Modiﬁed Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 7, No. 6, December 2017, pp. 3436 – 3445
ISSN: 2088-8708 3436
Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
Modified Projective Synchronization of Chaotic Systems
with Noise Disturbance, an Active Nonlinear Control
Method
Hamed Tirandaz, Hamidreza Tavakoli and Mohsen Ahmadnia
Electrical and Computer Engineering Faculty, Hakim Sabzevari University, Sabzevar, Iran
Article Info
Article history:
Received: Jan 15, 2017
Revised: Jul 25, 2017
Accepted: Aug 9, 2017
Keyword:
Active method
Nonlinear control
Modified Projective
synchronization (MPS)
Stability theorem
ABSTRACT
The synchronization problem of chaotic systems using active modified projective non-
linear control method is rarely addressed. Thus the concentration of this study is to
derive a modified projective controller to synchronize the two chaotic systems. Since,
the parameter of the master and follower systems are considered known, so active
methods are employed instead of adaptive methods. The validity of the proposed con-
troller is studied by means of the Lyapunov stability theorem. Furthermore, some
numerical simulations are shown to verify the validity of the theoretical discussions.
The results demonstrate the effectiveness of the proposed method in both speed and
accuracy points of views.
Copyright c 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Hamed Tirandaz
Hakim Sabzevari University
Electrical and Computer Engineering Faculty, Hakim Sabzevari University, Sabzevar, Iran.
Phone +98051-4401288
Email: tirandaz@hsu.ac.ir
1. INTRODUCTION
Master-slave synchronization of chaotic systems is strikely nonlinear, since the aperiodic and nonreg-
ular behavior of chaotic systems and their sensitivity to the initial conditions. Chaotic behavior may appear in
many physical systems. So, chaos synchronization subject has received a great deal of attention in the last to
decades, due to its potential applications in physics, chemistry, electrical engineering, secure communication
and so on[1]. Up to now, many types of controling methods are revealed and investigated for control and syn-
chronization of chaotic systems. Active method[2, 3, 4, 5, 6], adaptive method [7, 8, 9], linear feedback method
[10, 11], nonlinear feedback method [12, 14, 15], sliding mode method [16, 17, 18], impulsive method [19],
phase method [20], generalized method [21], robust synchronization [13] and projective method [22, 23, 24]
are some of the introduced methods by the researchers. Among these methods, synchronization with some
types of projective methods are extensively investigated in the last decades, since the faster synchronization
due to its synchronization scaling factors, which master and slave chaotic systems would be synchronized up
to a proportional rate. Projective lag method [25], modified projective synchronization (MPS) [26, 27, 28],
function projective synchronization (FPS)[29], modified function projective synchronization [30, 28], general-
ized function projective synchronization [31, 32] and modified projective lag synchronization[33, 34] are some
generalized schemes of projective method, which utilize some type of scaling factors.
When the parameters of a chaotic system are known beforehand, active related methods are preferably
chosen than adaptive methods. Active synchronization problem of two chaotic systems with known parameters
are vastly investigated by the researchers. For example, in [5, 3, 35], the active controlling method is studied
for synchronization of two typical chaotic systems. And also, in [2], an active method for controling the
behavior of a unified chaotic system is presented. Chaos synchronization of complex Chen and Lu chaotic
systems are addressed in citeMahmoud, with designing an active control method. Furthermore, in [36] active
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Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
, DOI: 10.11591/ijece.v7i6.pp3436-3445

IJECE ISSN: 2088-8708 3437
synchronization of two different fractional order chaotic system is studied.
Consequently, the modified projective synchronization of two chaotic systems with known system
parameters by acitve control method are rarely investigated by the researchers. Therefore, in the present study,
the modified projective synchronization problem is achieved by means of active nonlinear control method. An
appropriate feedback controller is designed to control the behavior the state variables of the follower system to
track the trajectories of the leader system state variables. In Section 2, the problem of chaos synchronization is
discussed. In addition, the validity of the proposed synchronization method is verified by means of Lyapunov
stability theorem. Then, in Section 3, some experiments are derived to show the effectiveness of the proposed
method. Moreover, some simulations are carried out. Finally, some concluding remarks are given in Section 4.
2. SYNCHRONIZATION
A wide variety of chaotic systems can be represented as follows:
˙x = f(x)Φ + F(x) + η (1)
Where x = (x1, x2, · · · , xn)T
is the state variables vector of the system (1). Φ = (φ1, φ2, · · · , φn)T
∈ Rn×1
and η = (η1, η2, · · · , ηn)T
∈ Rn×1
are two vectors denoting the unknown parameter vector of the system and
the external distributive noise of the system, respectively. f(x) ∈ Rn×n
and F(x) ∈ Rn×1
stand for the linear
and nonlinear matrix of functions, respectively. Let the dynamical system (1) as the leader system. Then the
follower system can be given by another chaotic function as follows:
˙y = g(y)ˆΦ + G(y) + u (2)
Where y = (y1, y2, · · · , yn)T
presents the state variables vector of the follower system (2). ˆΦ = (ˆφ1, ˆφ2, · · · , ˆφn) ∈
Rn×1
denotes the estimation of leader system parameters vector Φ. Moreover, g(y) ∈ Rn×n
and G(y) ∈
Rn×1
are the linear and nonlinear matrix of functions, respectively. In the proposed active nonlinear control
method, an appropriate controller u is designed which the states of leader system (1) are synchronized with
their corresponding states at the follower chaotic sytem (2), base on the modified projective synchronization
error that is defined as follows:
e = y − Λx (3)
Where Λ = diag{λ1, λ2, · · · , λn} represents the modified scaling factors and e = (e1, e2, · · · , en)T
∈ Rn×1
stands for synchronization error vector. Then the dynamical synchronizaton error can be obtatined as follows:
˙e = ˙y − Λ ˙x
= g(y)Φ + G(y) + u − f(x)Φ − ΛF(x) − Λη (4)
Where ¯η denotes the estimation of noise distrubance η.
Definition 1. For the leader system (1) and the follower system (2), the chaos synchronization would
be achieved if an appropriate control is designed to force the state variables of the follower system to track the
trajectories of the leader one, meanly, the synchronization error vector (3) converges to zero, as time goes to
infinity,i. e:
lim
t→∞
e(t) = 0
which . denotes 2-norm. Chaos synchronization can be achieved by deriving an appropriate feed-
back controller, which is the subject of the following theorem.
Theorem 1. The leader system (1) with the state variables vector x and the follower system (2) with
the state variables vector y, the parameters vector Φ and any noise disturbance vector η, would be synchronized
for any initial state variables x(0) and y(0), if the active feedback control law is defined as follows:
u = − g(y) − f(x) Φ − G(y) − ΛF(x) + Λ¯η − Ke (5)
Where ¯η can be estimated dynamically as follows:
˙¯η = −Λe − Ψ(¯η − η) (6)
Modified Projective Synchronization of Chaotic Systems with ... (Hamed Tirandaz)

3438 ISSN: 2088-8708
Where K = diag{k1, k2, · · · , kn} and Ψ = diag{ψ1, ψ2, · · · , ψn} are two diagonal matrix with positive
values for their main diagonal elements.
Proof. Let the Lyapunov stability function as follows:
V =
1
2
eeT
+
1
2
(¯η − η)(¯η − η)T
(7)
It is obvious that the Lyapunov function defined in (7) is positive definite. With calculating its time derivative,
we have:
˙V = ˙eeT
+ ˙¯η(¯η − η)T
(8)
Then, substituting the dynamical representation of synchronization error vector (4) and consequently consider-
ing the proposed feedback controller (5) and the noise estimation (6), one can get:
˙V = −KeeT
− Ψ(ˆη − η)(ˆη − η)T
(9)
Therefore, derivative of V is negative definite, when K and Ψ are diagonal matrix with positive elements
on their primary diagonal elements. In the following section, some numerical results are given to show the
effectiveness of the proposed synchronization method.
3. NUMERICAL SIMULATIONS
This section is devoted to the synchronization of two different chaotic or hyperchaotic systems. In the
following subsection, chaos synchronization between two chaotic systems, Zhang chaotic system and Lorenz
chaotic system is addressed. Then, the synchronizaton problem between two hyperchaotic system as Chen
hyperchaotic system and Lorenz hyperchaotic system is studied in the last subsection
3.1. chaotic systems
Chaos synchronization between Zhang chaotic system [14] and the Lü chaotic system [37] is addressed
in this subsection. The Zhang chaotic system is given by a three simple integer-based and nonlinear differential
equations that depends on the three positive real parameters as follows
˙x1 = a(x2 − x1) − x2x3
˙x2 = bx1 − x2
1 (10)
˙x3 = −cx3 + x2
2
Where x1, x2 and x3 are the state variables of the system and a, b, and c are the three constant parameters of
the system. When a = 10, b = 30 and c = 6, the behaviour of the system is chaotic. The phase portraits of the
system is shown in Fig. 1, with initial state variables x1(0) = 5, x2(0) = 2 and x3(0) = 30.
In addition, the Lü chaotic system can be described as follows:
˙y1 = α1(y2 − y1)
˙y2 = α2y2 − y1y3 (11)
˙y3 = y1y2 − α3y3
Where y1, y2 and y3 are the state variables of the system and α1, α2 and α3 are the parameter of the system.
The chaotic behavior of the Lü system is shown in Fig. 2, with system parameters as: α1 = 2.1, α2 = 30 and
α3 = 0.6, and state variables initial values as: x1(0) = 4.3, x2(0) = 7.2 and x3(0) = 5.8.
The Zhang chaotic system (10) can be rewritten based on the leader system (1) as follows:
˙x1 = a(x2 − x1) − x2x3 + η1
˙x2 = bx1 − x2
1 + η2 (12)
˙x3 = −cx3 + x2
2 + η3
IJECE Vol. 7, No. 6, December 2017: 3436 – 3445

IJECE ISSN: 2088-8708 3439
Figure 1. Phase portraits of hte Zhang chaotic system
Figure 2. Phase portraits of hte Lu chaotic system

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Where η1, η2 and η3 are the three noise disturbance corresponding to the state variables x1, x2 and x3, respec-
tively. Then, the L¨u chaotic system (11) can be represented as the follower system as follows:
˙y1 = a(y2 − y1) + u1
˙y2 = by2 − y1y3 + u2 (13)
˙y3 = y1y2 − cy3 + u3
According to the proposed control law (5) and noise disturbance estimation (6), we deﬁne the following feed-
back controller as:
u1 = −ay2 + λ1ax1 + ae1 − λ1x2x3 + λ1 ¯η1 − k1e1
u2 = −by2 + y1y3 + λ2(bx1 − x2
1) + λ2 ¯η2 − k2e2 (14)
u3 = −y1y2 + ce3 + λ3x2
2 + λ3 ¯η3 − k3e3,
and the noise disturbance estimation as:
˙¯η1 = −λ1e1 − ψ1(¯η1 − η1)
˙¯η2 = −λ2e2 − ψ2(¯η2 − η2) (15)
˙¯η3 = −λ3e3 − ψ3(¯η3 − η3)
Assume the parameter of the Zhang chaotic system as a = 10, b = 30 and c = 6 and the initial values
for the drive chaotic system (12) are taken as, x1(0) = 12, x2(0) = 5 , and, x3(0) = 6.5 . In additiion, the initial
values of the response L system (3) are selected as: y1(0) = 2, y2(0) = 15 and y3(0) = 0. Consider the nosie
disturbance values as η1 = 0.8, η2 = 0.6 and η3 = 0.3 and also their corresponding estimation ititial values as
¯η1 = 0.15, ¯η2 = 0.2 and ¯η3 = 0.1. Let the gain constants as k1 = 2, k2 = 2, k3 = 2, φ1 = 1.5, φ2 = 1.5 and
φ3 = 1.5.
The validify of the proposed synchronization method for contorling the behavior of the Lu chaotic
system (13) to track the motion trajectories of the Zhang chaotic system (12) and the noise disturbance estima-
tion are shown in Figure 3 and 4, respectively. Figure 3 shows that the state variables of the system (13) track
effectively the motion trajectories of the leader chaotic system. In addition, in Figure 4 exhibit that the distance
between noise disturbance and its estimation values converge to zero.
0 1 2 3 4 5 6 7 8
-20
-10
0
10
20
0 1 2 3 4 5 6 7 8
-20
0
20
40
0 1 2 3 4 5 6 7 8 9 10
-10
0
10
20
30
Figure 3. Time responce of the drive Zhang chaotic system and the response Lorenz chaotic system
3.2. Hyperchaotic systems
In this subsection, the synchronization between two hyperchaotic systems as Chen hyperchaotc system
and Lorenz hyperchaotic system is investivated via the proposed control method. The Chen hyperchaotic

IJECE ISSN: 2088-8708 3441
0 1 2 3 4 5 6 7 8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
noise disturbance errors
Figure 4. Time responce of the noise disturbance estimation
system is introduced in [38], as an extention of a three-dimensional Chen chaotic system as follows:
x1 = a(x2 − x1) + x4
x2 = dx1 + cx2 − x1x3
x3 = x1x2 − bx3 (16)
x4 = x1x2 + rx4
Where x1, x2, x3 and x4 are the state variables and a, b, c and d are the parameter of the system. The phase
prortait of the system (16) is shown in Fig. 5, with state variables x1(0) =, x2(0) =, x3(0) = and x4(0) = and
the parameters as a = 35, b = 3, c = 12, d = 7 and r=0.5 . As it can be seen the behavior of the system (16) is
hyperchaotic. The Lorenz hyperchaotic system, which was introduced in [39], can be described as follows:
y1 = α1(y2 − y1) + y4
y2 = −y1y3 + α3y1 − y2
y3 = y1y2 − α2y3 (17)
y4 = −y1y3 + α4y4
Where y1, y2, y3 and y4 are the state variabels, a, b, c and d are parameter of the system. The chaotic behavior
of the Lorenz hyperchaotic system is shown in Fig. 6, with initial values for the system state variables as
x1(0) =, x2(0) =, x3(0) = and x4(0) = and the system parameters as α1 = 36, α2 = 3, α3 = 20 and
α4 = 1.3
The leader system can be deﬁned based on the Chen hyperchaotic system (16) as follows:
x1 = a(x2 − x1) + x4 + η1
x2 = dx1 + cx2 − x1x3 + η2
x3 = x1x2 − bx3 + η3 (18)
x4 = x1x2 + rx4 + η4
Where η1, η2, η3 and η4 are the noise disturbances of the system. Then, consider the Lorenz hyperchaotic
system (17), as the follower system as follows:
y1 = a(y2 − y1) + y4 + u1
y2 = −y1y3 + dy1 − y2 + u2
y3 = y1y2 − by3 + u3 (19)
y4 = −y1y3 + cy4 + u4

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Where u1, u2, u3 and u4 are the feedback controller of the system.
The proposed chaos synchronization between the leader Chen hyperchaotic System (18) and the fol-
lower Lorenz hyperchaotic system (19) can be achived by designing an appropriate control law and noise
estimation law as follows:
u1 = −ay2 + λ1ax1 + ae1 − e4 + λ1 ¯η1 − k1e1
u2 = +y1y3 − λ2x1x3 − dy1 + λ2dx1 + y2 + cλ2x2 + λ2 ¯η2 − k2e2
u3 = −y1y2 + λ3x1x2 + be3 + λ3 ¯η3 − k3e3 (20)
u4 = y1y3 + λ4x1x2 − cy4 + λ4rx4 + λ4 ¯η4 − k4e4,
and,
˙¯η1 = −λ1e1 − ψ1(¯η1 − η1)
˙¯η2 = −λ2e2 − ψ2(¯η2 − η2) (21)
˙¯η3 = −λ3e3 − ψ3(¯η3 − η3)
˙¯η4 = −λ4e4 − ψ4(¯η4 − η4)
Now, some numerical results related to the proposed synchronization of two hyperchaotic systems are
given. Consider the parameter of the leaer Chen hyperchaotic system (18) as a = 35, b = 3, c = 12, d = 7
and r = 0.5 and its initial values are taken as, x1(0) = 11, x2(0) = 5, x3(0) = 9 , and, x4(0) = 13
. In additiion, the initial values of the response Lorenz hyperchaotic system (19) are selected as: y1(0) =
1, y2(0) = 11, y3(0) = 2 and y4(0) = 3. Consider the nosie disturbance values as η1 = 0.8, η2 = 0.6, η3 = 0.3
and η4 = 0.5. Let the gain constants as k1 = 2, k2 = 2, k3 = 2, k4 = 2, φ1 = 1.5, φ2 = 1.5, φ3 = 1.5 and
φ4 = 1.5.
The effectiveness of the synchronization method for the contorling behavior of the Lorenz hyper-
chaotic system (19) to track the motion trajectories of the Chen hyperchaotic system (18) and the noise distur-
bance estimation are illustrated in Figure 3 and 4, respectively. Figure 3 shows that the state variables of the
system (19) track effectively the motion trajectories of the leader chaotic system(18). In addition, in Figure 4
exhibit that the distance between noise disturbance and its estimation values converge to zero.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-20
0
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-20
0
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
20
40
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
200
400
Figure 5. Time responce of the drive Zhang chaotic system and the response Lorenz chaotic system
4. CONCLUSION
In this research, some results related to the modiﬁed projective synchronization of known chaotic/hyperchaotic
systems with noise disturbances are derived. Since the paramers of the leader system is considered knonwn,
an appropriated active nonlinear feedback control law with designed via modiﬁed projective synchronization
error. The validity of the proposed method is proved by means of Lyapunov stability theorem. Furtheremore,

IJECE ISSN: 2088-8708 3443
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 6. Time responce of the noise disturbance estimation
its effectiveness is verified by some numerical simulations of the chaotic and hyperchaotic systems. Finally,
some figures are shown to verify the accuracy of the theorical discussions. As it can be seen from these results,
the motion trajectories of the leader chaotic systems containing noise disturbances can effectively track by the
state variables of the follower chaotic systems state variabels, which affected by proposed control method.
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BIOGRAPHIES OF AUTHORS
Hamed Tirandaz received the B.Eng. Degree in Applied Mathematics from the University of
Sabzevar Tarbiat Moallem University, Sabzevar, Iran, in 2006, and PhD. degree in Mechatronics
Engineering from Semnan University, Semnan, Iran, in 2009. He is been working as a Lecturer at
Hakim Sabzevari University since 2010. His research interests include mainly Chaos control and
synchronization. He has published several papers in the above mentioned area.
Email: tirandaz@hsu.ac.ir
Mohsen Ahmadnia has received his PhD degree in Power Engineering at the People’s Friendship
university of Russia. He is currently with the Faculty of Electrical and Computer Engineering at
Hakim Sabzevari University, Sabzevar, Iran. His research interests are in nonlinear dynamics.
Email: m.ahmadnia@hsu.ac.ir
Hamidreza Tavakoli has obtained his PhD degree in Electrical Engineering at the Iran University
of Science and Technology and Ryerson University in Toronto, Canada. He is currently with the
Faculty of Electrical and Computer Engineering at Hakim Sabzevari University, Sabzevar, Iran. His
research interests are in Chaos and nonlinear dynamics.
Email: tavakoli@hsu.ac.ir

Modiﬁed Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

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Modiﬁed Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method