DYNAMICS, ADAPTIVE CONTROL AND EXTENDED SYNCHRONIZATION OF HYPERCHAOTIC SYSTEM AND ITS APPLICATION TO SECURE COMMUNICATION

International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.6, No.1/2/3/4, December 2017
DOI:10.5121/ijccms.2017.6501 1
DYNAMICS, ADAPTIVE CONTROL AND EXTENDED
SYNCHRONIZATION OF HYPERCHAOTIC SYSTEM
AND ITS APPLICATION TO SECURE
COMMUNICATION
Onma O. S1
and Akinlami J.O2
1
Nonlinear Dynamics Research Group, Department of Physics, Federal University of
Agriculture, Abeokuta, Nigeria
2
Department of Physics, Federal University of Agriculture Abeokuta, Nigeria
ABSTRACT
This work studies the dynamics, control and synchronization of hyperchaotic Lorenz-stenflo system and its
application to secure communication. The proposed designed nonlinear feedback controller control and
globally synchronizes two identical Lorenz-stenflo hyperchaotic systems evolving from different initial
conditions with unknown parameters. Adaptive synchronization results were further applied to secure
communication. The numerical simulation results were presented to verify the effectiveness of the designed
nonlinear controller and its success in secure communication application.
KEYWORDS
Hyperhaos, control, synchronization, Secure Communication
1. INTRODUCTION
Chaotic systems are characterized by their complex dynamical behaviuor particularly their
extreme sensitivity to initial conditions and parameters variation which make their behaviour
long-term unpredictable. Therefore, the idea to control and synchronize two chaotic systems has
been considered as illogical and impossible [1]. The year 1990 marked a turning point in the
study of chaotic dynamics when the idea of chaos control and synchronization was first proposed
by Pecoral and Carrol [2], and it was shown that synchronization is not only possible but it has
wide potential application in secure communication, engineering, physical, chemical, biological
and financial systems, and information science [3-5].
Synchronization involves the linking of one system trajectories to the corresponding trajectories
of the other such that both systems remain in step with each other (mimics) through the
transmission of signal [6]. The coupling of these systems is usually called drive (master) and
response (slave) systems [7]. Its numerous applications in secure communication has been one of
the main motivating factor, where the basic idea is to mask the information bearing signal to be
transmitted with chaotic signal that exhibits broad band feature [8-10].
A wide variety of approaches have been developed and reported in the literature to achieve chaos
synchronization in the couple chaotic and hyperchaotic systems such as; time-delay feedback
method [11], active control method [12-13], impulsive method [14], linear state error feedback
method [15], adaptive control method [16-19]), backstepping technique [20-23] (see the
references there in). Most of these approaches are based on exactly knowing the system structure

2
and parameters. But in many practical situations, some or all of these systems’ parameters are
unknown. Also, these parameters can change from time to time. Therefore, the derivative of
adaptive controller for control and synchronization of chaotic system in the presence of unknown
parameters is an important issue [24], that made adaptive control method more advantageous in
practice over the other approaches mention above. Though the adaptive control method is easy to
manipulate, it still has the following limitation; the controller designed by this method usually
have high or fixed signals strength and too large. As we know, the actual transmission of the
signals is always associated with the signals’ amplification and reduction. Also, one always
expected to achieve the final synchronization by using the relatively smaller signals. Thus the
adaptive control can be improved by reducing the signal strength and the complexity in controller
to minimize the energy consumption.In view of this, the present work is focused on the
following: the dynamics and qualitative properties of hyperchaotic Lorenz-Stenflo system,
designing of adaptive control method for controlling hyperchaoticLorenz-Stenflo system with
fully unknown parameters to control to equilibrium point, design via adaptive nonlinear extended
scheme control feedback input that capable of synchronizing two identical hyperchaotic Lorenz-
Stenflo systems evolving from different initial conditions and its application to secure
communication. The control and synchronization based on this technique are simple and easy to
implement in practical applications in some real systems.
2. DESCRIPTION OF LORENZ-STENFLO SYSTEM
The mathematical interest in this work is the hyperchaotic Lorenz-Stenflo system described by
the first order differential equations of the form given in equation (1) below;
4
1
4
3
2
1
3
2
3
1
2
4
1
2
1
)
(
)
(
ax
x
x
dx
x
x
x
x
x
c
x
x
bx
x
x
a
x
−
−
=
−
=
−
−
=
+
−
=
&
&
&
&
(1)
Where 3
2
1 ,
, x
x
x and 4
x are the state variables of the system and c
b
a ,
, and d are the real
constant parameter of the system.
System (1) exhibits hyperchaotic behavior when 0
.
26
,
5
.
1
,
0
.
1 =
=
= c
b
a and 7
.
0
=
d and the
phase portraits of the attractor is given in Fig.1.
(a) (b)
-25
-20
-15
-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
x
2
x1
0
5
10
15
20
25
30
35
40
45
-8 -6 -4 -2 0 2 4 6 8
x
3
x1

3
(c)
Figure1. Phase portraits of Loren- Stenflo hyperchaotic system attractor with the parameter
values; 0
.
26
,
5
.
1
,
0
.
1 =
=
= c
b
a and 7
.
0
=
d
3. BASIC DYNAMICS OF LORENZ-STENFLO HYPERCHAOTIC SYSTEM
To derive the equilibrium point for Lorenz-Stenflo hyperchaotic system (1), when 0
≠
a , 0
≠
b ,
0
≠
c and 0
≠
d , we solve the sets of equations in system (1). By solving system (1), we let;
0
1 =
x
& , 0
2 =
x
& , 0
3 =
x
& and 0
4 =
x
& (condition for stability)
Hence;
0
)
( 4
1
2 =
+
− bx
x
x
a (2)
0
)
( 2
3
1 =
−
− x
x
c
x (3)
0
3
2
1 =
− dx
x
x (4)
0
4
1 =
−
− ax
x (5)
Solving equations; (5), (3), (4) and (2) respectively we get;
4
1 ax
x −
= , )
( 3
4
2 x
c
ax
x −
−
= ,
d
x
c
ax
x
)
(
)
( 3
2
4
3
−
= and
b
x
x
a
x
)
( 1
2
4
−
−
=
3.1 CONDITION FOR EQUILIBRIUM
For hyperchaotic Lorenz-Stenflo system (1) to be in equilibrium, the following condition must be
satisfy; .
0
4
3
2
1 =
=
=
= x
x
x
x But solving the sets of equations in system (1) {i.e., equations,
(2), (3), (4) and (5)}, 4
3
2
1 x
x
x
x ≠
≠
≠ , provided that 0
≠
≠
≠
≠ d
c
b
a , which is a
contradiction. Thus, there is no equilibrium point for hyperchaotic Lorenz-Stenflo system (1).
Hence, control to equilibrium point E0 (0,0,0,0) and synchronization of hyperchaotic Lorenz-
Stenflo system (1) in the presence of system’s disturbances and unknown parameters is quiet
interesting that needed to be address.
-8 -6 -4 -2 0 2 4 6 8
x1 -20
-15
-10
-5
0
5
10
15
20
25
x2
0
5
10
15
20
25
30
35
40
45
x3

4
4. ADAPTIVE CONTROL OF HYPERCHAOTIC LORENZ-STENFLO (LS)
SYSTEM
4.1 DESIGN OF THE ADAPTIVE CONTROLLER
In order to design the control function )
(t
ui , ( 4
,
3
,
2
,
1
=
i ) using adaptive control method, we
assume that the real parameters c
b
a ,
, and d in equation (1) are unknown.
We now consider the controlled system as follow;
)
(
)
(
)
(
)
(
)
(
)
(
4
4
1
4
3
3
2
1
3
2
2
3
1
2
1
4
1
2
1
t
u
ax
x
x
t
u
dx
x
x
x
t
u
x
x
c
x
x
t
u
bx
x
x
a
x
+
−
−
=
+
−
=
+
−
−
=
+
+
−
=
&
&
&
&
(6)
Where )
(t
ui , )
4
,
23
,
1
( =
i are the control function to determined.
According to the Lyapunov stability theory, we choose the following Lyapunov function;
)
~
~
~
~
(
2
1 2
2
2
2
2
4
2
3
2
2
2
1 d
c
b
a
x
x
x
x
V +
+
+
+
+
+
+
= (7)
Where d
d
d
c
c
c
b
b
b
a
a
a
~
~
,
~
,
~
,
~ −
=
−
=
−
=
−
= and d
c
b
a ,
,
, are the estimated values of
these unknown parameter respectively.
The time derivative of equation (7) is given in equation (8) below.
d
d
c
c
b
b
a
a
x
x
x
x
x
x
x
x
V
&
&
&
&
&
&
&
&
& ~
~
~
~
~
~
~
~
4
4
3
3
2
2
1
1 +
+
+
+
+
+
+
= (8)
In order to ensure that the controlled system (6) converges at the origin )
0
,
0
,
0
,
0
( asymptotically
at a desire time, the controllers )
(t
ui are choosing as follows:
4
4
1
4
3
3
2
1
3
3
1
2
1
4
1
2
1
)
(
)
(
)
(
)
(
)
(
)
(
x
ax
x
t
u
x
dx
x
x
t
u
x
c
x
t
u
x
bx
x
x
a
t
u
−
+
=
−
+
−
=
−
−
=
−
−
−
−
=
(9)
)
(
~
)
(
~
)
(
~
)
(
~
)]
(
[
)]
(
[
)]
(
)
(
[
)]
(
)
(
[
4
4
1
4
3
3
2
1
3
2
2
3
1
2
1
4
1
2
1
d
d
c
c
b
b
a
a
t
u
ax
x
x
t
u
dx
x
x
x
t
u
x
x
c
x
x
t
u
bx
x
x
a
x
V
&
&
&
&
&
−
+
−
+
−
+
−
+
+
−
−
+
+
−
+
+
−
−
+
+
+
−
=
)
(
~
)
(
~
)
(
~
)
(
~
)]
(
[
)]
(
[
)]
(
)
(
[
)]
(
)
(
[
2
3
2
4
1
2
4
2
1
2
1
4
4
1
4
3
3
2
1
3
2
2
3
1
2
1
4
1
2
1
x
d
d
x
c
c
x
x
b
b
x
x
x
x
a
a
t
u
ax
x
x
t
u
dx
x
x
x
t
u
x
x
c
x
x
t
u
bx
x
x
a
x
−
−
+
+
−
+
+
−
+
−
−
+
−
+
+
−
−
+
+
−
+
+
−
−
+
+
+
−
=
&
&
&
&
(10)

5
The following parameter update laws were chosen;
2
3
2
2
1
2
4
2
1
2
1
x
d
x
c
x
x
b
x
x
x
x
a
−
=
=
=
−
−
=
(11)
Substituting equation (11) into equation (10) yields;
0
2
4
2
3
2
2
2
1 <
−
−
−
−
= x
x
x
x
V
& (12)
According to the Lyapunov stability theory, the above condition ensures that the controlled
system (6) stabilizes at the origin )
0
,
0
,
0
,
0
( at a chosen time with the designed controllers in
equation (9) and the parameter update laws given by equation (11).
4.2 NUMERICAL SIMULATION RESULTS
Appling fourth-order Runge-Kutta algorithm with the initialcondition 8
.
0
)
0
(
1 =
x , 6
.
0
)
0
(
2 =
x ,
9
.
0
)
0
(
3 −
=
x , 8
.
0
)
0
(
4 −
=
x , a time sizes of 0.001 and fixing the parameter values as in figure
1.
The result obtained show that the state variables move hyperchaotically with time when the
controllers are deactivated and when the controllers are switched on at 50
=
t the state variables
are converges or stabilized at the origin )
0
,
0
,
0
,
0
( according to the Lyapunov stability theory. The
results are display in Figure 2.
(a)
(b)
-8
-6
-4
-2
0
2
4
6
8
0 10 20 30 40 50 60 70 80 90 100
x
1
time
-20
-15
-10
-5
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80 90 100
x
2
time

6
(c)
(d)
Figure2. Time responses of the state variables ( 4
3
2
1 ,
,
, x
x
x
x ) for hyperchaotic Loren-Stenflo
system when the controller is activated at 50
=
t to stabilize at the origin.
5. ADAPTIVE CONTROL FOR SYNCHRONIZATION OF HYPERCHAOTIC
LORENZ-STENFLO (LS) SYSTEM
5.1 DESIGN OF ADAPTIVE CONTROLLER FOR SYNCHRONIZATION OF HYPERCHAOTIC
LORENZ-STENFLO SYSTEM
In this section, we synchronized two identical hyperchaotic Loren-Stenflo systems evolving from
two different initial conditions.
In order to obtain a partial strict- feedback, system (1) is rearranged as follows;
3
2
1
3
2
3
1
2
4
1
2
1
4
1
4
)
(
)
(
dx
x
x
x
x
x
c
x
x
bx
x
x
a
x
ax
x
x
−
=
−
−
=
+
−
=
−
−
=
&
&
&
&
(13)
We let 3
2
2
1
1
4 ,
, x
x
x
x
x
x =
=
= and 4
3 x
x = , then equation (13) takes the following forms;
4
3
2
4
3
4
2
3
1
2
3
2
1
2
1
)
(
)
(
dx
x
x
x
x
x
c
x
x
bx
x
x
a
x
ax
x
x
−
=
−
−
=
+
−
=
−
−
=
&
&
&
&
(14)
-5
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70 80 90 100
x
3
time
-4
-3
-2
-1
0
1
2
3
4
0 10 20 30 40 50 60 70 80 90 100
x
4
time

7
Equation (14) is called the drive (master) and equation (15) is the response (slave).
)
(
)
(
)
(
)
(
)
(
)
(
4
4
3
2
4
3
3
4
2
3
2
1
2
3
2
1
1
2
1
t
u
dy
y
y
y
t
u
y
y
c
y
y
t
u
by
y
y
a
y
t
u
ay
y
y
+
−
=
+
−
−
=
+
+
−
=
+
−
−
=
&
&
&
&
(15)
Where )
(t
ui )
4
,
3
,
2
,
1
( =
i are the control functions to be designed with the assumption that the
parameters c
b
a ,
, and d are unknown constant.
Our objective is to design an appropriate adaptive control inputs vector )
(t
ui to stabilize the
errors vector of the state variables at the origin )
0
,
0
,
0
,
0
( and also to make the state variables of
the response (slave) system (15) to track the ones of the drive (master) system (14) with the
parameter update laws at any chosen time.
We define the synchronization errors vector between the drive (master) system (14) and the
response (slave) system (15) as follows;
i
i
i x
y
e −
= or i
i
i x
e
y +
= (16)
Where 4
,
3
,
2
,
1
=
i
The subtraction of equation (14) from equation (15) is the result of the following errors vector:
)
(
)
(
)
(
)
(
)
(
)
(
4
4
3
2
2
3
3
2
4
3
3
4
2
2
4
4
2
2
3
2
1
2
3
2
1
1
2
1
t
u
de
e
e
e
x
e
x
e
t
u
e
e
e
e
x
e
x
ce
e
t
u
be
e
e
a
e
t
u
ae
e
e
+
−
+
+
=
+
−
+
+
−
=
+
+
−
=
+
−
−
=
&
&
&
&
(17)
We consider the Lyapunov function as follows;
)
~
~
~
~
(
2
1 2
2
2
2
2
4
2
3
2
2
2
1 d
c
b
a
e
e
e
e
V +
+
+
+
+
+
+
= (18)
Where a
a
a −
=
~ , b
b
b −
=
~
, c
c
c −
=
~ and d
d
d −
=
~
are the estimate values these unknown
parameters c
b
a ,
, and d respectively. The time derivative of equation (18) along the trajectories
of the errors vector and estimate values of the constant parameter gives equation (19) below.
d
d
c
c
b
b
a
a
e
e
e
e
e
e
e
e
V
&
&
&
&
&
&
&
&
& ~
~
~
~
~
~
~
~
4
4
3
3
2
2
1
1 +
+
+
+
+
+
+
= (19)
We substituted equation (17) in equation (19) to give equation (20) below.
)
(
~
)
(
~
)
(
~
)
(
~
)]
(
[
)]
(
)
(
[
)]
(
)
(
[
)]
(
[
4
4
3
2
2
3
3
2
4
3
3
4
2
2
4
4
2
2
3
2
1
2
3
2
1
1
2
1
d
d
c
c
b
b
a
a
t
u
de
e
e
e
x
e
x
e
t
u
e
e
e
e
x
e
x
ce
e
t
u
be
e
e
a
e
t
u
ae
e
e
V
&
&
&
&
&
−
+
−
+
−
+
−
+
+
−
+
+
+
+
−
+
+
+
−
+
+
+
−
+
+
−
−
=

8
)
(
~
)
(
~
)
(
~
))
(
(
~
)]
(
[
)]
(
)
(
[
)]
(
)
(
[
)]
(
[
2
4
3
2
2
1
2
3
2
2
1
4
4
3
2
2
3
3
2
4
3
3
4
2
2
4
4
2
2
3
2
1
2
3
2
1
1
2
1
e
d
d
e
e
c
c
e
e
b
b
e
e
e
e
a
a
t
u
de
e
e
e
x
e
x
e
t
u
e
e
e
e
x
e
x
ce
e
t
u
be
e
e
a
e
t
u
ae
e
e
−
−
+
+
−
+
+
−
+
−
+
−
−
+
+
−
+
+
+
+
−
+
+
−
+
+
+
−
+
+
−
−
=
&
&
&
&
(20)
c
b
a ,
, and d takes the place of c
b
a ,
, and d respectively.
To ensure that the errors vector converges to equilibrium point )
0
,
0
,
0
,
0
(
0
E asymptotically, the
condition 0
<
V
& must be satisfied. From equation (17), we selected the controllers as follows:
4
3
2
2
3
3
2
4
4
4
2
2
4
4
2
2
3
2
1
2
3
2
1
1
2
1
)
(
)
(
)
(
)
(
)
(
)
(
e
e
e
e
x
e
x
de
t
u
e
e
e
x
e
x
ce
t
u
e
be
e
e
a
t
u
e
ae
e
t
u
−
+
+
−
=
+
+
+
−
=
−
−
−
−
=
−
+
=
(21)
And the parameter update estimation laws as;
d
e
d
c
e
e
c
b
e
e
b
a
e
e
e
e
a
−
−
=
−
=
−
=
−
−
+
−
=
2
4
3
2
2
1
2
3
2
2
1 )
(
&
&
&
&
(22)
Substituting equation (22) into equation (20) yield;
0
~
~
~
~ 2
2
2
2
2
4
2
3
2
2
2
1 <
−
−
−
−
−
−
−
−
= d
c
b
a
e
e
e
e
V
& (23)
In order to minimize energy consumption and controller complexity, we introduce the weight i
ε
to the designed controller in equation (21) as follows;
4
4
3
2
2
3
3
2
4
4
3
4
2
2
4
4
2
2
3
2
2
1
2
3
2
1
1
1
2
1
]
)
(
[
)
(
]
[
)
(
]
)
(
[
)
(
]
[
)
(
ε
ε
ε
ε
e
e
e
e
x
e
x
de
t
u
e
e
e
x
e
x
ce
t
u
e
be
e
e
a
t
u
e
ae
e
t
u
−
+
+
−
=
+
+
+
−
=
−
−
−
−
=
−
+
=
(24)
Where i
ε ( 4
,
3
,
2
,
1
=
i ) are the weight added to the designed controllers. For simplicity,
ε
ε
ε
ε
ε =
=
=
= 4
3
2
1 was chosen.
The errors vector converges to the origin )
0
,
0
,
0
,
0
( asymptotically in line with the Lyapunov
stability theory. Also the drive (master) system (14) is synchronized with the response (slave)
system (15) with the controllers (21) and (24)and the parameter update law (22).

9
5.2 NUMERICAL SIMULATION RESULTS
To verify the feasibility and effectiveness of the designed controllers (21) and (24) and the
parameter update laws (22), we simulate the dynamics of the drive (master) system (14) and the
response (slave) system (15) by employing fourth-order Runge-Kutta integration method to solve
the systems of differential equations given by equations (14) and (15) respectively with the
following initial conditions )
8
.
0
,
6
.
0
,
6
.
0
,
5
.
0
(
)
,
,
,
( 4
3
2
1 =
x
x
x
x ,
)
0
.
9
,
2
.
0
,
0
.
1
,
0
.
9
(
)
,
,
,
( 4
3
2
1 −
=
y
y
y
y , a time size of 0.001 and fixed the real parameter values of
the system as in figure 1. The results obtained shows that the error vector variable moves
hyperchaotically with time when the controllers are switched off and when the controllers are
activated at 50
=
t as shown in figures 3 and 4, the error vector variables converges to the origin
)
0
,
0
,
0
,
0
( and thereby guaranteeing the synchronization of systems (14) and (15). This is
confirmed by the synchronization norme given by;
2
4
2
3
2
2
2
1 e
e
e
e
e +
+
+
= (25)
We observed from the numerical simulation results that the control strength and its complexity
are reduced by 30% ( 7
.
0
=
ε ) when the proposed extended adaptive control approach is applied
(Figure 4), i.e. only about 70% of the strength of the usual adaptive control is needed to achieve
complete synchronization. Thus this new method proposed in this work is an evident that the
control strength weightε is important by produces economics controllerswith low energy
consumption which may be useful for practical application.
However, the initial values of the parameter update laws (22) are chosen as follows; 0
.
4
)
0
(
1 =
a ,
0
.
6
)
0
(
1 =
b , 0
.
18
)
0
(
1 =
c and 0
.
3
1 =
d . The parameter estimation values a , b , c and d
converges to 0
.
1
=
a , 5
.
1
=
b , 0
.
26
=
c and 7
.
0
=
d respectively (see Fig. 5) as ∞
→
t .
(a)
(b)
-10
-8
-6
-4
-2
0
2
4
6
0 10 20 30 40 50 60 70 80 90 100
e
1
time
-20
-15
-10
-5
0
5
10
15
0 10 20 30 40 50 60 70 80 90 100
e
2
time

10
(c)
(d)
(e)
Figure 3: Error dynamics between two identical hyperchaotic Loren-Stenflo systems when the
controller is activated at 50
=
t .
(a)
-40
-30
-20
-10
0
10
20
30
0 10 20 30 40 50 60 70 80 90 100
e
3
time
-30
-20
-10
0
10
20
30
0 10 20 30 40 50 60 70 80 90 100
e
4
time
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70 80 90 100
e
e
time
-10
-8
-6
-4
-2
0
2
4
6
0 10 20 30 40 50 60 70 80 90 100
e
1
time

11
(b)
(c)
(d)
(e)
Fig. 4: Error dynamics between two identical hyperchaotic Loren-Stenflo systems when the
controller is activated at 50
=
t with 7
.
0
=
ε .
-20
-15
-10
-5
0
5
10
15
0 10 20 30 40 50 60 70 80 90 100
e
2
time
-40
-30
-20
-10
0
10
20
30
0 10 20 30 40 50 60 70 80 90 100
e
3
time
-30
-20
-10
0
10
20
30
0 10 20 30 40 50 60 70 80 90 100
e
4
time
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70 80 90 100
e
e
time

12
(a)
(b)
(c)
(d)
Figure 5: Time responses of the state variables of drive (master) and response (slave) systems
when the controller is switch on at 50
=
t
Note: Red is the drive (master) system and Green is the response (slave) system.
-10
-8
-6
-4
-2
0
2
4
0 10 20 30 40 50 60 70 80 90 100
x
1
,y
1
time
-10
-8
-6
-4
-2
0
2
4
6
8
0 10 20 30 40 50 60 70 80 90 100
x
2
,y
2
time
-20
-15
-10
-5
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80 90 100
x
3
,y
3
time
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70 80 90 100
x
4
,y
4
time

13
Figure 6: The time responses of the parameter estimation errors.
6. APPLICATION TO SECURE COMMUNICATION
One of the most important applications of chaos synchronization in engineering is in secure
communication, where the basic idea is to mask or modulate (encrypt) the information bearing
signal to be transmitted with chaotic signal that exhibits broad band feature (see references [8-
10]).The information signal is added i.e. masked to the output of a chaotic oscillator to produce
unpredictable signal which is transmitted from the transmitter to the receiver. At the receiver the
pseudo-random is generated through the inverse operation, original message is retrieved. In order for
this scheme to properly work, the receiver must synchronize robustly enough so as to admit the small
perturbation in the drive signal due to the addition of the message [25-26]. The power of information signal
must be lower than that of chaotic signal to effectively bury the information signal. The signal from the
master controls the slave system so as to synchronize it with the master and to carry the information
signal, just like any other communication scheme. The purpose of chaos secure communication is
to hide message during transmission.
In this chaotic masking scheme, encryption is achieved by combing the information signal with
the chaotic carrier signal using mixing algorithm which is simply a function of information and
chaotic carrier signals [27]. Here we demonstrate our secure communication scheme using the
additive encryption masking scheme. The information signal is chosen to be a periodic function
t
t
i 3
.
0
sin
5
)
( = , with this choice the chaotic carrier 1
x remain chaotic. The encrypted
information is given by the masking algorithm 1
)
(
)
( x
t
i
t
ie +
= . Consequently, the decrypted
information )
(t
id is given by the inverse function 1
)
(
)
( y
t
i
t
i e
d −
= . The chaotic signal 1
x of the
master is transmitted to the slave via a coupling channel for synchronization between the master
and the slave, the information signal t
t
i 3
.
0
sin
5
)
( = is masked in the encrypted signal )
(t
ie and
transmitted to the receiver. The decrypted information )
(t
id is extracted by inverse function. The
numerical simulation results of the communication scheme are displayed in Figure 6.
0
5
10
15
20
25
30
0 10 20 30 40 50 60 70 80 90 100
a,b,c,d
time

14
(a)
(b)
(c)
-6
-4
-2
0
2
4
6
0 10 20 30 40 50 60 70 80 90 100
i(t)
time
-8
-6
-4
-2
0
2
4
6
8
0 10 20 30 40 50 60 70 80 90 100
ie(t)
time
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80 90 100
id(t)
time

15
(d)
Figure7: (a) information signal )
(t
i ; (b) encrypted signal )
(t
ie ; (c) decrypted signal (t=50), )
(t
id
(d) decrypted error )
(
)
( t
i
t
i
er e
−
= .
7. CONCLUSION
In this work the adaptive control method has been applied to control and synchronize Lorenz-
stenflo system with fully unknown parameters in the presence of system’s disturbancesand further
extended the synchronization results to secure communication. The designed adaptive nonlinear
controllers control Lorenz-stenflo system to equilibrium point )
0
,
0
,
0
,
0
(
0
E according to the
Lyapunov stability theory and globally synchronized two identical Lorenz-stenflo systems
evolving from different initial conditions. The presence of the weight (ε i.e., the control signal)
introduced to the controllers play a major role in the synchronization by overcomes the
limitations of large signals and controller complexity in the normal adaptive control. This
approach suggested that only about 70% ( 7
.
0
=
ε ) of the strength of the usual adaptive control is
enough to achieve complete synchronization. We also observed from the numerical simulation
results that the speed of synchronization depend on the value ofε , for large value of ε , the speed
for achieve synchronization is high and to minimize the expenses, small value of the control
strength ε should be adopted. In addition, one can easily change the value of ε according to the
actual requirement. Hence, the introduction of weight to the usual controller as a signal control is
of economic benefit that is suitable for practical implementation. Numerical simulations were
given to demonstrate the effectiveness and efficiency of the proposed scheme. Thus, practical
implementation of the proposed scheme shall be very useful and the future work shall focus on
addressing this problem.
REFERENCES
[1] Liu S.Y, X and Zhu, S.J, Study on the chaos anti control technology in nonlinear vibration system.
Journal of sound and vibration 310 (4-5): 855-864 (2008).
[2] L.M. Pecora, T.L. Corral, Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821-824 (1990).
[3] C. Kyrtsou and W.C. Labys,Evidence for chaotic dependence between US inflation and commodity
prices. Journal of Macroeconomics, vol. 28 (1), PP 256-266 (2006).
[4] C. Kryrtsou and W.C. Labys,Detecting positive feedback in multivariate time series and US inflation,
Physica A: Statistical Mechanics and its applications, vol.377 (1), PP 227-229 (2007).
[5] S.H Kellert,In the wake of Chaos: Unpredictable order in Dynamical systems, University of Chicago
Press, (1993).
-10
-8
-6
-4
-2
0
2
4
6
0 10 20 30 40 50 60 70 80 90 100
e
r
time

16
[6] Nastaran Vasegh, Faarhad Khella, Chaos synchronization of chemical model. Journal of Applied
science and technology 1(5), (2011).
[7] Njah, A.N., Tracking control and synchronization of the new hyperchaotic Liu system via
backstepping techniques. Nonlinear Dyn. 61, PP 1-9 (2009).
[8] Kocarev, L., Chaos –based cryptography: A brief overview, IEEE Circuits and systems magazine,
vol. 1, pp. 6-21 (2001).
[9] Hasler, M., Synchronization of chaotic systems and transmission of information. International Journal
of Bifurcation and chaos, vol.8, PP 647-659 (1998).
[10] Silva, C.P., A. M. Young, Introduction to chaos-based communications and signal processing, “Pro.
IEEE Aerospace Conference”, PP.279-299 (2000).
[11] Dianchen L, Aicheng W, Xiandon T., Control and synchronization of a new hyperchaotic system with
unknown parameters. International journal of nonlinear science 6(3), PP 224-229 (2008).
[12] Vincent, U. E., Chaos synchronization using active control and backstepping control: A Comparative
Analysis. Non-linear Analysis: Theory and applications, 13, PP 256-261 (2008).
[13] Bai, E. W., Longngren, K. E., Synchronization of two Lorenz systems using active control. Chaos
Solitons and fractals 8(1), PP 51-58 (1997).
[14] Li, C., Liao, X., Wong, K-W, Lag synchronization of hyperchaos with applications to secure
communications, Chaos Solitons Fractals 183-193 (2005).
[15] O.I. Olusola, U.E. Vincent, A.N. Njah, Synchronization, multistability and basin crisis in coupled
pendula. Journal of sound and vibration, 329 PP 443-456 (2010).
[16] M.T Yassen, Adaptive control and synchronization of a modified chua’s circuit system. Applied
mathematics and computation 135, PP 113-128 (2003).
[17] M. M. El-Dessoky, M.T. Yassen, Adaptive feedback control for chaos control and synchronization
for new chaotic Dynamical system. Mathematical problems in Engineering. Vol. 2012, Article ID
347210, 12pages (2012).
[18] Xingyuan Wang, Yaqin Wang, Adaptive control for synchronization of a four-dimensional chaotic
system via a single variable. Nonlinear Dyn. 65 PP 311-316 (2011).
[19] O.S. Onma, O.I. Olusola, A.N. Njah, Control and Synchronization of chaotic and hyperchaotic
Lorenz system via extended adaptive control method. Far East Journal of Dynamical Systems, 28 (1)
PP 1-32 (2016).
[20] S. Onma Okpabi, Adelakun A.Oke, J.O. Akinlami, S.T Opeifa, Backstepping Control and
Synchronization of hyperchaotic Lorenz-Stenflo system with application to secure communication.
Far East Journal of Dynamical Systems, 29 (1) PP 1-23 (2017).
[21] O. S. Onma, O.I. Olusola, A.N. Njah, Control and synchronization of chaotic and hyperchaotic
Lorenz systems via extended backstepping techniques. Journal of nonlinear dynamics, vol. 2014,
Article ID861727, 15 Pages (2014).
[22] Olasunkanmi I., Olusola, Uchechukwu E. Vincent, Abdulahi N. Njah, Emad Ali, Control and
synchronization of chaos in biological systems via backstepping design. International journal of
nonlinear science. 11(1)PP121-128 (2011).
[23] S. Mascolo, ” Backstepping design for controlling Lorenz chaos”, in proceedings of the 36th IEEE
Conference on Decision and Control, , San Diego, Calif, USA, PP 1500-1501 (1997).

17
[24] Zhigong, S., Wenzhi, Z., Gangquan, S., Adaptive synchronization design for uncertain systems in the
presence of unknown parameters: Nonlinear Dyn. 72 PP 729-749 (2013).
[25] Adelakun A.Oke, Adelakun A. Adisa, Onma O. Sunday, Bidirectional synchronization of two
identical Jerk oscillators with memristor. International journal of Electrical, Electronics and
Telecommunication Engineering, 45 (2) PP 2051-3240 (2014).
[26] Aceng Sambas, Mada Sanjaya WS, Mustafa Mamat, Bidirectional coupling scheme of chaotic
systems and its application in secure communication system. Journal of Engineering Science and
Technology Review, 8 (2) PP 89-95 (2015).
[27] M. Liu, J. Feng, C.K. Tse, A new hyperchaotic system and its circuit implementation. International
Journal of Bifurcation and Chaos, 20 (4) PP 1201-1208 (2010).
[28] Chittaranjan Pradhan, Ajay Kumar Bisoi’ Chaotic Vibrations AES Algorithm international journal of
chaos , control modeling and simulation 2 ( 2013)
Authors
Onma O.S Has B.Sc., (Physics), M.Sc in Theoretical Physics and a PhD Student of
Theoretical/condensed matter Physics, department of physics, Federal university of
Agriculture Abeokuta. His primary research interest is in nonlinear dynamics with
application to chaos control, synchronization and secure communication.
J.O AKINLAMI Has B.Sc in Physics M.Sc and PhD in condensed matter physics.
His primary research area are in theoretical and computational condensed matter
physics. A senior lecturer in department of physics, Federal university of Agriculture,
Abeokuta Nigeria

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