CHAOS SYNCHRONIZATION: A LAGRANGE PROGRAMMING …chua/papers/Suykens00b.pdf · be regarded as a control problem where a feedback matrix Khas to be designed such that the slave is

International Journal of Bifurcation and Chaos, Vol. 10, No. 4 (2000) 797–810c© World Scientific Publishing Company

CHAOS SYNCHRONIZATION: A LAGRANGEPROGRAMMING NETWORK APPROACH

J. A. K. SUYKENS∗ and J. VANDEWALLEKatholieke Universiteit Leuven,

Department of Electrical Engineering,ESAT-SISTA, Kardinaal Mercierlaan 94,

B-3001 Leuven (Heverlee ), Belgium

Received May 14, 1999; Revised August 30, 1999

In this paper we interpret chaos synchronization schemes within the framework of Lagrangeprogramming networks, which form a class of continuous-time optimization methods for solvingconstrained nonlinear optimization problems. From this study it follows that standard synchro-nization schemes can be regarded as a Lagrange programming network with soft constraining,where synchronization between state vectors is defined as a constraint to the dynamical systems.New schemes are proposed then which implement synchronization by hard and soft constraintswithin Lagrange programming networks. A version is derived which takes into account synchro-nization errors within the problem formulation. Furthermore Lagrange programming networksfor achieving partial and generalized synchronization are given. The methods assume the ex-istence of potential functions for the given systems. The proposed Lagrange programmingnetworks with hard and soft constraining show improved performance on many simulation ex-amples for identical and nonidentical chaotic systems. The schemes are illustrated on Chua’scircuit, Lorenz attractor and n-scroll circuits.

1. Introduction

Chaos synchronization has been intensively stud-ied in recent years [Chen & Dong, 1998]. Schemeswith mutual coupling and master–slave synchro-nization have been analyzed especially for ad hocexamples of general interest such as Chua’s cir-cuit and Lorenz attractor or for classes of non-linear systems, e.g. Lur’e systems [Chen & Dong,1998; Wu & Chua, 1994; Kapitaniak & Chua,1994; Kapitaniak et al., 1994; Kocarev et al.,1993; Suykens et al., 1997a, 1999]. Synchroniza-tion schemes have been investigated for identicaland nonidentical chaotic systems, including aspectsas local and global synchronization, robust syn-chronization, partial synchronization and general-ized synchronization. Many of the synchronization

methods have been applied to secure communica-tions using chaos, where a message signal is definedas an external input of the scheme [Chen & Dong,1998; Wu & Chua, 1994; Suykens et al., 1997c].

In this paper we make an interpretation ofchaos synchronization schemes within the frame-work of Lagrange programming networks [Zhang &Constantinides, 1992]. Being related to the Hopfieldnetwork [Hopfield, 1984] and other recurrent neu-ral networks for nonlinear optimization [Kennedy& Chua, 1988], Lagrange programming networkshave been proposed as a class of continuous-timenonlinear optimization [Cichocki & Unbehauen,1994] for solving constrained nonlinear optimiza-tion problems with equality and/or inequality con-straints [Fletcher, 1987]. Synchronization between

∗Author for correspondence.E-mail: [email protected]

797

798 J. A. K. Suykens & J. Vandewalle

state vectors of two nonlinear dynamical systems isdefined then as a constraint to the dynamical sys-tem, assuming that there exists a potential functionfor the given systems, which is motivated by [Zubov,1997] and potential function theory [Hazewinkel,1991] with related results in circuit theory [Weisset al., 1998].

As a result of the study we propose Lagrangeprogramming networks for chaos synchronizationthat implement state vector synchronization byhard and soft constraints based upon an augmentedLagrangian. The synchronization constraint isrealized in an asymptotical sense, which is dif-ferent from the theory of differential-algebraicequations [Beardmore & Song, 1998] where syn-chronization could be considered as an algebraicconstraint. Certain standard synchronizationschemes can be regarded as Lagrange programmingnetworks with soft constraining only. New schemesare presented for full state synchronization, partialsynchronization and generalized synchronization[Kocarev & Parlitz, 1996]. The Lagrange program-ming networks for synchronization are applied toidentical and nonidentical chaotic systems. Oftensynchronization is only achievable up to a smallsynchronization error, as shown in e.g. [Suykenset al., 1999] on robust synchronization of noniden-tical Lur’e systems. A Lagrange programming net-work for chaos synchronization is proposed whichtakes into account synchronization errors in theproblem formulation. In order to demonstrate themethods, Chua’s circuit [Chua et al., 1986; Chua,1994; Madan, 1993], Lorenz system [Lorenz, 1963]and n-scroll circuits [Suykens et al., 1997b; Yalcinet al., 2000] have been taken as simulation ex-amples. In many cases the Lagrange program-ming network shows an improved performance com-pared to standard schemes that correspond to softconstraining. Theoretically speaking smooth ver-sions to Chua’s circuit and n-scroll circuits [Huanget al., 1996; Ponomarenko & Matrosov, 1996]should be considered. However, simulation resultssuggest that the piecewise-linear nature of the cir-cuits does not pose a problem for the investigatedexamples. At this point it is important to stressthat no explicit construction of the underlying po-tential functions has to be made in order to simulateall of the synchronization schemes; only the exis-tence of a potential function is needed, eventuallyin a generalized sense of potential function theory.

This paper is organized as follows. In Sec. 2we discuss some standard synchronization schemes.In Sec. 3 Lagrange programming networks forconstrained nonlinear optimization are shortly re-viewed. In Sec. 4 Lagrange programming networksfor chaos synchronization are proposed. The casesof partial and generalized synchronization are dis-cussed in Sec. 5. Simulation examples are presentedin Sec. 6.

2. Synchronization Schemes

In the context of synchronization of chaotic systemsone studies schemes of the form [Chen & Dong,1998; Wu & Chua, 1994]{

x = f(x) +M (x− z)z = g(z) +N (x− z)

(1)

with f(·) : Rn → Rn, g(·) : Rn → Rn, state vec-tors x, z ∈ Rn and coupling matrices M, N ∈ Rn×nwhich are often chosen diagonal. In the case of (1)the coupling between the two dynamical systems ismutual. For master–slave synchronization on theother hand one has the scheme{M : x = f(x)

S : z = g(z) +K(x− z)(2)

with master system M, slave system S and cou-pling matrix K ∈ Rn×n. Master–slave schemes areespecially of interest in applications for secure com-munications using chaos, where the message signalis defined as an external input to the scheme [Wu& Chua, 1994; Suykens et al., 1997].

Loosely speaking, for (1) and (2) synchroniza-tion means that ‖x− z‖2 → 0 as t→∞. Synchro-nization is often studied by stability analysis of theerror system

e = f(x)− g(z) −Ke (3)

for (2) where e = x−z is the error signal. From theequations it is immediately clear that the analysis issimplified in the case of identical master and slavesystems. For ad hoc examples such as Chua’s circuitor Lorenz attractor synchronization has been stud-ied in detail [Kocarev et al., 1993; Kapitaniak et al.,1994]. For the example of identical Lur’e systemsa sufficient condition for global asymptotic stabilityof the error system may be derived from a quadratic

Chaos Synchronization: A Lagrange Programming Network Approach 799

Lyapunov function. The class of Lur’e systems in-cludes e.g. Chua’s circuit [Chua et al., 1986; Chua,1994; Madan, 1993], n-scroll circuits [Suykens et al.,1997b] and coupled Chua’s circuits [Kapitaniak &Chua, 1994].

More specifically, for identical Lur’e systemsone has the scheme{

x = Ax+Bσ(Cx) +M(x− z)z = Az +Bσ(Cz) +N(x− z)

(4)

with error system

e = (A+M −N)e+Bη(e; z) (5)

where η(e; z) = σ(Ce + Cz) − σ(Cz) with σ(·) :Rl → Rl and A ∈ Rn×n, B ∈ Rn×l, C ∈ Rl×n. Forη and σ one assumes a sector condition [0, k]. Thismeans that for η the inequality ηTΛ[η − kCe] ≤ 0holds ∀ e, z with Λ a positive definite diagonal ma-trix. Employing a quadratic Lyapunov functionV (e) = eTPe with P = P T > 0 it is straightfor-ward to derive the following sufficient condition forglobal asymptotic stability of the error system:[P (A+M−N)+(A+M−N)TP PB+kCTΛ

BTP+kΛC −2Λ

]

< 0 , (6)

where < 0 denotes a negative definite matrix. Inthe master–slave scheme (2) synchronization maybe regarded as a control problem where a feedbackmatrix K has to be designed such that the slave issynchronized to the master system. The control isbased upon the full state vectors x, z. In case thereis a lack of full state information dynamic outputfeedback could be applied to the scheme [Suykenset al., 1997a]. Also observer-based synchronizationdesign is relevant in this context [Morgul & Solak,1997].

For nonidentical Lur’e systems within themaster–slave scheme{M : x = A1x+B1σ(C1x)

S : z = A2z +B2σ(C2z) +K(x− z) ,(7)

with nonidentical matrices A1, A2, B1, B2 and C1,C2, conditions for robust synchronization have beenderived in [Suykens et al., 1999]. Master and slavesystems that behave in qualitative different ways

with respect to each other (stable points, limit cy-cles versus chaos), can be synchronized up to a smallsynchronization error within (7).

In addition to synchronization of the type ‖x−z‖2 → 0 as t → ∞, one can also consider gen-eralized synchronization [Kocarev & Parlitz, 1996]which means that there exists a transformation H :Rn → Rm such that a manifold {(x, z) : z = H(x)}is reached as t → ∞ with a nonempty basin ofattraction.

3. Lagrange Programming Networks

In this section we shortly review Lagrange pro-gramming networks as introduced by Zhang andConstantinides [1992]. Lagrange programming net-works form a class of continuous-time optimizationmethods for solving constrained nonlinear program-ming problems with equality and/or inequality con-straints. Here we review the equality constraintcase.

Consider the optimization problem

minξ∈Rr

J(ξ) such that h(ξ) = 0 (8)

with given functions J(·) : Rp → R and h(·) : Rr →Rs, where r ≤ s. The functions J and h are assumedto be twice continuously differentiable. Based uponthe Lagrangian

L(ξ, λ) = J(ξ) + λT h(ξ) (9)

where λ ∈ Rr denotes the vector of Lagrange multi-pliers, the Lagrange programming network for find-ing a local minimum to (8) is given by

ξ = −∂L(ξ, λ)

∂ξ

λ =∂L(ξ, λ)

∂λ

(10)

which yields the equationsξi = −∂J

∂ξi−

s∑j=1

λj∂hj∂ξi

, i = 1, 2, . . . , r

λj = hj , j = 1, 2, . . . , s .

(11)

For this network it can be shown that when (ξ∗, λ∗)is a stationary point of L(ξ, λ) and the Hessian∇2L(ξ∗, λ∗) is positive definite, then (ξ∗, λ∗) isan asymptotically stable point of (11). Under the


assumption that the Hessian ∇2L(ξ, λ) is posi-tive definite everywhere, V (ξ, λ) = 1/2‖∂L/∂ξ‖22 +1/2‖h(ξ)‖22 is a Lyapunov function for the network.

The problem (8) can be convexified by consid-ering the optimization problem

minξ∈Rr

J(ξ) +1

2γ‖h(ξ)‖22 such that h(ξ) = 0

(12)

with augmented Lagrangian

Lγ(ξ, λ) = J(ξ) + λT h(ξ) +1

2γ ‖h(ξ)‖22 . (13)

In this case the local convexity assumption∇2L(ξ∗, λ∗) > 0 is weakened. This augmented La-grangian yields the Lagrange programming network

ξi = −∂J∂ξi−

s∑j=1

γ hj∂hj∂ξi−

s∑j=1

λj∂hj∂ξi

,

i = 1, 2, . . . , r

λj = hj , j = 1, 2, . . . , s .

(14)

This network makes use of hard and soft constraintsin order to solve the optimization problem, while in(11) only hard constraints are implemented.

4. Chaos Synchronization andLagrange Programming Networks

Now, let us establish the link between the previoustwo sections and interpret the chaos synchroniza-tion schemes with respect to Lagrange program-ming networks.

For a system

x = f(x) (15)

with right-hand side defined in an n-dimensionalEuclidean space and f being a real-valued, contin-uous and continuously differentiable function in x,Poincare and Lyapunov have shown that (15) canbe transformed to the form

x =∂ϕ(x)

∂x(16)

by introducing new independent variables [Zubov,1997] (potential systems of differential equationspp. 298–301). Consider again the vector fields f(x),

g(z) as in (1). In the sequel we have to assume thenthe existence of potential functions U1(x) and U2(z)for the given vector fields f(x), g(z) such that

f(x) = −∂U1(x)

∂x

g(z) = −∂U2(z)

∂z

(17)

without the need actually for constructing U1(x),U2(z). This assumption is reasonable for a largeclass of nonlinear systems being further motivatedin the area of circuit theory by the work onBrayton–Moser mixed potential functions [Weisset al., 1998] and potential function theory[Hazewinkel et al., 1991]. According to [Hazewinkelet al., 1991] the test for the fields to be the gra-dient of a potential is that their Jacobian matrixis symmetric, which indicates that the field is ro-tation free. However, in the sequel we will assumethe existence of a potential function also in the caseof nonsymmetric Jacobian matrices, thereby rely-ing on [Zubov, 1997] and further possible general-izations of potential functions [Hazewinkel et al.,1991]. The latter are characterized then in anotherway than (17) and would require a different nota-tion than (17) in the further derivations.

Let us consider then the optimization problem

minx, z∈Rn

J(x, z) = U1(x) + U2(z) +1

2γ ‖x− z‖22

such that x− z = 0 (18)


Lγ(x, z, λ) = U1(x) + U2(z)

+1

2γ ‖x− z‖22 + λT (x− z) (19)

where γ is a positive real constant. The synchro-nization constraint x = z is expressed in terms ofhard and soft constraints by the terms λT (x − z)and 1/2 γ ‖x− z‖22 respectively.

The corresponding Lagrange programming net-work for (18) and (19) is given by

x = f(x)− γ(x− z)− λz = g(z) + γ(x− z) + λ

λ = x− z

(20)


after application of (10) and (17). We modify thisfurther into the following synchronization schemewith mutual coupling

Σ1{x, z, λ|γ1, γ2} :

x = f(x)− γ1(x− z)− λz = g(z) + γ2(x− z) + λ

λ = x− z

(21)

which is characterized by two positive real couplingconstants γ1, γ2 instead of γ in (20). Master–slavesynchronization is a special case of (21) by takingone of the constants γ1, γ2 equal to zero. HereΣ{ν|θ} denotes a system of differential equationswith state vector ν for given parameters θ.

Hence, classical synchronization schemes of theform

Σ2{x, z|γ1, γ2} :

{x = f(x)− γ1(x− z)z = g(z) + γ2(x− z)

(22)

could be considered as a special case of (21) withsoft constraining only, instead of hard together withsoft constraints.

As illustrated for example in [Suykens et al.,1999], synchronization can often only be achievedup to a small synchronization error. In this sensethe performance of the Lagrange programming net-work can be enhanced by modifying the problemformulation (18) into:

minx, z, ε∈Rn

J(x, z, ε) = U1(x) + U2(z)

+1

2γ ‖x− z‖22 +

1

2c εT ε

such that x− z + ε = 0 (23)


Lγ, c(x, z, ε, λ) = U1(x) + U2(z) +1

2γ ‖x− z‖22

+1

2c εT ε+ λT (x− z + ε)

(24)

where c is a positive real constant. This formu-lation tolerates a synchronization error ε which isminimized within the cost function J .

From (23) and (24) one obtains the Lagrangeprogramming network

Σ3{x, z, ε, λ|γ1, γ2, c} :

x = f(x)− γ1(x− z)− λz = g(z) + γ2(x− z) + λ

ε = −c ε− λλ = x− z + ε

(25)

after applying the master–slave heuristic of (21).We have to emphasize that in order to simulate eachof the synchronization schemes Σ no construction ofthe underlying potential functions U1, U2 has to bemade. The schemes are applied with given systemsf, g.

Furthermore, note that the synchronizationconstraint in (18) is an algebraic constraint whichthe Lagrange programming network has to real-ize. This constraint is only achieved in an asymp-totical sense which is basically different from theway similar problems are analyzed in the theory ofdifferential-algebraic equations [Beardmore & Song,1998], with applications e.g. to constrained Hamil-tonian systems.

5. Partial andGeneralized Synchronization

In the case of partial synchronization one aims atsynchronizing e.g. only one state variable instead ofthe full state vector. We illustrate how this can bedone within the Lagrange programming framework.

Let us define two output variables p, q ∈ R with{x = f(x)

p = wTx

{z = g(z)

q = wT z .(26)

Similar to (24), defining the Lagrangian

Lγ, c(x, z, ε, λ) = U1(x) + U2(z) +1

2γ (p− q)2

+1

2c ε2 + λ (p − q + ε) (27)

for (26) yields the Lagrange programming network

Σ4{x, z, ε, λ|γ1, γ2, c} :

x = f(x)− γ1(p− q)w − λwz = g(z) + γ2(p− q)w + λw

ε = −c ε− λλ = p− q + ε

(28)


where ε(t), λ(t) ∈ R and γ has been split into theconstants γ1, γ2. This network aims at achievingpartial synchronization |p− q| → 0 as t→∞.

Finally, generalized synchronization can be ex-pressed by the optimization problem

minx, z∈Rn, ε∈Rm

J(x, z, ε)

= U1(x) + U2(z) +1

2γ ψ(x, z)Tψ(x, z) +

1

2c εT ε

such that ψ(x, z) + ε = 0 (29)

where ψ(·, ·) : Rn×Rn → Rm characterizes the syn-chronization manifold that has to be reached. Fromthe Lagrangian

Lγ, c(x, z, ε, λ)

= U1(x) + U2(z) +1

2γ ψ(x, z)Tψ(x, z)

+1

2c εT ε+ λT [ψ(x, z) + ε] (30)

one obtains the Lagrange programming network

Σ5{x, z, ε, λ|γ1, γ2, c} :

x = f(x)− γ1

(∂ψ

∂x

)Tψ

−(∂ψ

∂x

)Tλ

z = g(z) − γ2

(∂ψ

∂z

)Tψ

−(∂ψ

∂z

)Tλ

ε = −c ε− λ

λ = ψ(x, z) + ε

(31)

after applying the master–slave heuristic.

6. Simulation Examples

In this section we present simulation exam-ples of the Lagrange programming networksΣ1, Σ2, . . . , Σ5 by taking Chua’s circuit, Lorenz at-tractor and n-scroll circuits as examples for f, g.Chua’s circuit is given by the equations

x1 = a [x2 − φ(x1)]

x2 = x1 − x2 + x3

x3 = −b x2

(32)

with nonlinear characteristic

φ(x1) = m1x1 +1

2(m0 −m1) (|x1 + 1| − |x1 − 1|)

(33)

and parameters a = 9, b = 14.286, m0 = −1/7,m1 = 2/7, in order to obtain the double-scroll at-tractor [Chua et al., 1986; Chua, 1994; Madan,1993]. For a = 8 and a = 5 one obtains the case ofperiodic and sink behavior respectively. By intro-ducing additional breakpoints in the PWL charac-teristic (33) one obtains a generalized Chua’s circuitfor (32) with nonlinearity

φ(x1) = m2q−1x1

+1

2

2q−1∑i=1

(mi−1 −mi)(|x1 + ci| − |x1 − ci|)

(34)

which exhibits n-scroll attractors where n de-notes a natural number and for q = 1 one ob-tains Chua’s circuit (32) and (33) [Suykens et al.,1997b; Yalcin et al., 1999]. In order to obtaina 5-scroll attractor one sets q = 3 with m =[0.9/7; −3/7; 3.5/7; −2.7/7; 4/7; −2.4/7] and c =[1; 2.15; 3.6; 6.2; 9] in (34). The Lorenz system[Lorenz, 1963; Chen & Dong, 1998] is given by theequations

x1 = σl(x2 − x1)

x2 = −x1x3 + rlx1 − x2

x3 = x1x2 − blx3

(35)

with the Lorenz attractor obtained by σl = 10,rl = 28, bl = 8/3. We apply the synchronizationschemes by assuming the existence of a potentialfunction for each of the subsystems, eventually in amore generalized sense than characterized by (17).Strictly speaking, in fact smooth versions to Chua’scircuit and n-scroll circuits [Huang et al., 1996;Ponomarenko & Matrosov, 1996] should be consid-ered. However, simulation results suggest that thepiecewise-linear nature of the circuits does not posea problem in the examples that have been investi-gated.

In Fig. 1 a comparison is made between thestandard scheme Σ2 and the Lagrange program-ming networks Σ1, Σ3 that implement synchroniza-tion as a hard constraint. The example makes


Fig. 1. Comparison of master–slave synchronization schemes with hard and soft constraining applied to Chua’s circuits:[Right] Σ1

{x, z, λ|0, 1} (blue dashed), Σ2{x, z|0, 1} (red dotted), Σ3

{x, z, ε, λ|0, 1, 1} (green solid) where Σ3 has the best performance;[Left] in each of the cases f(x) corresponds to a Chua’s circuit with a = 9 (double-scroll) and g(z) to Chua’s circuits witha = 9 (double-scroll) [Top], a = 8 (periodic) [Middle] and a = 5 (sink) [Bottom]. In all cases the initial states were chosen asx(0) = [0.1; 0; −0.1] and z(0) = [0.2; 0.1; 0.3].


(a) (b)

(c) (d)

Fig. 2. Master–slave synchronization by hard and soft constraining applied to Chua’s circuit and Lorenz attractor: (a) Chua’scircuit with double-scroll as master system f(x); (b) Lorenz attractor as uncontrolled slave system g(z); (c) Lorenz attractorcontrolled to a double-scroll by synchronization. In these simulations the Lagrange programming network Σ3

{x, z, ε, λ|0, 1000, 1}has been applied; (d) Comparison between Σ3

{x, z, ε, λ|0, 1000, 1} and Σ2{x, z|0, 1000}. The performance is comparable due to the

large value of γ2. In all simulations the initial states were chosen as x(0) = [0.1; 0; −0.1] and z(0) = [1; 0; −1].


(a) (b)

(c) (d)

Fig. 3. Master–slave synchronization by hard and soft constraining applied to 5-scroll circuit and Chua’s circuit: (a) 5-scroll circuit as master system f(x); (b) Chua’s circuit with double-scroll as uncontrolled slave system g(z); (c) double-scrollcontrolled to a 5-scroll by synchronization. In these simulations the Lagrange programming network Σ3

{x, z, ε, λ|0, 100, 1} hasbeen applied; (d) x(t)− z(t) for the synchronized system. Synchronization is achieved up to a small synchronization error. Asinitial states were taken x(0) = [−0.2; 0; 0.2] and z(0) = [0.1; 0; −0.1].


Fig. 4. (Continued) Master–slave synchronization by hard and soft constraining applied to 5-scroll circuit and Chua’s circuit:comparison between Σ3

{x, z, ε, λ|0, 10, 1} and Σ2{x, z|0, 10, 1} which shows that a better performance in terms of synchronization

error is obtained by applying Lagrange programming networks Σ3.

use of Chua’s circuits with a double-scroll as mastersystem and a double-scroll, periodic behavior andsinks as the slave system. The Lagrange program-ming network Σ3 shows the best performance. Thesynchronization scheme Σ3 that has been simulatedfor this example of two Chua’s circuits is

x1 = a [x2 − φ(x1)]− γ1(x1 − z1)− λ1

x2 = x1 − x2 + x3 − γ1(x2 − z2)− λ2

x3 = −b x2 − γ1(x3 − z3)− λ3

z1 = a [z2 − φ(z1)] + γ2(x1 − z1) + λ1

z2 = z1 − z2 + z3 + γ2(x2 − z2) + λ2

z3 = −b z2 + γ2(x3 − z3) + λ3

ε1 = −c ε1 − λ1

ε2 = −c ε2 − λ2

ε3 = −c ε3 − λ3

λ1 = x1 − z1 + ε1

λ2 = x2 − z2 + ε2

λ3 = x3 − z3 + ε3 .

(36)

In Fig. 2 master–slave synchronization with hardand soft constraining between Chua’s circuit andLorenz attractor is shown using Σ3. When the valueof γ2 is taken as large (γ1 = 0 for master–slavesynchronization) the performance of Σ2 and Σ3 iscomparable in this example. In Fig. 3 master–slave synchronization between a 5-scroll attractorand the double-scroll is shown by applying Σ3 fora smaller value of γ2. Synchronization is achievedup to a small synchronization error. Figure 4 showsthe benefit of applying Lagrange programming net-works Σ3 which results in smaller synchronizationerrors compared to Σ1. In Fig. 5 partial synchro-nization is illustrated on identical and nonidenticalChua’s circuits by applying Σ4. For larger values ofγ2 in the master–slave scheme the complete statevector becomes synchronized, while for smallervalues only x2, z2 synchronize when w = [0; 1; 0].Figures 6 and 7 show generalized synchronizationusing Lagrange programming networks Σ5 for iden-tical and nonidentical Chua’s circuits. The tar-get synchronization manifold here is characterizedby ψ(x, z) = (x − z)T (x − z) − 1 = 0. Thismanifold is reached up to a small synchronizationerror.


Fig. 5. Partial synchronization using Lagrange programming networks Σ4{x, z, ε, λ|0, 1, 1} (Left) and Σ4

{x, z, ε, λ|0, 10, 1} (Right)illustrated on Chua’s circuits. In all cases a double-scroll attractor is chosen for f(x). [Top] double scroll (a = 9) as slavesystem g(z); [Middle] Chua’s circuit with a = 8 (periodic) as slave system g(z); [Bottom] Chua’s circuit with a = 5 (sink) asslave system g(z). In the simulations w = [0; 1; 0] has been taken. This leads to partial synchronization of x2 − z2 (green).For large values of γ2 all state variables become synchronized.


(a) (b)

(c) (d)

Fig. 6. Generalized synchronization using Lagrange programming networks Σ5{x, z, ε, λ|0, 100, 1} illustrated on identical Chua’s

circuits with double-scroll attractors for f(x) and g(z). The target synchronization manifold is characterized by ψ(x, z) =(x−z)T (x−z)−1 = 0. (a) Chua’s circuit with double-scroll as master system f(x); (b) controlled double-scroll at slave systemafter synchronization in the generalized sense; (c) ε(t), λ(t); (d) ‖ψ(x(t), z(t))‖2 showing that the synchronization manifoldis reached up to a small synchronization error.


Fig. 7. (Continued) Similar results as in Fig. 6 but with double-scroll master system as f(x) and Chua’s circuit with a = 8(periodic) as g(z) slave system.

7. Conclusions

We introduced methods of Lagrange programmingnetworks for chaos synchronization. The main dif-ference with other synchronization methods is thatin Lagrange programming network approach syn-chronization is implemented both as a hard andsoft constraint. Classical schemes may be regardedas soft constraining only. We have shown onsimulation examples of full state synchronization,partial and generalized synchronization how theperformance can be improved by applying Lagrangeprogramming networks. This has been shown foridentical as well as nonidentical chaotic systemswith Chua’s circuit, Lorenz system and n-scroll cir-cuits as examples. From the investigated Lagrangeprogramming networks, the version which takes intoaccount an explicit formulation of the synchroniza-tion error yields the best performance. All this isbased upon a new interpretation of synchronization

problems within a Lagrangian framework. The con-tribution of this paper is mainly conceptual in thesense of establishing new links between synchroniza-tion, neural networks and constrained optimization.Although the results of this paper are supported bymany simulation results, a better understanding ofthe existence of potential functions in the gener-alized sense of potential function theory is neededwithin this context.

Acknowledgments

This research work was carried out at the ESAT lab-oratory and the Interdisciplinary Center of NeuralNetworks ICNN of the Katholieke Universiteit Leu-ven, in the framework of the Belgian Programmeon Interuniversity Poles of Attraction, initiated bythe Belgian State, Prime Minister’s Office for Sci-ence, Technology and Culture (IUAP P4-02) and in


the framework of a Concerted Action Project MIPS(Modelbased Information Processing Systems) ofthe Flemish Community. J. Suykens is a postdoc-toral researcher with the National Fund for Scien-tific Research FWO–Flanders.

ReferencesBeardmore, R. E. & Song, Y. H. [1998] “Differential-

algebraic equations: A tutorial review,” Int. J. Bifur-cation and Chaos 8(7), 1399–1411.

Chen, G. & Dong, X. [1998] From Chaos to Order —Perspectives, Methodologies, and Applications (WorldScientific, Singapore).

Chua, L. O. [1994] “Chua’s circuit 10 years later,”Int. J. Circuit Th. Appl. 22, 279–305.

Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “Thedouble scroll family,” IEEE Trans. Circuits Syst. I33(11), 1072–1118.

Cichocki, A. & Unbehauen, R. [1994] Neural Net-works for Optimization and Signal Processing (Wiley,Chichester).

Fletcher, R. [1987] Practical Methods of Optimization(John Wiley, Chichester, NY).

Hazewinkel, M. (ed.) [1991] Encyclopaedia of Mathemat-ics (Kluwer Academic Publishers, Boston).

Hopfield, J. J. [1984] “Neurons with graded responsehave collective properties like those of two-state neu-rons,” Proc. Natl. Acad. Sci. 81, pp. 3088–3092.

Huang, A., Pivka, L., Wu, C.-W. & Franz, M. [1996]“Chua’s equation with cubic nonlinearity,” Int. J. Bi-furcation and Chaos 6(8), 2175–2222.

Kapitaniak, T. & Chua, L. O. [1994] “Hyperchaotic at-tractors of unidirectionally-coupled Chua’s circuits,”Int. J. Bifurcation and Chaos 4(2), 477–482.

Kapitaniak, T., Chua, L. O. & Zhong, G.-Q. [1994]“Experimental synchronization of chaos using contin-uous control,” Int. J. Bifurcation and Chaos 4(2),483–488.

Kennedy, M. P. & Chua, L. O. [1988] “Neural networksfor nonlinear programming,” IEEE Trans. CircuitsSyst. 35, 554–562.

Kocarev, L. & Parlitz, U. [1996] “Generalized synchro-nization, predictability, and equivalence of unidirec-tionally coupled dynamical systems,” Phys. Rev. Lett.76(11), 1816–1819.

Kocarev, L., Shang, A. & Chua, L. O. [1993] “Transitionsin dynamical regimes by driving: A unified method ofcontrol and synchronization of chaos,” Int. J. Bifur-cation and Chaos 3(2), 479–483.

Lorenz, E. N. [1963] “Deterministic nonperiodic flow,” J.Atmos. Sci. 20, 130–141.

Madan, R. N. (Guest Editor) [1993] Chua’s Circuit: AParadigm for Chaos (World Scientific, Singapore).

Morgul, O. & Solak E. [1997] “On the synchronization ofchaotic systems by using state observers,” Int. J. Bi-furcation and Chaos 7(6), 1307–1322.

Ponomarenko V. P. & Matrosov V. V. [1996] “Nonlineardynamics of multistable Chua’s circuits,” Int. J. Bi-furcation and Chaos 6(11), 2087–2096.

Suykens, J. A. K., Curran, P. F. & Chua, L. O. [1997a]“Master–slave synchronization using dynamic out-put feedback,” Int. J. Bifurcation and Chaos 7(3),671–679.

Suykens, J. A. K., Huang, A. & Chua, L. O. [1997b] “Afamily of n-scroll attractors from a generalized Chua’scircuit,” Archiv fur Elektronik und Ubertragungstech-nik (Int. J. Electron. Commun.) 51(3), 131–138.

Suykens, J. A. K., Curran, P. F., Vandewalle, J. & Chua,L. O. [1997c] “Robust nonlinear H∞ synchroniza-tion of chaotic Lur’e systems,” IEEE Trans. CircuitsSyst. I, Special issue on Chaos Synchronization, Con-trol and Applications 44(10), 891–904.

Suykens, J. A. K., Curran, P. F. & Chua, L. O. [1999]“Robust synthesis for master–slave synchronization ofLur’e systems,” IEEE Trans. Circuits Syst. I 46(7),841–850.

Weiss L., Mathis W. & Trajkovic Lj. [1998] “A general-ization of Brayton–Moser’s mixed potential function,”IEEE Trans. Circuits Syst. I 45(4), 423–427.

Wu, C. W. & Chua, L. O. [1994] “A unified framework forsynchronization and control of dynamical systems,”Int. J. Bifurcation and Chaos 4(4), 979–989.

Yalcin, M. E., Suykens, J. A. K. & Vandewalle, J.[2000] “Experimental confirmation of 3- and 5-scrollattractors from a generalized Chua’s circuit,” IEEETrans. Circuits Syst. I, to appear.

Zhang, S. & Constantinides, A. G. [1992] “Lagrangeprogramming neural networks,” IEEE Trans. CircuitsSyst. II 39(7), 441–452.

Zubov, V. I. [1997] Mathematical Theory of the MotionStability (Saint-Petersburg, Russia).

Documents

CHAOS SYNCHRONIZATION: A LAGRANGE PROGRAMMING …chua/papers/Suykens00b.pdf · be regarded as a control problem where a feedback matrix Khas to be designed such that the slave is