Chaos synchronization in a 6-D hyperchaotic system with self-excited attractor

TELKOMNIKA Telecommunication, Computing, Electronics and Control
Vol. 18, No. 3, June 2020, pp. 1483~1490
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v18i3.13672  1483
Journal homepage: http://journal.uad.ac.id/index.php/TELKOMNIKA
Chaos synchronization in a 6-D hyperchaotic system
with self-excited attractor
Ahmed S. Al-Obeidi, Saad Fawzi Al-Azzawi
Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul, Iraq
Article Info ABSTRACT
Article history:
Received Jul 22, 2019
Revised Jan 29, 2020
Accepted Feb 23, 2020
This paper presented stability application for chaos synchronization using a
6-D hyperchaotic system of different controllers and two tools: Lyapunov
stability theory and Linearization methods. Synchronization methods based
on nonlinear control strategy is used. The selecting controller's methods have
been modified by applying complete synchronization. The Linearization
methods can achieve convergence according to the of complete synchronization.
Numerical simulations are carried out by using MATLAB to validate the
effectiveness of the analytical technique.Keywords:
6-D hyperchaotic system
Chaos synchronization
Lyapunov stability theory
Nonlinear control strategy
Self-excited attractor This is an open access article under the CC BY-SA license.
Corresponding Author:
Saad Fawzi AL-Azzawi,
Department of Mathematics, College of Computer Science and Mathematics,
University of Mosul, Mosul, Iraq.
Email: saad_fawzi78@yahoo.com, saad_alazawi@uomosul.edu.iq
1. INTRODUCTION
In recent years, the dynamical system has attracted significant attention due to its widespread
applications in engineering and different scientific research as lasers, nonlinear circuits biological [1, 2],
engineering [3, 4] and secure communications [5, 6]. Lorenz system is the first physical and mathematical
model of a chaotic system contains real variables only which discovered in 1963 and open the way to find
another chaotic system such as Chen system, Lu system, Liu system and Pan system [7-9]. Each system has a
3-D of differential equations and just one positive Lyapunov exponent [10]. One important application in the
field of engineering is secure communication i.e., the messages which are made by such simple chaotic
systems are not always safe [6, 11, 12]. It is suggested that this problem can be overcome by using
higher-dimensional hyperchaotic systems, which have increased randomness and higher unpredictability.
In 1979, Rössler discovers the first 4-D hyperchaotic system including real variables with two
positive Lyapunov exponents and followed to discover another 4-D, as well as 5-D hyperchaotic with three
positive Lyapunov exponents [10, 13-15] and some other systems, have been revealed. The dynamical
systems with higher dimensions are effective and interesting compared with the low dimensions [16-18].
In 2015, Yang et al., proposes a 6-D hyperchaotic system including real variables and has four positive
Lyapunov exponents [19].
These days, the synchronization of the mentioned systems witnessed large attention by researchers
because of its important applications in the is secure communication [20-22]. Many of the papers that relate
to this topic are increasing, and numerous research devoted to investigating CS of high-dimensional
hyperchaotic systems based on traditional Lyapunov stability theory [23-25]. Lyapunov stability theory is

 ISSN: 1693-6930
TELKOMNIKA Telecommun Comput El Control, Vol. 18, No. 3, June 2020: 1483 - 1490
1484
extensively utilized in the phenomena of synchronization because the Lyapunov function can deliver
accurately and speed data of the system convergence. However, Lyapunov function in some time is incapable
of meeting the convergence requirements of error dynamics system owing to suffers from its drawbacks of
modified the function itself. To achieve synchronization of good performance, the Linearization tool is
preferred. So the Linearization and nonlinear control strategy integration can achieve higher performance.
The contributions of this research can be summarized in the following points.
a. Chaos synchronization between identical 6-D hyperchaotic systems is studied and used to find the error
dynamics between them and its secure communication is then presented theoretically.
b. Designs of three different controllers of complete synchronization are done by a nonlinear control
strategy based on the Lyapunov stability theory, Linearization method.
c. Compare between the Lyapunov and Linearization method.
2. SYSTEM DESCRIPTION
The Lorenz system was the first 3-D chaotic system to be modeled and one of the most widely
studied. The original system was modified into a 4-D and 5-D hyperchaotic systems by introducing a linear
feedback controller. In 2015, Yang constructed a 6-D hyperchaotic system which contains four positive
Lyapunov Exponents 𝐿𝐸1 = 1.0034, 𝐿𝐸2 = 0.57515, 𝐿𝐸3 = 0.32785, 𝐿𝐸4 = 0.020937, and two negative
Lyapunov Exponents 𝐿𝐸5 = −0.12087, 𝐿𝐸6 = −12.4713. The 6-D system which is described by
the following mathematical form [19]:
{
𝑥̇1 = 𝑎( 𝑥2 − 𝑥1) + 𝑥4
𝑥̇2 = 𝑐𝑥1 − 𝑥2 − 𝑥1 𝑥3 + 𝑥5
𝑥̇3 = −𝑏𝑥3 + 𝑥1 𝑥2
𝑥̇4 = 𝑑𝑥4 − 𝑥1 𝑥3
𝑥̇5 = −𝑘𝑥2
𝑥̇6 = ℎ𝑥6 + 𝑟𝑥2
(1)
where 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 are real state variables and 𝑎, 𝑏, 𝑐, 𝑑, 𝑘, ℎ, 𝑟 are all positive real parameters which
equals (10 ,8/3 ,28 , 2, 8.4, 1, 1) respectively. This system is rich in dynamic properties. Figure 1 (a) shows
the 3-D attractor of the system (1), while Figure 1 (b) shows the 2-D attractor of the same system.
(a) (b)
Figure 1. The attractor of the system (1), (a) In the 3-D(𝑥1, 𝑥3, 𝑥6) space, (b) In the 2-D (𝑥1 , 𝑥3) plane
3. CHAOS SYNCHRONIZATION BETWEEN TWO IDENTICAL LORENZ SYSTEM
In this section, two systems are needed, the first system is called the drive system which represents
the picture or message information will be sent while the second system is called response system represents
the noise that followed this information to ensure that they are not penetrated. Assume that the system (1) is
the drive system and can be written as

TELKOMNIKA Telecommun Comput El Control 
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractor (Ahmed S. Al-Obeidi)
1485
[
𝑥̇1
𝑥̇2
𝑥̇3
𝑥̇4
𝑥̇5
𝑥̇6]
=
[
−𝑎
𝑐
0
0
0
0
𝑎
−1
0
0
−𝑘
𝑟
0
0
−𝑏
0
0
0
1
0
0
𝑑
0
0
0
1
0
0
0
0
0
0
0
0
0
ℎ]⏟
𝐴
[
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑥6]
+
[
0 0 0
1 0 0
0
0
0
0
1
0
0
0
0
1
0
0]⏟
𝐵
[
−𝑥1 𝑥3
𝑥1 𝑥2
−𝑥1 𝑥3
]
⏟
𝐶
(2)
𝐴 and the product 𝐵. 𝐶 represents parameters matrix and nonlinear part of the system (1), respectively.
While the response system is as follows:
[
𝑦̇1
𝑦̇2
𝑦̇3
𝑦̇4
𝑦̇5
𝑦̇6]
= 𝐴1
[
𝑦1
𝑦2
𝑦3
𝑦4
𝑦5
𝑦6]
+
(
𝐵1 [
−𝑦1 𝑦3
𝑦1 𝑦2
−𝑦1 𝑦3
]
⏟
𝐶1
+
[
𝑢1
𝑢2
𝑢3
𝑢4
𝑢5
𝑢6] )
(3)
and let 𝑈 = [𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5, 𝑢6] 𝑇
is the nonlinear controller to be designed. The synchronization error
dynamics between the 6-D hyperchaotic system (2) and system (3) is defined as 𝑒𝑖 = 𝑦𝑖 − 𝑥𝑖 , 𝑖 = 1,2, … ,6
and satisfied that, lim
𝑡→∞
𝑒𝑖 = 0. The error dynamics is calculated as the following:
{
𝑒̇1 = 𝑎( 𝑒2 − 𝑒1) + 𝑒4 + 𝑢1
𝑒̇2 = c𝑒1 − 𝑒2 − 𝑒1 𝑒3 − 𝑥3 𝑒1 − 𝑥1 𝑒3 + 𝑒5 + 𝑢2
𝑒̇3 = −b𝑒3 + 𝑒1 𝑒2 + 𝑥2 𝑒1 + 𝑥1 𝑒2 + 𝑢3
𝑒̇4 = d𝑒4 − 𝑒1 𝑒3 − 𝑥3 𝑒1 − 𝑥1 𝑒3 + 𝑢4
𝑒̇5 = −𝑘𝑒2 + 𝑢5
𝑒̇6 = ℎ𝑒6 + 𝑟𝑒2 + 𝑢6
(4)
If the matrices 𝐴1 and 𝐵1 as
𝐴1 = 𝐴 and 𝐵1 = 𝐵, then refer for identical synchronization.
𝐴1 ≠ 𝐴 or 𝐵1 ≠ 𝐵 , then refer for non-identical synchronization.
Based on Linearization method, The system (4) is unstable and the characteristic equation and eigenvalues
are respectively as
λ6
+
32
3
λ5
−
4069
15
λ4
+
1658
15
λ3
+
24004
15
λ2
−
9496
5
λ − 448 = 0
{
λ1 = 2
λ2 = 1
λ3 = −8/3
λ4 = 11.3659 − 8.10−9
𝑖
λ5 = −22.6916 − 3.92820323010−9
𝑖
λ6 = 0.3257 + 9.92820323010−9
𝑖
Now, different controllers are designed based on Lyapunov and Linearization methods and we
compare them.
Theorem 1. If the control 𝑈 of system (4) is design as the following:
{
𝑢1 = 𝑒4( 𝑥3 − 1) − 𝑒2(a + 𝑐 − 𝑥3)
𝑢2 = −𝑟𝑒6
𝑢3 = −𝑥2 𝑒1
𝑢4 = 𝑒3( 𝑒1 + 𝑥1) − 3𝑑𝑒4
𝑢5 = −𝑒2(1 − 𝑘) − 𝑒5
𝑢6 = −2ℎ𝑒6
(5)
Then the system (3) can be followed by the system (2) by two methods.
Proof. Substitute above control in the error dynamics system (4) we have (6).

 ISSN: 1693-6930
1486
{
𝑒̇1 = −𝑎𝑒1 + 𝑥3 𝑒4 − 𝑐𝑒2 + 𝑥3 𝑒2
𝑒̇2 = c𝑒1 − 𝑒2 − 𝑒1 𝑒3 − 𝑥3 𝑒1 − 𝑥1 𝑒3 + 𝑒5 − 𝑟𝑒6
𝑒̇3 = −b𝑒3 + 𝑒1 𝑒2 + 𝑥1 𝑒2
𝑒̇4 = −2d𝑒4 − 𝑥3 𝑒1
𝑒̇5 = −𝑒2 − 𝑒5
𝑒̇6 = 𝑟𝑒2 − ℎ𝑒6
(6)
In the first method (Linearization method), the characteristic equation and eigenvalues as
λ6
+
32
3
λ5
+
2488
3
λ4
+
20696
3
λ3
+
59225
3
λ2
+
66172
3
λ +
25184
3
= 0
{
λ1 = −4
λ2 = −1
λ3 = − 1
λ4 = −8/3
λ5 = − 1 + √786𝑖
λ6 = − 1 − √786𝑖
All real parts of eigenvalues are negative, the linearization method is realized the chaos
synchronization between system (2) and system (3). If the Lyapunov function is constructed as (7).
𝑉( 𝑒𝑖) =
1
2
∑ 𝑒𝑖
26
𝑖=1 = 𝑒𝑖
𝑇
𝑃𝑒𝑖 , 𝑃 = 𝑑𝑎𝑖𝑔(0.5, 0.5, 0.5, 0.5, 0.5, 0.5) (7)
The derivative of the above function 𝑉( 𝑒𝑖) is
𝑉̇ ( 𝑒𝑖) = 𝑒1 𝑒̇1 + 𝑒2 𝑒̇2 + 𝑒3 𝑒̇3 + 𝑒4 𝑒̇4 + 𝑒5 𝑒̇5 + 𝑒6 𝑒̇6
𝑉̇ ( 𝑒𝑖) = 𝑒1(− 𝑎𝑒1 + 𝑥3 𝑒4 − 𝑐𝑒2 + 𝑥3 𝑒2) + 𝑒2( 𝑐𝑒1 − 𝑒2 − 𝑒1 𝑒3 − 𝑥3 𝑒1 − 𝑥1 𝑒3 + 𝑒5 − 𝑟𝑒6) +
𝑒3(−b𝑒3 + 𝑒1 𝑒2 + 𝑥1 𝑒2) + 𝑒4( −2d𝑒4 − 𝑥3 𝑒1) + 𝑒5(−𝑒2 − 𝑒5) + 𝑒6(𝑟𝑒2 − ℎ𝑒6)
𝑉̇ ( 𝑒𝑖) = −𝑎𝑒1
2
− 𝑒2
2
− 𝑏𝑒3
2
− 2𝑑𝑒4
2
− 𝑒5
2
− ℎ𝑒6
2
= −𝑒𝑖
𝑇
𝑄 𝑒𝑖 (8)
where 𝑄 = 𝑑𝑖𝑎𝑔(𝑎, 1, 𝑏, 2𝑑, 1, ℎ) , so 𝑄 > 0. Consequently, 𝑉̇ ( 𝑒𝑖) is negative definite on 𝑅6
. The nonlinear
controller is suitable and the complete synchronization is achieved. Now, we will take the initial values as
(1,0,2,4,1, −1) and (−8, −7, −15,12,20,1) to illustrate the complete synchronization that happened between
(2) and (3) numerically. Figure 2 shows verify these results numerically.
Figure 2. Complete synchronization between systems (2) and (3) with control (5)

1487
Theorem 2. If the nonlinear control 𝑈 of error dynamical system (4) is designed (9).
{
𝑢1 = −𝑐𝑒2 − 𝑥2 𝑒3 + 𝑥3( 𝑒4 + 𝑒2)
𝑢2 = −𝑎𝑒1 − 𝑟𝑒6
𝑢3 = 𝑥1 𝑒4
𝑢4 = 𝑒1( 𝑒3 − 𝑑) − 2𝑑𝑒4
𝑢5 = −𝑒5
𝑢6 = −2ℎ𝑒6
(9)
Then the system (3) can be followed by the system (2) by two methods.
Proof. From the above control (9) with the error system (4), we get (10).
{
𝑒̇1 = 𝑎𝑒2 − 𝑎𝑒1 + 𝑒4−𝑐𝑒2 − 𝑥2 𝑒3 + 𝑥3 𝑒4 + 𝑥3 𝑒2
𝑒̇2 = c𝑒1 − 𝑒2 − 𝑒1 𝑒3 − 𝑥3 𝑒1 − 𝑥1 𝑒3 + 𝑒5 − 𝑎𝑒1 − 𝑟𝑒6
𝑒̇3 = −𝑏𝑒3 + 𝑒1 𝑒2 + 𝑥2 𝑒1 + 𝑥1 𝑒2 + 𝑥1 𝑒4
𝑒̇4 = −𝑑𝑒4 − 𝑥3 𝑒1 − 𝑥1 𝑒3 − 𝑑𝑒1
𝑒̇5 = −𝑘𝑒2 − 𝑒5
𝑒̇6 = 𝑟𝑒2 − ℎ𝑒6
(10)
Based on the first method (Linearization method), the characteristic equation and eigenvalues as:
λ6
+
53
3
λ5
+
2172
5
λ4
+
38594
15
λ3
+
91112
15
λ2
+
93856
15
λ +
35072
15
= 0
{
λ1 = −1
λ2 = −8/3
λ3 = −1.3438
λ4 = −1.9026
λ5 = − 5.3768 + 17.7207 𝑖
λ6 = − 5.3768 − 17.7207 𝑖
all real parts of eigenvalues are negative. The linearization method is succeeded to achieve complete
synchronization. In Lyapunov approach, the Lyapunov function is taken as the same form in theorem1, the
derivative Lyapunov function with control (9) becomes
𝑉̇ ( 𝑒) = −𝑎𝑒1
2
− 𝑒2
2
− 𝑏𝑒3
2
− 𝑑𝑒4
2
− 𝑒5
2
− ℎ𝑒6
2
+ 𝑒1 𝑒4(1 − 𝑑) + 𝑒2 𝑒5(1 − 𝑘) = −𝑒 𝑇
𝑄1 𝑒 (11)
where
𝑄1 =
[
𝑎
0
0
−(1 − 𝑑)/2
0
0
0
1
0
0
−(1 − 𝑘)/2
0
0
0
𝑏
0
0
0
−(1 − 𝑑)/2
0
0
𝑑
0
0
0
−(1 − 𝑘)/2
0
0
1
0
0
0
0
0
0
ℎ ]
Note that 𝑄1 is not a diagonal matrix. If all the following five inequalities are satisfied, then the 𝑄1 is
positive definite:
{
1. 𝑎 > 0
2. 𝑏 > 0
3. ℎ > 0
4. (𝑎𝑑 −
(1−𝑑)2
4
) > 0
5. (𝑎𝑑 (1 −
(1−𝑘)2
4
) −
(1−𝑑)2
4
(1 −
(1−𝑘)2
4
)) > 0
(12)
Fifth inequality is not correct with given parameters. Therefore, this control is failed. If update the matrix 𝑃
with the same control as:
𝑃1 = 𝑑𝑖𝑎𝑔(1 2⁄ , 1 2⁄ , 1 2⁄ , 1 4⁄ , 5/84 ,1) (13)

 ISSN: 1693-6930
1488
Then, the derivative of Lyapunov function as:
𝑉̇ ( 𝑒𝑖) = −10𝑒1
2
−𝑒2
2
−
8
3
𝑒3
2
− 𝑒4
2
−
5
42
𝑒5
2
− 𝑒6
2
= −𝑒 𝑇
𝑄2 𝑒 (14)
where 𝑄2 = 𝑑𝑖𝑎𝑔(10,1,8/3,1,5/42,1) is a positive definite. Figure 3 shows verify these results numerically.
Theorem 3. If the nonlinear control 𝑈 of error dynamical system (4) is designed as:
{
𝑢1 = −𝑐𝑒2 − 𝑎(𝑒5 + 𝑒2)
𝑢2 = −𝑟𝑒6 + 𝑥3 𝑒1
𝑢3 = 𝑒4(𝑥1 + 𝑒1) − 𝑥2 𝑒1
𝑢4 = −𝑒1 − 2𝑑𝑒4 + 𝑥3 𝑒1
𝑢5 = −𝑒2 − 𝑒5 + 𝑘(2𝑒1 + 𝑒2)
𝑢6 = −2ℎ𝑒6
(15)
then the system (3) can be followed by the system (2) by linearization method only.
Proof. Rewrite system (4) with control (15) as follows (16).
{
𝑒̇1 = −𝑎𝑒1 + 𝑒4−𝑐𝑒2 − 𝑎𝑒5
𝑒̇2 = c𝑒1 − 𝑒2 − 𝑒1 𝑒3 − 𝑥1 𝑒3 + 𝑒5 − 𝑟𝑒6
𝑒̇3 = −b𝑒3 + 𝑒1 𝑒2 + 𝑥1 𝑒2+𝑥1 𝑒4 + 𝑒1 𝑒4
𝑒̇4 = −d𝑒4 − 𝑒1 𝑒3 − 𝑥1 𝑒3−𝑒1
𝑒̇5 = −𝑒2 − 𝑒5 + 2𝑘𝑒1
𝑒̇6 = 𝑟𝑒2 − ℎ𝑒6
(16)
Based on the Lyapunov stability theory, we obtain
𝑉̇ ( 𝑒) = −𝑎𝑒1
2
− 𝑒2
2
− 𝑏𝑒3
2
− 𝑑𝑒4
2
− 𝑒5
2
− ℎ𝑒6
2
+ 𝑒1 𝑒5(2𝑘 − 𝑎) = −𝑒 𝑇
𝑄3 𝑒 (17)
where
𝑄3 =
[
𝑎
0
0
0
(𝑎 − 2𝑘)/2
0
0
1
0
0
0
0
0
0
𝑏
0
0
0
0
0
0
𝑑
0
0
−(𝑎 − 2𝑘)/2
0
0
0
1
0
0
0
0
0
0
ℎ ]
(18)
So 𝑄3 is not a diagonal matrix. The necessary conditions to make 𝑄3 is positive definite, the following
inequalities must hold.

1489
{
1. 𝑎 > 0
2. 𝑏 > 0
3. 𝑑 > 0
4. ℎ > 0
5. 𝑎 >
(𝑎−2𝑘)2
4
(19)
Note all inequalities are realized except the fifth inequality. So, the matrix 𝑄3 is a negative
definition, and failed to achieve complete synchronization. Therefore modified the matrix 𝑃 as follows:
{
𝑃3,1 = 𝑑𝑖𝑎𝑔(21/25,1/2,1/2,1/2,1/2,1/2)
𝑃3,2 = 𝑑𝑖𝑎𝑔(1/2,1/2,1/2,1/2,25/84, 1 2⁄ )
𝑃3,3 = 𝑑𝑖𝑎𝑔(1/20,1/2,1/2,1/2,5/168,1 2⁄ )
all the above matrices are not diagonal 𝑄3, therefore Lyapunov method failed. Based on Linearization
method, the characteristic equation and eigenvalues as
λ6
+
53
3
λ5
+ 1054λ4
+
34142
5
λ3
+
83193
5
λ2
+
53173
3
λ +
35784
5
= 0
{
λ1 = −8/3
λ2 = −1.9967
λ3 = − 1.1097 − 0.4060 𝑖
λ4 = − 1.1097 + 0.4060 𝑖
λ5 = − 5.3920 − 30.5554 𝑖
λ6 = − 5.3920 + 30.5554 𝑖
Note that all eigenvalues with negative real parts, and thus the Linearization method has succeeded
in achieving complete synchronization between systems (2) and (3) without any update compared to the
Lyapunov method and thus the proof has been completed. These results are justified numerically in Figure 4.
4. CONCLUSION
In this paper, complete synchronization of a 6-D hyperchaotic system with a self-excited attractor is
proposed. based on nonlinear control strategy and two analytical methods; first is Lyapunov's, and the second
is the Linearization method. Through these two approaches we have found the difference between them and
what is the appropriate method in each approach for achieving complete synchronization and thus we showed
the best way observed that the Linearization method does not need to a auxiliary function or modifying this
function as a method Lyapunov. Thus the linearization method is better than the Lyapunov method in
achieving the desired one. Numerical results have been found to be the same results as we proposed.

 ISSN: 1693-6930
1490
ACKNOWLEDGEMENTS
The authors are very grateful to University of Mosul/College of Computer Sciences and
Mathematics for their provided facilities, which helped to improve the quality of this work.
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