let Λ = diag{λi} be a diagonal ma-trix with λi ≥ 0 for i = 1, 2, . . . , nh and τ∗ > 0 be ascalar, then a sufficient condition for global asymp-totic stability of error system (E) from (8) for anyconstant time-delay τ satisfying 0 ≤ τ ≤ τ∗ is givenby the matrix inequality

Y =

[Z PB + kCΛ

BTP + kΛCT r3τBTB − 2Λ

]< 0 (9)

where Z = P (A + F ) + (A + F )TP + r1τATA +

r2τFTF + ((1/r1) + (1/r2) + (1/r3))τPFFTP .

Proof. We have

e(t− τ) = e(t)−∫ 0

−τe(t+ θ)dθ (10)

Substituting e(t+ θ) into Eq. (10) one obtains

e(t− τ) = e(t)−∫ 0

−τ{Ae(t + θ) +Bη(Ce(t+ θ);

y(t+ θ)) + Fe(t+ θ − τ)}dθ

Substituting e(t− τ) back into Eq. (2)

e = (A+ F )e+Bη(Ce; y)

−F{∫ 0

−τ{Ae(t+ θ) +Bη(Ce(t+ θ);

y(t+ θ)) + Fe(t+ θ − τ)}dθ}

(11)

Taking the time derivative of the Lyapunov functionEq. (8) along Eq. (11) we obtain

V3(e) = eT (P (A+ F ) + (A+ F )TP )e+ eTPBη(Ce; y) + ηT (Ce; y)BTPe

− eTPF∫ 0

−τAe(t+ θ)dθ − eTPF

∫ 0

−τBη(Ce(t+ θ); y(t+ θ))dθ

− eTPF∫ 0

−τFe(t+ θ − τ)dθ −

∫ 0

−τeT (t+ θ)ATFTPedθ

−∫ 0

−τηT (Ce(t+ θ); y(t+ θ))BTFTPedθ

−∫ 0

−τeT (t+ θ − τ)FTFTPedθ + r1τe

TATAe+ r2τeTFTFe

+ r3τηT (Ce; y)BTBη(Ce; y)−

∫ 0

−τr1[eT (t+ θ)ATAe(t+ θ)]dθ

−∫ 0

−τr2[eT (t+ θ − τ)FTFe(t+ θ − τ)]dθ

−∫ 0

−τr3[ηT (Ce(t+ θ); y(t+ θ))BTBη(e(t+ θ); y(t+ θ))]dθ

Master–Slave Synchronization with Time-Delay 1711

We have the following inequalities from [Mahmoud,2000, pp. 33–34, p. 401]

−∫ 0

−τeTPFAe(t+ θ)dθ

−∫ 0

−τeT (t+ θ)ATFTPedθ

≤ r−11

∫ 0

−τeTPFFTPedθ

+ r1

∫ 0

−τeT (t+ θ)ATAe(t+ θ)dθ

and

−∫ 0

−τeTPFFe(t+ θ − τ)dθ

−∫ 0

−τeT (t+ θ − τ)FTFTPedθ

≤ r−12

∫ 0

−τeTPFFTPedθ

+ r2

∫ 0

−τeT (t+ θ − τ)FTFe(t− τ + θ)dθ

and also

−∫ 0

−τeTPFBη(e(t+ θ); y(t+ θ))dθ

−∫ 0

−τηT (e(t+ θ); y(t+ θ))BTFTPedθ

≤ r−13

∫ 0

−τeTPFFTPedθ + r3

∫ 0

−τηT (e(t+ θ);

y(t+ θ))BTBη(e(t+ θ); y(t+ θ))dθ

Using the above inequalities, we get

V3(e) ≤ eTZe+ eTPBη(Ce; y) + ηT (Ce; y)BTPe

+ r3τηT (Ce; y)BTBη(Ce; y) .

Applying the S-procedure, by using the inequalitiesfrom the nonlinearities, gives

V3(e) ≤ eTZe+ eTPBη(Ce; y) + ηT (Ce; y)BTPe

+ r3τηT (Ce; y)BTBη(Ce; y)

+∑i

λiηi(ηi − kcT∗ie)

≤ ξTY ξ

where ξ = [e; η].

If V3 < 0 for τ∗, then the following inequalityis satisfied for all τ ∈ [0, τ∗]

τ∗{eT(r1A

TA+ r2FTF +

(1

r1+

1

r2+

1

r3

)Υ

)e

+ r3ηT (Ce; y)BTBη(Ce; y)

}≤ −eT (P (A+ F ) + (A+ F )TP )e

− ηT (Ce; y)BTPe− eTPBη(Ce; y)

where Υ = PFFTP . �

The matrix inequality (9) includes informationon the delay. Therefore, this result is a delay-dependent stability criterion for synchronization.Note that a necessary condition for synchronizationhere is that A+ F must be strictly Hurwitz.

5. Examples

We illustrate Theorem 2 for the examples ofChua’s circuits, n-scroll attractors and hyperchaoticattractors.

5.1. Chua’s circuit

Let us take the following representation of Chua’sCircuit

x = α(y − h(x))

y = x− y + z

z = −βy

with nonlinear characteristic

h(x) = m1x+1

2(m0 −m1)(|x+ c| − |x− c|)

and parameters m0 = −(1/7), m1 = (2/7), α = 9,β = 14.28, c = 1 in order to obtain the doublescroll attractor [Chua et al., 1986; Madan, 1993].The system can be represented in Lur’e form by

A =

−αm1 α 0

1 −1 1

0 −β 0

,B = [−α(m0 −m1); 0; 0] , C = [1 0 0]

and σ(ξ) = (1/2)(|ξ+c|−|ξ−c|) belonging to sector[0, k] with k = 1.

1712 M. E. Yalcin et al.

The matrix inequality (9) is employed as follows

minΛ,P,F,r1,r2,r3

λmax[Y (P, F, Λ, r1, r2, r3)]

such that

P = P T > 0

Λ ≥ 0

r1, r2, r3 > 0

(12)

according to e.g. Suykens et al. [1997]. Sequen-tial quadratic programming has been applied inMatlab’s optimization toolbox for different valuesof τ . The matrix G = [6.0229; 1.3367; −2.1264]stabilizes the error system for τ ∈ [0 0.039]. Nofeasible points were found for τ > 0.039. In the ex-periments H = [1 0 0] was chosen which meansthat the master system is connected to the slavesystem with the first state variable only.

The synchronization scheme (Fig. 1) has beenmodeled in Matlab Simulink and the following sim-ulation results have been obtained. The first ob-servation is given in Fig. 2 for maximum delayτ∗ = 0.039. A second observation is given in Fig. 3for a smaller delay τ = 0.01 than the maximum de-lay. During the simulations it has been observedthat two Chua’s circuits synchronize until τ = 0.21for the same feedback matrix (Fig. 4). Synchro-nization could not be observed for τ bigger than0.21. The simulation result for τ = 0.22 is givenin Fig. 5. As initial conditions of the master andslave systems were taken x(0) = [−0.2; −0.33; 0.2],y(0) = [0.5; −0.1; 0.66] in the simulations.

5.2. 5-scroll attractors

A more complete family of n-scroll [Suykens & Van-dewalle, 1993] instead of double and n-double scrollattractors has been obtained from a generalizedChua’s circuit proposed in [Suykens et al., 1997].An experimental confirmation of 3- and 5-scroll at-tractors has been given by Yalcin et al. [2000]. Then-scroll circuit is given by

x = α(y − h(x))

y = x− y + z

z = −βy(13)

Fig. 2. Simulation result for master–slave synchronizationof two identical Chua’s circuits. The master system is cou-pled to the slave system with the first state variable and delay(τ = 0.039): Three-dimensional view on the double scroll at-tractor generated for (a) master system and (b) slave system.(c) Error signal (x1(t)− y1(t)) with respect to time.

(a)

(b)

(c)

Master–Slave Synchronization with Time-Delay 1713

(a)

(b)

(c)

Fig. 3. Delay (τ = 0.01).

(a)

(b)

(c)

Fig. 4. Delay (τ = 0.21).

1714 M. E. Yalcin et al.

(a) (b)

(c)

Fig. 5. Delay (τ = 0.22).

with a nonlinear characteristic having additionalbreak points

h(x) = m2q−1x+1

2

2q−1∑i=1

(mi−1−mi)(|x+ci|−|x−ci|)

(14)

where q denotes a natural number [Suykens et al.,1997]. Here, we will consider the 5-scroll attrac-tor, which is obtained for m = [0.9/7, −3/7, 3.5/7,−2.7/7, 4/7, −2.4/7], c = [1, 2.15, 3.6, 6.2, 9],α = 9, β = 14.28. The system is represented inanother Lur’e form than given in [Suykens et al.,

1997], based upon [Guzelis, 1993] with −(1 + δ)x+f(x) = −h(x) and

A =

−α(1 + δ) α 0

1 −1 1

0 −β 0

,B = [−α; 0; 0] , C = [1 0 0]

and δ = 1 where f(x) belong to sector [0, k] withk = 2.5. This representation results in nh = 1.

The same optimization procedure has beenapplied as for Chua’s circuit. For τ > 0.04 no

Master–Slave Synchronization with Time-Delay 1715

feasible points were found such that Y negativedefinite. The feedback vector G = [0.6945; 0.7002;−0.1645] found for τ = 0.04 has stabilized the errorsystem between τ ∈ [0 0.04]. In the experimentsH = [1 0 0] was chosen. Figure 6 shows sim-ulation results for the maximum delay τ∗ = 0.04.Results for a smaller delay τ = 0.01 are given inFig. 7. In Matlab Simulink, synchronization hasbeen observed until τ = 0.14 (Fig. 8) for the samefeedback matrix. Synchronization has not been ob-served when the delay was bigger than 0.14. Fig-ure 9 shows results for τ = 0.15. During the sim-ulations the initial conditions of master and slavesystems are taken as x(0) = [−1.7; −0.4; −0.2],y(0) = [0.2; −0.2; 0.33].

5.3. Hyperchaotic system

We consider the following system which consists oftwo unidirectionally coupled Chua circuits

x1 = a(x2 − h(x1))

x2 = x1 − x2 + x3

x3 = −bx2

x4 = a(x5 − h(x5))

x5 = x4 − x5 + x6 +K(x5 − x2)

x6 = −bx5

(15)

with nonlinear characteristic

h(xi) = m1xi +1

2(m0 −m1)(|xi + c|

− |xi − c|), i = 1, 4 (16)

and parameters m0 = −(1/7), m1 = 2/7, a =9, b = 14.28, c = 1, K = 0.01. The sys-tem exhibits hyperchaotic behavior with a double–double scroll attractor [Kapitaniak & Chua, 1995].This system was represented in Lur’e form by

Fig. 6. Simulation result for master–slave synchronizationof two identical 5-scroll attractors. The master system iscoupled to the slave system with the first state variable anddelay (τ = 0.04): Three-dimensional view on the 5-scroll at-tractors generated at (a) master system and (b) slave system.(c) Error signal (x1(t)− y1(t)) with respect to time.

(a)

(b)

(c)

1716 M. E. Yalcin et al.

(a)

(b)

(c)

Fig. 7. Delay (τ = 0.01).

(a)

(b)

(c)

Fig. 8. Delay (τ = 0.14).

Master–Slave Synchronization with Time-Delay 1717

(a) (b)

(c)

Fig. 9. Delay (τ = 0.15).

Suykens et al. [1998]. h(x) belongs to sector [0, 1]and

A =

−am1 a 0 0 0 0

1 −1 1 0 0 0

0 −b 0 0 0 0

0 0 0 −am1 a 0

0 −K 0 1 −1 +K 1

0 0 0 0 −b 0

,

B =

−a(m0 −m1) 0

0 0

0 0

0 −a(m0 −m1)

0 0

0 0

,

C =

[1 0 0 0 0 0

0 0 0 1 0 0

].

Sequential quadratic programming has been applied

1718 M. E. Yalcin et al.

(a) (b)

(c)

Fig. 10. Simulation result for master–slave synchronization of two identical hyperchaotic systems. The master system iscoupled to the slave system with the first and fourth state variable and delay (τ = 0.038): Double-double scroll attractorgenerated for (a) master system (projection onto the x1 − x4 plane) and (b) slave system (projection onto the y1 − y4 plane).(c) Error signal (x1(t)− y1(t)) with respect to time.

similar to the other examples. For τ > 0.038 no feasible points were found such that Y is negative definite.The feedback matrix

G =

[7.6909 2.1313 −3.9865 −0.3491 0.1811 −0.5180

−1.0520 0.0835 0.3455 8.0879 1.8021 −4.8256

]

which is found for the maximum delay τ∗ = 0.038,has stabilized the error system for τ ∈ [0 0.038].In the experiments H = [1 0 0 0 0 0; 0 0 0 1 0 0]has been taken. In Fig. 10 the result is given for

the maximum obtained delay τ∗ = 0.038. Sim-ulation results for a smaller delay τ = 0.01 areshown in Fig. 11. In Matlab Simulink, synchroniza-tion has been observed until τ = 0.17 (Fig. 12) for

Master–Slave Synchronization with Time-Delay 1719

(a)

(b)

(c)

Fig. 11. Delay (τ = 0.01).

(a)

(b)

(c)

Fig. 12. Delay (τ = 0.17).

1720 M. E. Yalcin et al.

(a) (b)

(c)

Fig. 13. Delay (τ = 0.18).

the same feedback matrix. Synchronization has notbeen observed when the delay was larger than 0.17.Figure 13 shows the result for τ = 0.18. Duringthe simulations, the initial conditions of the masterand slave systems are taken as x(0) = [−0.2; −0.2;−0.33; 0.2; 0.9; 0.33], y(0) = [0.2; −0.2; 0.33; 0.2;−0.2; 0.33].

6. Conclusion

In this paper a master–slave synchronizationscheme for Lur’e systems has been investigated

for a known delay existing between master andslave systems. Synchronization criteria have beenclassified into two categories: delay-independentand delay-dependent synchronization criteria. Suf-ficient conditions for global asymptotic stabilityof the error system have been given for thesetwo categories. Delay-independent criteria havebeen applied to Chua’s circuit, 5-scroll attrac-tors and hyperchaotic attractors but feasible pointscould not be found. Therefore, a new Lyapunov–Krasovskii function has been introduced, whichgives a delay-dependent synchronization criterion.

Master–Slave Synchronization with Time-Delay 1721

This condition has been successfully applied to thechaotic and hyperchaotic systems.

Acknowledgments

This research work was carried out at the ESATlaboratory and the Interdisciplinary Center of Neu-ral Networks ICNN of the Katholieke UniversiteitLeuven, in the framework of the Belgian Pro-gramme on Interuniversity Poles of Attraction, ini-tiated by the Belgian State, Prime Minister’s Of-fice for Science, Technology and Culture (IUAPP4-02), the Concerted Action Project MEFISTOof the Flemish Community and ESPRIT IV 27077(DICTAM). Johan Suykens is a postdoctoral re-searcher with the National Fund for ScientificResearch FWO – Flanders.

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