SYNCHRONIZATION OF MODIFIED CHUA’S CIRCUITS IN STAR

COUPLED NETWORKS

O. R. Acosta del Campo

Engineering Faculty, UABC, Km. 103 Carretera Tij-Ens, Ensenada, M´exico

C. Cruz-Hern´andez

Electronics and Telecommunications Department, CICESE, Km. 107 Carretera Tij-Ens, Ensenada, M´exico

R. M. L´opez-Guti´errez , E. E. Garc´ıa-Guerrero

Engineering Faculty, UABC, Km. 103 Carretera Tij-Ens, Ensenada, M´exico

Keywords:

Hyperchaos synchronization, Modiﬁed fourth-order Chua’s circuit, Generalized Hamiltonian forms, Ob-

servers, Star coupled networks.

Abstract:

In this paper, we use Generalized Hamiltonian systems approach to synchronize dynamical networks of modi-

ﬁed fourth-order Chua’s circuits, which generate hyperchaotic dynamics. Network synchronization is obtained

among a single master node and two slave nodes, with the slave nodes being given by observers.

1 INTRODUCTION

The synchronization problem of two chaotic oscilla-

tors has received a lot of attention in last decades,

see e.g. this example in order to achieve the high-

est quality possible (Pecora and Carroll, 1990); (Ni-

jmeijer and Mareels, 1997); (L´opez-Mancilla and

Cruz-Hern´andez, 2005); (L´opez-Mancilla and Cruz-

Hern´andez, 2008); (Cruz-Hern´andez and Nijmeijer,

2000); (Boccaleti and et. al., 2002); (Luo, 2008);

(Cruz-Hern´andez, 2004) and references therein. This

interest increases by practical applications in differ-

ent ﬁelds, particularly in secure communications, see

e.g. (Cruz-Hern´andez, 2004); (L´opez-Mancilla and

Cruz-Hern´andez, 2005); (Aguilar-Bustos and Cruz-

Hern´andez, 2008); (Cruz-Hern´andez and N.Romero-

Haros, 2008). Hyperchaotic dynamics characterized

by more than one positive Lyapunov exponent are

advantageous over simple chaotic dynamics. How-

ever, hyperchaos synchronization is a much more dif-

ﬁcult problem, see e.g. (Aguilar-Bustos and Cruz-

Hern´andez, 2008) for two coupled oscillators.

In (Posadas-Castillo and et.al.(a), 2007) was de-

veloped an experimental study on practical realization

to synchronize dynamical networks of Chua’s circuits

globally coupled. While in recent works (Posadas-

Castillo and et. al.(b), 2007); (Posadas-Castillo and

et. al., 2008); (H. Serrano Guerrero, 2009) was ob-

tained synchronization in coupled star networks with

chaotic nodes given by Nd:YAG lasers and 3D CNNs,

respectively; by using the approach given in (Wang,

2002). Some literature devoted on synchronization

of complex networks (Manrubia and et.al., 2004);

(Pogromsky and Nijmeijer, 2001); (Wang, 2002).

Network synchronization of coupled star nodes

can be applied to transmit encrypted messages, from

a single transmitter to multiple receivers in network

communication systems, if the coupled nodes are

chaotics. The aim of this paper is to synchronize three

modiﬁed fourth-order Chua’s circuits (which exhibit

hyperchaotic behavior) studied in (Thamilmaran and

et.al., 2004) in star coupled networks via Generalized

Hamiltonian forms and observer design proposed in

(Sira-Ram´ırez and Cruz-Hern´andez, 2001). This ap-

proach presents several advantages over the existing

synchronization methods reported in the current liter-

ature.

2 PROBLEM SETTING

Consider the following set of N interconnected iden-

162

R. Acosta del Campo O., Cruz-Hernández C., E. García-Guerrero E. and M. López-Gutiérrez R. (2009).

SYNCHRONIZATION OF MODIFIED CHUA’S CIRCUITS IN STAR COUPLED NETWORKS.

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 162-167

DOI: 10.5220/0002218401620167

 SciTePress

tical dynamical systems

= f(x

) + u

, i = 1,2,...,N, (1)

where x

= (x

,...,x

)

∈ R

is the state vector

and u

= u

∈ R is the input signal of the system i,

deﬁned by

∑

j=1

Γx

, i = 1,2, ...,N, (2)

the constant c > 0 represents the coupling strength,

and Γ ∈ R

n×n

is a constant 0-1 matrix linking cou-

pled states.Whereas, A = (a

) ∈ R

n×n

is the coupling

matrix, which represents the coupling conﬁguration

in (1)-(2). If there is a connection between node i

and node j, then a

= 1; otherwise, a

= 0 for i 6= j.

Note that, if u

= 0, i = 1, 2,..., N, in (1) we have a set

of N isolated dynamical systems, operating with their

own dynamics. While, if u

6= 0 the set constitutes

a dynamical network and each dynamical system i is

called nodo i; and under appropiates u

the dynami-

cal networks can be achieve collective behaviors. It is

clear that, the input singal u

determines the kind of

coupling among nodes in the networks. The coupling

matrix for star coupled networks is given by

A =







N − 1 −1 −1 ··· −1

−1 0 0 ··· 0







(3)

The star coupled conﬁguration for N nodes is

shown in Fig. 1, with the common or central node

Figure 1: Star coupled conﬁguration with N nodes.

The complex dynamical network (1) is said to

achieve (asymptotically) synchronization, if (Wang

2002):

(t) = x

(t) = ... = x

(t) as t → ∞. (4)

The synchronization state in (1) can be an equi-

librium point, a periodic orbit or, a chaotic attrac-

tor. This paper addresses the synchronization prob-

lem of dynamical networks (1) with coupled nodes

in star topologies. In particular, by choosing a mas-

ter node with the objective of to impose a particular

collective behavior in (1). For illustrative purposes

only, we consider three isolated nodes (N = 3) to be

synchronized (which are described in Section 4), see

Fig. 2(a). In Fig. 2(b) is shown this dynamical net-

work with master node N1 and two slave nodes N2

and N3. Our objective is the synchronization of this

network, when the coupled nodes are given by mod-

iﬁed fourth-order Chua’s circuits, to be described in

Section 4. This particular coupling topology is impor-

tant for its application to network communication sys-

tems, to transmit messages from a single transmitter

to multiple receivers (Chow T.W.W. and Ng, 2001).

(a)

(b)

Figure 2: (a) Three isolated nodes. (b) Star coupled network

with master node N1.

S1 M S2

y y

Figure 3: Single master node M and two slave nodes S1 and

S2 conﬁguration.

3 SYNCHRONIZATION VIA

HAMILTONIAN FORMS

To solve the network synchronization problem stated

in previous section, we appeal to synchronization (of

two chaotic oscillators) via Hamiltonian forms and

observer design reported in (Sira-Ram´ırez and Cruz-

Hern´andez, 2001). In the sequel, we show that this

approach is appropriate to synchronize a coupled star

network with three nodes shown in Fig. 2(b). By us-

ing the proposed synchronization scheme shown in

Fig. 3, where M is given in Hamiltonian form (Eq.

(7)) and S1 and S2 being two observers for M given

by Eq. (8).

Consider the following isolated dynamical system

˙x = f(x), (5)

SYNCHRONIZATION OF MODIFIED CHUA'S CIRCUITS IN STAR COUPLED NETWORKS

163

where x(t) ∈ R

is the state vector, f : R

→ R

is a

nonlinear function.

In (Sira-Ram´ırez and Cruz-Hern´andez, 2001) is

reported how the dynamical system (5) can be written

in the following Generalized Hamiltonian canonical

form,

˙x = J (x)

∂H

∂x

+ S (x)

∂H

∂x

+ F (x), x ∈ R

, (6)

H(x) denotes a smooth energy function which is glob-

ally positive deﬁnite in R

. The gradient vector of H,

denoted by ∂H/∂x, is assumed to exist everywhere.

We use quadratic energy function H(x) = (1/2)x

with M being a, constant, symmetric positive deﬁnite

matrix. In such case, ∂H/∂x = Mx. The matrices,

J (x) and S(x) satisfy, for all x ∈ R

, the properties:

J (x) + J

(x) = 0 and S(x) = S

(x). The vector ﬁeld

J (x)∂H/∂x exhibits the conservative part of the sys-

tem and it is also referred to as the workless part,

or work-less forces of the system; and S(x) depict-

ing the working or nonconservative part of the sys-

tem. For certain systems, S(x) is negative deﬁnite or

negative semideﬁnite. Thus, the vector ﬁeld is con-

sidered as the dissipative part of the system. If, on the

other hand, S(x) is positive deﬁnite, positive semidef-

inite, or indeﬁnite, it clearly represents, respectively,

the global, semi-global, and local destabilizing part

of the system. In the last case, we can always (al-

though nonuniquely) descompose such an indeﬁnite

symmetric matrix into the sum of a symmetric nega-

tive semideﬁnite. matrix R(x) and a symmetric pos-

itive semideﬁnite matrix N(x). Finally, F(x) repre-

sents a locally destabilizing vector ﬁeld.

In the context of observer design, we consider a

special class of Generalized Hamiltonian forms (to be

considered as the master node M) with linear output

map y(t), given by

˙x = J (y)

∂H

∂x

+ (I + S )

∂H

∂x

+ F (y), x ∈ R

, (7)

y = C

∂H

∂x

, y ∈ R

where S is a constant symmetric matrix, not necessar-

ily of deﬁnite sign. The matrix I is a constant skew

symmetric matrix, and C is a constant matrix.

We denote the estimates of the state x(t) by ˆx

(t),

i = 1,2 and consider the Hamiltonian energy func-

tion H( ˆx

) to be the particularization of H in terms

of ˆx

(t). Similarly, we denote by η

(t), i− 1, 2 the es-

timated outputs, computed in terms of the estimated

states ˆx

(t). The gradient vector ∂H( ˆx

)/∂ˆx

is natu-

rally, of the form M ˆx

with M being a, constant, sym-

metric positive deﬁnite matrix.

Two nonlinear state observers for M (7) are given

ˆx

= J (y)

∂H

∂ˆx

+(I +S )

∂H

∂ˆx

+F (y)+K

(y−η

), (8)

= C

∂H

∂ˆx

, η

∈ R

, i = 1,2,

with ˆx

∈ R

and K

is the observer gain.

The state estimation errors, deﬁned as e

(t) =

x(t) − ˆx

(t) and the output estimation error, deﬁned

as e

(t) = y(t) − η

(t), are governed by

˙e

= J (y)

∂H

∂e

+ (I + S − KC )

∂H

∂e

, e

∈ R

, (9)

= C

∂H

∂e

, e

∈ R

, i = 1,2,

where the vectors ∂H/∂e

actually stands, with some

abuse of notation, for the gradient vector of the

modiﬁed energy functions, ∂H(e

)/∂e

= ∂H/∂x −

∂H/∂ˆx

= M(x − ˆx

) = Me

. We set, when needed,

I + S =W .

A necessary and sufﬁcient condition for global

asymptotic stability to zero of the estimation errors

(9) is given by the following theorem.

Theorem 1 (Sira-Ram´ırez and Cruz-Hern

andez,

2001). The state x(t) of the master node M (7) can

be globally, exponentially, asymptotically estimated,

by the states ˆx

(t), i = 1,2 of the observers (8) if and

only if, there exist constant matrices K

such that the

symmetric matrices

[W − K

C ] + [W − K

C ]

= [S − K

C ]+[S − K

C ]

= 2



S −

C + C

)



are negative deﬁnite.

4 HYPERCHAOTIC CHUA’S

CIRCUIT LIKE NODE

Consider the modiﬁed fourth-order Chua’s circuit de-

scribed by (Thamilmaran and et.al., 2004):

˙x

= α

− f (x

)),

˙x

= −α

− x

˙x

= β

− x

˙x

= β

(10)

with nonlinear function given by

f (x

) = bx

(a− b)(|x

+ 1|− |x

− 1|). (11)

With the paramerter values: α

= 2.1429, α

−12.83, β

= 0.0393, β

= 0.0015, a = −0.0299, and

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

164

b = 1.995 the modiﬁed Chua’s circuit (10)-(11) ex-

hibits hyperchaotic behavior, with two positive Lya-

punov exponents. By using the initial conditions

x(0) = (1.1,0.1,−0.5,0.01), Figs. 1, 2, 3, and 4 show

the hyperchaotic attractors x

vs x

, x

vs x

, x

vs x

and x

vs x

, respectively.

−4 −3 −2 −1 0 1 2 3 4

−1.5

−1

−0.5

0.5

1.5

Figure 4: Hyperchaotic attractor projected onto the (x

, x

plane.

−4 −3 −2 −1 0 1 2 3 4

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

Figure 5: Hyperchaotic attractor projected onto the (x

, x

plane.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.2

−0.15

−0.1

−0.05

0.05

0.1

0.15

Figure 6: Hyperchaotic attractor projected onto the (x

, x

plane.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15

−1.5

−1

−0.5

0.5

1.5

Figure 7: Hyperchaotic attractor projected onto the (x

, x

plane.

Next, we show the arrangement for star dynami-

cal network by using as coupled node to hyperchaotic

Chua’s circuit deﬁned by (10)-(11).

5 SYNCHRONIZATION OF

HYPERCHAOTIC CHUA’S

CIRCUITS IN A STAR

NETWORK

In this section, we show the synchronization of three

hyperchaotic Chua’s circuits in a star coupled net-

work, via Generalized Hamiltonian forms and ob-

server design proposed in (Sira-Ram´ırez & Cruz-

Hern´andez 2001). Firstly, we rewrite the modiﬁed

fourth-order Chua’s circuit (10)-(11) for the master

node as follows.

Taking as Hamiltonian energy function to

H (x) =



+ x



. (12)

Modiﬁed fourth-order Chua’s circuit (10)-(11) in

Generalized Hamiltonian form (master node, M) ac-

cording to Eq. (7) is given by







˙x













0 0 α

0 0 −β

−β

−α

0 0

0 β

0 0







∂H

∂x

+ (13)







0 0 0 0

0 −α

0 0

0 0 −β

0 0 0 0







∂H

∂x







−α

f (x

)







The destabilizing vector ﬁeld calls for x

(t) to be

used as the output y(t), of the master node M (13).

The matrices C, S, and I are given by













, S =







0 0 0 0

0 −α

0 0

0 0 −β

0 0 0 0







I =







0 0 α

0 0 −β

−β

−α

0 0

0 β

0 0







Next, we design two state observers (slave nodes S1

and S2, see Fig. 3) for master node (13). The ﬁrst

nonlinear state observer for the Generalized Hamilto-

nian system (13) (according to Eq. (8) as slave node

SYNCHRONIZATION OF MODIFIED CHUA'S CIRCUITS IN STAR COUPLED NETWORKS

165

S1 is given by







ˆx













0 0 α

0 0 −β

−β

−α

0 0

0 β

0 0







∂H

∂ˆx

+ (14)







0 0 0 0

0 −α

0 0

0 0 −β

0 0 0 0







∂H

∂ˆx







−α

f (x

)



















= ˆx

the second state observer (slave S2) is described by







ˆx













0 0 α

0 0 −β

−β

−α

0 0

0 β

0 0







∂H

∂ˆx

+ (15)







0 0 0 0

0 −α

0 0

0 0 −β

0 0 0 0







∂H

∂ˆx







−α

f (x

)



















= ˆx

where e

= x

− ˆx

= y − η

) and e

= x

−

ˆx

= y− η

). From master node (13) and slave

nodes (14) and (15), we have that the synchronization

error dynamics among the master node and two slave

nodes (observers) is governed by







˙e













−

0 −β

−β

−γ

0 0

−

0 0







∂H

∂e







−k

−

−α

0 0

−

0 −β

−

0 0 0







∂H

∂e

(16)

where γ

= α

. Where the synchronization

errors are deﬁned by e

and e

among master M

and slaves 1 and 2, respectively. One may now

choose the observer gains K

= (k

)

, i =

1,2 in order to guarantee asymptotic exponential sta-

bility to zero of the synchronization errors e

(t) =

(t),e

(t)), i = 1, 2 as will be shown

in the next section.

6 SYNCHRONIZATION

CONDITIONS

Now, we examine the stability of the synchroniza-

tion errors (16) for the network constructed with mas-

ter (13) and two slaves (14) and (15), with modi-

ﬁed Chua’s circuits as coupled nodes. Thus, we in-

voke Theorem 1, which guarantees global asymp-

totic stability to zero of e

(t), i = 1,2. In par-

ticular, for modiﬁed Chua’s circuit, the matrices



S −

C + C

)



, i = 1,2 shown in Theorem

1, are give by







−2k

−k

−2α

0 0

−k

0 −2β

−k

0 0 0







, i = 1,2

(17)

by applying the Sylvester’s Criterion -which provides

a test for negative deﬁnite of a matrix- thus, we have

that the mentioned matrices will be negative deﬁ-

nite matrices, if we choose K

= (k

)

i = 1, 2 such that the following conditions are satis-

ﬁed:

≤ 1, (18)

− k

≥ 0,





− 4k



+ k



≥ 0,

= 0.

We have used K

= (3.3,1.5,0.39,0)

and

= (2.3,1,0.3,0)

with initial conditions:

for M, x(0) = (1.1,0.1,−0.5,0.01) and for

S1, ˆx

(0) = (0.5,0.3,−0.4,0) and for S2,

ˆx

(0) = (1,0, −0.2,0.04). Fig. 8 shows the

synchronization among master node (13) and two

slave nodes (14) and (15).

−1.5 −1 −0.5 0 0.5 1 1.5

−2

Master vs Slave 1

−1.5 −1 −0.5 0 0.5 1 1.5

−2

Master vs Slave 2

−3 −2 −1 0 1 2 3 4

−5

−3 −2 −1 0 1 2 3 4

−5

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−1

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−1

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08

−0.1

0.1

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08

−0.1

0.1

Figure 8: Complete synchronization among states of hyper-

chaotic master node M and slave nodes S1 and S2.

7 CONCLUSIONS

In this paper, we have presented multiple synchro-

nization of coupled modiﬁed fourth-order Chua’s cir-

cuit, in particular by using star coupled networks.

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

166

We have achieve synchronization of three hyper-

chaotic Chua’s circuit (used as fundamental node) in

star complex networks, via Generalized Hamiltonian

forms and observer design considering a single mas-

ter node and two slave nodes. This result is particu-

larly interesting given its application in communica-

tion network systems, where is required that a single

sender transmits simultaneously information to many

receivers via a public channel.

ACKNOWLEDGEMENTS

This work was supported by CONACYT, M´exico un-

der Research Grant Nos. J49593-Y, and P50051-Y.

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