Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 193138, 12 pages
http://dx.doi.org/10.1155/2013/193138
Research Article
Enhanced Symplectic Synchronization between Two Different
Complex Chaotic Systems with Uncertain Parameters
Cheng-Hsiung Yang
Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, 43 Section 4,
Keelung Road, Taipei 106, Taiwan
Correspondence should be addressed to Cheng-Hsiung Yang; chyang123123@mail.ntust.edu.tw
Received 12 October 2012; Accepted 13 April 2013
Academic Editor: Haydar Akca
Copyright © 2013 Cheng-Hsiung Yang. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An enhanced symplectic synchronization of complex chaotic systems with uncertain parameters is studied. The traditional chaos
synchronizations are special cases of the enhanced symplectic synchronization. A sufficient condition is given for the asymptotical
stability of the null solution of error dynamics. The enhanced symplectic synchronization may be applied to the design of secure
communication. Finally, numerical simulations results are performed to verify and illustrate the analytical results.
1. Introduction
A synchronized mechanism that enables a system to maintain
a desired dynamical behavior (the goal or target) even when
intrinsically chaotic has many applications ranging from
biology to engineering [1–4]. Thus, it is of considerable
interest and potential utility to devise control techniques
capable of achieving the desired type of behavior in nonlinear
and chaotic systems. Many approaches have been presented
for the synchronization of chaotic systems [5–10]. There are
a chaotic master system and either an identical or a different
slave system. Our goal is the synchronization of the chaotic
master and the chaotic slave by coupling or by other methods.
The symplectic chaos synchronization concept [11]
𝑦 = 𝐻 (𝑡, 𝑥, 𝑦) + 𝐹 (𝑡)
(1)
is studied, where 𝑥, 𝑦 are the state vectors of the master
system and of the slave system, respectively, and 𝐹(𝑡) is a
given function of time in different form. The 𝐹(𝑡) may be a
regular motion function or a chaotic motion function. When
𝐻(𝑡, 𝑥, 𝑦) + 𝐹(𝑡) = 𝑥 and 𝐻(𝑡, 𝑥, 𝑦) = 𝑥, (1) reduces to the
generalized chaos synchronization and the traditional chaos
synchronization given in [1–3], respectively. In this paper, a
new enhance symplectic chaos synchronization:
𝑦 = 𝐻 (𝑡, 𝑥,̇ 𝑦,̇ 𝑥, 𝑦) + 𝐹 (𝑡) .
(2)
As numerical examples, we select hyperchaotic Chen
system [12] and hyperchaotic Lorenz system [13] as the master
system and the slave system, respectively.
This paper is organized as follows. In Section 2, by
the Lyapunov asymptotical stability theorem, a symplectic
synchronization scheme is given. In Section 3, various feedbacks of nonlinear controllers are designed for the enhanced
symplectic synchronization of a hyperchaotic Chen system
with uncertain parameters and a hyperchaotic Lorenz system.
Numerical simulations are also given in Section 3. Finally,
some concluding remarks are given in Section 4.
2. Enhanced Symplectic
Synchronization Scheme
There are two different nonlinear chaotic systems. The partner 𝐴 controls the partner 𝐵 partially. The partner 𝐴 is given
by
𝑥̇ = 𝑓 (𝑡, 𝑥, 𝐴 (𝑡)) ,
(3)
where 𝑥 = [𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ]𝑇 ∈ 𝑅𝑛 is a state vector, 𝐴(𝑡) =
[𝐴 1 (𝑡), 𝐴 2 (𝑡), . . . , 𝐴 𝑀(𝑡)]𝑇 ∈ 𝑅𝑀 is a vector of uncertain
coefficients in 𝑓, and 𝑓 is a vector function.
2
Abstract and Applied Analysis
The partner 𝐵 is given by
𝑦̇ = 𝑔 (𝑡, 𝑦, 𝐵 (𝑡)) ,
(4a)
where 𝑦 = [𝑦1 , 𝑦2 , . . . , 𝑦𝑛 ]𝑇 ∈ 𝑅𝑛 is a state vector, 𝐵(𝑡) =
[𝐵1 (𝑡), 𝐵2 (𝑡), . . . , 𝐵𝑚 (𝑡)]𝑇 ∈ 𝑅𝑚 is a vector of uncertain
coefficients in 𝑔, and 𝑔 is a vector function different from 𝑓.
After a controller 𝑢(𝑡) is added, partner 𝐵 becomes
𝑦̇ = 𝑔 (𝑡, 𝑦, 𝐵 (𝑡)) + 𝑢 (𝑡) ,
(4b)
3. Numerical Results for the Enhanced
Symplectic Chaos Synchronization of Chen
System with Uncertain Parameters and
Hyperchaotic Lorenz System
To further illustrate the effectiveness of the controller, we
select hyperchaotic Chen system and hyperchaotic Lorenz
system as the master system and the slave system, respectively.
Consider
𝑥1̇ = 𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ,
where 𝑢(𝑡) = [𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡)]𝑇 ∈ 𝑅𝑛 is the control
vector.
Our goal is to design the controller 𝑢(𝑡) so that the
state vector 𝑦 of the partner 𝐵 asymptotically approaches
𝐻(𝑡, 𝑥,̇ 𝑦,̇ 𝑥, 𝑦) + 𝐹(𝑡), a given function 𝐻(𝑡, 𝑥,̇ 𝑦,̇ 𝑥, 𝑦) plus a
given vector function 𝐹(𝑡) = [𝐹1 (𝑡), 𝐹2 (𝑡), . . . , 𝐹𝑛 (𝑡)]𝑇 which
is a regular or a chaotic function. Define error vector 𝑒(𝑡) =
[𝑒1 , 𝑒2 , . . . , 𝑒𝑛 ]𝑇 :
𝑒 = 𝐻 (𝑡, 𝑥,̇ 𝑦,̇ 𝑥, 𝑦) − 𝑦 + 𝐹 (𝑡) ,
(5)
lim 𝑒 = 0
(6)
𝑡→∞
is demanded.
From (5), it is obtained that
𝑒̇ =
𝜕𝐻
+ ∇𝐻𝑇 Ψ̇ − 𝑦̇ + 𝐹̇ (𝑡) ,
𝜕𝑡
(7)
where Ψ̇ = [𝑥̈ 𝑦̈ 𝑥̇ 𝑦]̇ 𝑇 .
Using (3), (4a), and (4b), (7) can be rewritten as
𝑒̇ =
𝜕𝐻
𝜕𝐻
𝜕𝐻 𝜕𝐻
+
𝑥̈ +
𝑦̈ +
𝑓 (𝑡, 𝑥, 𝐴 (𝑡))
̇
̇
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
+
𝜕𝐻
𝑔 (𝑡, 𝑦, 𝐵 (𝑡)) − 𝑔 (𝑡, 𝑦, 𝐵 (𝑡)) − 𝑢 (𝑡) + 𝐹̇ (𝑡) .
𝜕𝑦
(8)
𝜕𝐻
𝑔 (𝑡, 𝑦, 𝐵 (𝑡))
𝜕𝑦
𝑥4̇ = 𝑟𝑥4 + 𝑥2 𝑥3 ,
𝑦1̇ = 𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 ,
𝑦2̇ = 𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 ,
(12)
𝑦3̇ = − 𝑐1 𝑦3 + 𝑦1 𝑦2 ,
𝑦4̇ = 𝑑1 𝑦4 − 𝑦1 𝑦3 ,
where 𝑎, 𝑏, 𝑐, 𝑑, 𝑟, 𝑎1 , 𝑏1 , 𝑐1 , and 𝑑1 are parameters. The
parameters of master system and slave system are chosen as
𝑎 = 31, 𝑏 = 3.5, 𝑐 = 11, 𝑑 = 7.7, 𝑟 = 0.1, 𝑎1 = 11, 𝑏1 =
28, 𝑐1 = 2.8, and 𝑑1 = 1.2.
The controllers 𝑢1 , 𝑢2 , 𝑢3 , and 𝑢4 are added to the four
equations of (12), respectively as follows:
𝑦2̇ = 𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 + 𝑢2 ,
(13)
𝑦3̇ = −𝑐1 𝑦3 + 𝑦1 𝑦2 + 𝑢3 ,
(9)
𝜕𝐻 𝜕𝐻
𝜕𝐻
𝜕𝐻
𝑉̇ (𝑒) = 𝑒𝑇 {
+
𝑥̈ +
𝑦̈ +
𝜕𝑡
𝜕𝑥̇
𝜕𝑦 ̇
𝜕𝑥
× 𝑓 (𝑡, 𝑥, 𝐴 (𝑡)) +
(11)
𝑥3̇ = − 𝑏𝑥3 + 𝑥1 𝑥2 ,
𝑦1̇ = 𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 + 𝑢1 ,
Proof. A positive definite Lyapunov function 𝑉(𝑒) is chosen
[14, 15] as
1
𝑉 (𝑒) = 𝑒𝑇 𝑒.
2
Its derivative along any solution of (8) is
𝑥2̇ = 𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 ,
(10)
− 𝑔 (𝑡, 𝑦, 𝐵 (𝑡)) + 𝐹̇ (𝑡) − 𝑢 (𝑡) }.
̇
In (10), the 𝑢(𝑡) is designed so that 𝑉(𝑒)
= 𝑒𝑇 𝐶𝑛×𝑛 𝑒, where
𝐶𝑛×𝑛 is a diagonal negative definite matrix. The 𝑉̇ is a negative
definite function of 𝑒.
Remark 1. Note that 𝑒 approaches zero when time approaches
infinitly, according to Lyapunov theorem of asymptotical stability. The enhanced symplectic synchronization is obtained
[12, 13, 16–19].
𝑦4̇ = 𝑑1 𝑦4 − 𝑦1 𝑦3 + 𝑢4 .
The initial values of the states of the Chen system and of
the Lorenz system are taken as 𝑥1 (0) = 11, 𝑥2 (0) = 13, 𝑥3 (0)
= 12, 𝑥4 (0) = 12, 𝑦1 (0) = −11, 𝑦2 (0) = −13, 𝑦3 (0) = −12,
and 𝑦4 (0) = −12.
Case 1 (a symplectic synchronization). We take 𝐹1 (𝑡) = 𝑥43 (𝑡),
𝐹2 (𝑡) = 𝑥13 (𝑡), 𝐹3 (𝑡) = 𝑥23 (𝑡), and 𝐹4 (𝑡) = 𝑥33 (𝑡). They are
chaotic functions of time. 𝐻𝑖 (𝑥, 𝑦, 𝑡) = −𝑥𝑖2 𝑦𝑖 (𝑖 = 1, 2, 3, 4)
are given. By (6), we have
= lim (−𝑥𝑖2 𝑦𝑖 − 𝑦𝑖 + 𝑥𝑗3 ) = 0,
lim 𝑒
𝑡→∞ 𝑖
𝑡→∞
4,
𝑖 = 1, 2, 3, 4; 𝑗 = {
𝑖 − 1,
𝑖 = 1,
𝑖 ≠ 1.
(14)
From (7), we have
𝑒𝑖̇ = −2𝑥𝑖̇ 𝑥𝑖 𝑦𝑖 − 𝑥𝑖2 𝑦𝑖̇ − 𝑦𝑖̇ + 3𝑥𝑗̇ 𝑥𝑗2 ,
𝑖 = 1, 2, 3, 4; 𝑗 = {
4,
𝑖 = 1,
𝑖 − 1, 𝑖 ≠ 1.
(15)
Abstract and Applied Analysis
3
Equation (8) can be expressed as
According to (10), we get the controller
𝑢1 = − 2𝑦1 𝑥1 [𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ] + 3𝑥42 (𝑟𝑥4 + 𝑥2 𝑥3 )
𝑒1̇ = − 2𝑦1 𝑥1 [𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ]
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 ] + 𝑒1 ,
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 ]
𝑢2 = − 2𝑦2 𝑥2 (𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 ) + 3𝑥12 [𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ]
− 𝑢1 + 3𝑥42 (𝑟𝑥4 + 𝑥2 𝑥3 ) ,
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 ) + 𝑒2 ,
𝑒2̇ = − 2𝑦2 𝑥2 (𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 )
𝑢3 = − 2𝑦3 𝑥3 (−𝑏𝑥3 + 𝑥1 𝑥2 ) + 3𝑥22 (𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 )
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 )
+ (1 + 𝑥32 ) × (𝑐1 𝑦3 − 𝑦1 𝑦2 ) + 𝑒3 ,
− 𝑢2 + 3𝑥12 [𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ] ,
𝑢4 = − 2𝑦4 𝑥4 (𝑟𝑥4 + 𝑥2 𝑥3 ) + 3𝑥32 (−𝑏𝑥3 + 𝑥1 𝑥2 )
𝑒3̇ = − 2𝑦3 𝑥3 (−𝑏𝑥3 + 𝑥1 𝑥2 )
− (1 + 𝑥42 ) × (𝑑1 𝑦4 − 𝑦1 𝑦3 ) + 𝑒4 .
− (1 + 𝑥32 ) × (−𝑐1 𝑦3 + 𝑦1 𝑦2 )
(19)
− 𝑢3 + 3𝑥22 (𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 ) ,
Equation (18) becomes
𝑒4̇ = − 2𝑦4 𝑥4 (𝑟𝑥4 + 𝑥2 𝑥3 )
− (1 +
𝑥42 )
𝑉̇ = − (𝑒12 + 𝑒22 + 𝑒32 + 𝑒42 ) < 0,
× (𝑑1 𝑦4 − 𝑦1 𝑦3 )
− 𝑢4 + 3𝑥32 (−𝑏𝑥3 + 𝑥1 𝑥2 ) ,
(16)
where 𝑒1 = −𝑥12 𝑦1 − 𝑦1 + 𝑥43 , 𝑒2 = −𝑥22 𝑦2 − 𝑦2 + 𝑥13 , 𝑒3 =
−𝑥32 𝑦3 − 𝑦3 + 𝑥23 , and 𝑒4 = −𝑥42 𝑦4 − 𝑦4 + 𝑥33 .
Choose a positive definite Lyapunov function as
𝑉 (𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 ) =
1 2 2 2 2
(𝑒 + 𝑒 + 𝑒 + 𝑒 ) .
2 1 2 3 4
(17)
Its time derivative along any solution of (16) is
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization
of the Chen system and the Lorenz system is achieved. The
numerical results are shown in Figures 1, 2, and 3. After 1
second, the motion trajectories enter a chaotic attractor.
Case 2 (a symplectic synchronization with uncertain parameters). The master Chen system with uncertain variable
parameters is
𝑥1̇ = 𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ,
𝑥2̇ = 𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 ,
𝑉̇ = 𝑒1 {−2𝑦1 𝑥1 [𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ]
𝑥3̇ = −𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 ,
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 ] − 𝑢1
where 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), and 𝑟(𝑡) are uncertain parameters.
In simulation, we take
+ 𝑒2 {−2𝑦2 𝑥2 (𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 )
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 )
𝑎 (𝑡) = 𝑎 (1 + 𝑘1 sin 𝜔1 𝑡) ,
𝑏 (𝑡) = 𝑏 (1 + 𝑘2 sin 𝜔2 𝑡) ,
𝑐 (𝑡) = 𝑐 (1 + 𝑘3 sin 𝜔3 𝑡) ,
𝑑 (𝑡) = 𝑑 (1 + 𝑘4 sin 𝜔4 𝑡) ,
𝑟 (𝑡) = 𝑟 (1 + 𝑘5 sin 𝜔5 𝑡) ,
− 𝑢2 + 3𝑥12 [𝑎 (𝑥2 − 𝑥1 ) + 𝑥4 ] }
(22)
+ 𝑒3 {−2𝑦3 𝑥3 (−𝑏𝑥3 + 𝑥1 𝑥2 )
where 𝑘1 , 𝑘2 , 𝑘3 , 𝑘4 , 𝑘5 , 𝜔1 , 𝜔2 , 𝜔3 , 𝜔4 , and 𝜔5 are constants.
Take 𝑘1 = 0.3, 𝑘2 = 0.5, 𝑘3 = 0.2, 𝑘4 = 0.4, 𝑘5 = 0.6, 𝜔1 =
13, 𝜔2 = 17, 𝜔3 = 19, 𝜔4 = 23, and 𝜔5 = 29. So, (21) is
chaotic system, shown in Figure 4.
We take 𝐹1 (𝑡) = 𝑥43 (𝑡), 𝐹2 (𝑡) = 𝑥13 (𝑡), 𝐹3 (𝑡) = 𝑥23 (𝑡), and
𝐹4 (𝑡) = 𝑥33 (𝑡). They are chaotic functions of time. 𝐻𝑖 (𝑥, 𝑦, 𝑡) =
−𝑥𝑖2 𝑦𝑖 (𝑖 = 1, 2, 3, 4) are given. By (6), we have
− (1 + 𝑥32 ) × (−𝑐1 𝑦3 + 𝑦1 𝑦2 ) − 𝑢3
+ 3𝑥22 (𝑑𝑥1 + 𝑐𝑥2 − 𝑥1 𝑥3 ) }
+ 𝑒4 {−2𝑦4 𝑥4 (𝑟𝑥4 + 𝑥2 𝑥3 )
− (1 + 𝑥42 ) × (𝑑1 𝑦4 − 𝑦1 𝑦3 )
− 𝑢4 +
(21)
𝑥4̇ = 𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 ,
+ 3𝑥42 (𝑟𝑥4 + 𝑥2 𝑥3 ) }
3𝑥32
(20)
lim 𝑒
𝑡→∞ 𝑖
(−𝑏𝑥3 + 𝑥1 𝑥2 ) } .
(18)
= lim (−𝑥𝑖2 𝑦𝑖 − 𝑦𝑖 + 𝑥𝑗3 ) = 0,
𝑡→∞
𝑖 = 1, 2, 3, 4; 𝑗 = {
4,
𝑖 = 1,
𝑖 − 1, 𝑖 ≠ 1.
(23)
Abstract and Applied Analysis
40
100
30
50
𝑥4 , 𝑦4
𝑥3 , 𝑦3
4
20
10
0
−50
0
20
𝑥2 , 10
𝑦2
0
−10
−10
0
20
10
−100
30
𝑥 1 , 𝑦1
20
𝑥2 , 10
𝑦2
0
100
100
50
50
0
0
−50
−50
−100
50
−100
50
40
𝑥3 ,
30
𝑦3
20
𝑥 1 , 𝑦1
(b)
𝑥4 , 𝑦4
𝑥4 , 𝑦4
(a)
0
−10
−10
20
10
10 −15 −10
−5
5
0
𝑥 1 , 𝑦1
10
15
20
40
𝑥3 , 30
𝑦3
20
10 −20
−10
10
0
𝑥 2 , 𝑦2
20
30
Master system
Slave system
Master system
Slave system
(c)
(d)
Figure 1: Projections of phase portrait for master system (11) and slave system (12).
𝑒3̇ = −2𝑦3 𝑥3 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
From (7), we have
− (1 + 𝑥32 ) × (−𝑐1 𝑦3 + 𝑦1 𝑦2 + 𝑢3 )
𝑒𝑖̇ = −2𝑥𝑖̇ 𝑥𝑖 𝑦𝑖 − 𝑥𝑖2 𝑦𝑖̇ − 𝑦𝑖̇ + 3𝑥𝑗̇ 𝑥𝑗2 ,
𝑖 = 1, 2, 3, 4; 𝑗 = {
4,
𝑖 − 1,
𝑖 = 1,
𝑖 ≠ 1.
Equation (8) can be expressed as
𝑒1̇ = −2𝑦1 𝑥1 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
(24)
+ 3𝑥22 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 ) ,
𝑒4̇ = −2𝑦4 𝑥4 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
− (1 + 𝑥42 ) × (𝑑1 𝑦4 − 𝑦1 𝑦3 + 𝑢4 )
+ 3𝑥32 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) ,
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 + 𝑢1 ]
+ 3𝑥42 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 ) ,
𝑒2̇ = −2𝑦2 𝑥2 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 + 𝑢2 )
+
3𝑥12
[𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ] ,
(25)
where 𝑒1 = −𝑥12 𝑦1 − 𝑦1 + 𝑥43 , 𝑒2 = −𝑥22 𝑦2 − 𝑦2 + 𝑥13 , 𝑒3 =
−𝑥32 𝑦3 − 𝑦3 + 𝑥23 , and 𝑒4 = −𝑥42 𝑦4 − 𝑦4 + 𝑥33 .
Choose a positive definite Lyapunov function as
𝑉 (𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 ) =
1 2 2 2 2
(𝑒 + 𝑒 + 𝑒 + 𝑒 ) .
2 1 2 3 4
(26)
Abstract and Applied Analysis
5
×104
4
20
0
3
𝑦4
𝑦3
−20
−40
1
−60
−80
800
2
600
400
𝑦2
200
0
−200
−400
0
−5
5
𝑦1
0
1000
10
×105
500
𝑦2
0
−500
(a)
×104
−1
−0.5
1
×106
𝑦1
×104
4
3
3
2
2
1
0
100
−1.5
0.5
(b)
𝑦4
𝑦4
4
−1000
0
1
50
𝑦3
0
−50
−100 −1.5
−1
0
−0.5
𝑦1
1
0.5 ×106
(c)
0
100
50
𝑦3
0
−50
−100 −1000
−500
0
500
1000
𝑦2
(d)
Figure 2: Projections of the phase portrait for chaotic system (13) of Case 1.
Its time derivative along any solution of (25) is
According to (10), we get the controller
𝑉̇ = 𝑒1 {−2𝑦1 𝑥1 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
𝑢1 = − 2𝑦1 𝑥1 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 ] − 𝑢1
+ 3𝑥42 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
+ 3𝑥42 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 ) }
− (1 + 𝑥12 ) [𝑎1 (𝑦2 − 𝑦1 ) − 𝑦4 ] + 𝑒1 ,
+ 𝑒2 {−2𝑦2 𝑥2 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
𝑢2 = − 2𝑦2 𝑥2 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 ) − 𝑢2
+ 3𝑥12 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
+ 3𝑥12 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ] }
− (1 + 𝑥22 ) (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 ) + 𝑒2 ,
+ 𝑒3 {−2𝑦3 𝑥3 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
(27)
𝑢3 = − 2𝑦3 𝑥3 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
− (1 + 𝑥32 ) × (−𝑐1 𝑦3 + 𝑦1 𝑦2 ) − 𝑢3
+ 3𝑥22 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
+ 3𝑥22 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 ) }
+ (1 + 𝑥32 ) (𝑐1 𝑦3 − 𝑦1 𝑦2 ) + 𝑒3 ,
+ 𝑒4 {−2𝑦4 𝑥4 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
𝑢4 = − 2𝑦4 𝑥4 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
− (1 + 𝑥42 ) × (𝑑1 𝑦4 − 𝑦1 𝑦3 )
+ 3𝑥32 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
− 𝑢4 + 3𝑥32 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) } .
− (1 + 𝑥42 ) (𝑑1 𝑦4 − 𝑦1 𝑦3 ) + 𝑒4 .
(28)
6
Abstract and Applied Analysis
×105
10000
6
8000
4
6000
2
4000
𝑥2 , 𝑦2 , 𝐹2 , 𝐻2
𝑥1 , 𝑦1 , 𝐹1 , 𝐻1
8
0
−2
2000
0
−4
−2000
−6
−4000
−8
0
2
1
4
3
7
5
6
Time (s)
8
9
−6000
10
0
1
𝐻1
𝐹1
𝑥1
𝑦1
2
3
×10
5
8
9
10
𝐻2
𝐹2
×104
0
𝑥4 , 𝑦4 , 𝐹4 , 𝐻4
−1
−2
−3
−5
−10
−15
−4
0
1
2
3
4
5
6
Time (s)
7
8
9
−20
10
0
1
𝐻3
𝐹3
𝑥3
𝑦3
2
3
4
5
6
Time (s)
7
𝐻4
𝐹4
𝑥4
𝑦4
(c)
(d)
2 ×10
5
1.5
1
𝑒1 , 𝑒2 , 𝑒3 , 𝑒4
𝑥3 , 𝑦3 , 𝐹3 , 𝐻3
7
(b)
4
0
−5
5 6
Time (s)
𝑥2
𝑦2
(a)
1
4
0.5
0
−0.5
−1
−1.5
0
1
2
3
4
5
6
Time (s)
𝑒1
𝑒2
7
8
9
10
𝑒3
𝑒4
(e)
Figure 3: Time histories of states, state errors, 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 for Case 1.
8
9
10
7
40
100
30
50
20
0
𝑥4
𝑥3
Abstract and Applied Analysis
10
0
30
−50
20
𝑥2
10
0
10
𝑥1
0
−10 −10
20
30
−100
60
𝑥2
20
0 −20
(a)
100
50
50
𝑥4
0
𝑥4
0
20
30
(b)
100
−50
0
−50
−100
−150
40
−10
10
𝑥1
−100
40
35
30
𝑥3
25
20
−10
15
0
10
20
30
𝑥3
𝑥1
20
10 −20
(c)
−10
0
10
20
𝑥2
(d)
Figure 4: Projections of the phase portrait for chaotic system (21).
Equation (27) becomes
From (7) we have
𝑉̇ = − (𝑒12 + 𝑒22 + 𝑒32 + 𝑒42 ) < 0,
(29)
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The symplectic synchronization of
the Chen system with uncertain parameters and the Lorenz
system is achieved. The numerical results are shown in
Figures 5 and 6. After 1 second, the motion trajectories enter
a chaotic attractor.
Case 3 (an enhanced symplectic synchronization with
uncertain parameters). We take 𝐹1 (𝑡) = 𝑥43 (𝑡), 𝐹2 (𝑡) =
𝑥13 (𝑡), 𝐹3 (𝑡) = 𝑥23 (𝑡), and 𝐹4 (𝑡) = 𝑥33 (𝑡). They are chaotic
functions of time. 𝐻𝑖 (𝑥,̇ 𝑦,̇ 𝑥, 𝑦, 𝑡) = −𝑥𝑖2 𝑦𝑖 − 𝑥̇ − 𝐾𝑦̇ (𝑖 =
1, 2, 3, 4) are given. The 𝐾 value is 0.0001. By (6), we have
𝑡→∞
4,
𝑖 = 1, 2, 3, 4; 𝑗 = {
𝑖 − 1,
𝑖 = 1,
𝑖 ≠ 1.
4,
𝑖 = 1, 2, 3, 4; 𝑗 = {
𝑖 − 1,
𝑖 = 1,
𝑖 ≠ 1.
Equation (8) can be expressed as
𝑒1̇ = −2𝑦1 𝑥1 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 + 𝑢1 ]
− 𝑥1̈ − 𝐾𝑦1̈ + 3𝑥42 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 ) ,
𝑒2̇ = −2𝑦2 𝑥2 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 + 𝑢2 )
− 𝑥2̈ − 𝐾𝑦2̈ + 3𝑥12 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ] ,
lim 𝑒𝑖 = lim (−𝑥𝑖2 𝑦𝑖 − 𝑥𝑖̇ − 𝐾𝑦𝑖̇ − 𝑦𝑖 + 𝑥𝑗3 ) = 0,
𝑡→∞
𝑒𝑖̇ = −2𝑥𝑖 𝑥𝑖̇ 𝑦𝑖 − 𝑥𝑖2 𝑦𝑖̇ − 𝑥𝑖̈ − 𝐾𝑦𝑖̈ − 𝑦𝑖̇ + 3𝑥𝑗̇ 𝑥𝑗2 ,
(30)
𝑒3̇ = −2𝑦3 𝑥3 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
− (1 + 𝑥32 ) × (−𝑐1 𝑦3 + 𝑦1 𝑦2 + 𝑢3 )
(31)
8
Abstract and Applied Analysis
60
4
40
3
𝑦4
𝑦3
20
0
2
1
−20
−40
2000
×104
0
2000
1000
𝑦2
0
−1000
−2000
−4
−2
0
2
×106
1000
𝑦2
𝑦1
0
−1000
(a)
×104
4
3
3
2
2
1
1
0
100
0
100
50
0
𝑦3
−50 −4
−2
𝑦1
(b)
𝑦4
𝑦4
4
−2000 −4
2
×106
0
−2
0
2
×106
𝑦1
×104
50
𝑦3
0
−50 −2000
(c)
−1000
0
𝑦2
1000
2000
(d)
Figure 5: Projections of the phase portrait for chaotic system (13) of Case 2.
− 𝑥3̈ − 𝐾𝑦3̈ + 3𝑥22 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 ) ,
− 𝑥1̈ − 𝐾𝑦1̈ + 3𝑥42 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 ) }
𝑒4̇ = −2𝑦4 𝑥4 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
− (1 +
𝑥42 )
+ 𝑒2 {−2𝑦2 𝑥2 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
× (𝑑1 𝑦4 − 𝑦1 𝑦3 + 𝑢4 )
− (1 + 𝑥22 ) × (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 ) + 𝑢2
− 𝑥4̈ − 𝐾𝑦4̈ + 3𝑥32 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) ,
(32)
where 𝑒1 = −𝑥12 𝑦1 − 𝑦1 − 𝑥1̇ − 𝐾𝑦1̇ + 𝑥43 , 𝑒2 = −𝑥22 𝑦2 − 𝑦2 −
𝑥2̇ − 𝐾𝑦2̇ + 𝑥13 , 𝑒3 = −𝑥32 𝑦3 − 𝑦3 − 𝑥3̇ − 𝐾𝑦3̇ + 𝑥23 , and 𝑒4 =
−𝑥42 𝑦4 − 𝑦4 − 𝑥4̇ − 𝐾𝑦4̇ + 𝑥33 .
Choose a positive definite Lyapunov function as
𝑉 (𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 ) =
1 2 2 2 2
(𝑒 + 𝑒 + 𝑒 + 𝑒 ) .
2 1 2 3 4
Its time derivative along any solution of (32) is
𝑉̇ = 𝑒1 {−2𝑦1 𝑥1 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
− (1 + 𝑥12 ) × [𝑎1 (𝑦2 − 𝑦1 ) + 𝑦4 ] + 𝑢1
(33)
− 𝑥2̈ − 𝐾𝑦2̈ + 3𝑥12 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ] }
+ 𝑒3 {−2𝑦3 𝑥3 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
− (1 + 𝑥32 ) × (−𝑐1 𝑦3 + 𝑦1 𝑦2 ) + 𝑢3
− 𝑥3̈ − 𝐾𝑦3̈ + 3𝑥22 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 ) }
+ 𝑒4 {−2𝑦4 𝑥4 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
− (1 + 𝑥42 ) × (𝑑1 𝑦4 − 𝑦1 𝑦3 ) + 𝑢4
− 𝑥4̈ − 𝐾𝑦4̈ + 3𝑥32 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) } .
(34)
Abstract and Applied Analysis
×106
1.5
2
1
1.5
0.5
1
𝑥2 , 𝑦2 , 𝐹2 , 𝐻2
𝑥1 , 𝑦1 , 𝐹1 , 𝐻1
9
0
−0.5
0.5
0
−0.5
−1
−1.5
×104
0
2
1
3
4
5
7
6
8
9
−1
10
0
1
2
3
4
Time (s)
𝐻1
𝐹1
𝑥1
𝑦1
×10
0.5
9
10
×10
7
8
9
10
5
0
−1
𝑥4 , 𝑦4 , 𝐹4 , 𝐻4
−0.5
−2
−3
−4
−1
−1.5
−2
−5
−2.5
−6
0
1
2
3
4
5
6
7
8
9
−3
10
0
1
2
3
4
Time (s)
𝐻3
𝐹3
𝑥3
𝑦3
5
6
Time (s)
𝐻4
𝐹4
𝑥4
𝑦4
(c)
(d)
4
×10
5
3
2
𝑒1 , 𝑒2 , 𝑒3 , 𝑒4
𝑥3 , 𝑦3 , 𝐹3 , 𝐻3
8
(b)
4
0
−7
7
𝐻2
𝐹2
𝑥2
𝑦2
(a)
1
5
6
Time (s)
1
0
−1
−2
−3
0
1
2
3
4
5
6
Time (s)
𝑒1
𝑒2
7
8
9
10
𝑒3
𝑒4
(e)
Figure 6: Time histories of states, state errors, 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 for Case 2.
10
Abstract and Applied Analysis
60
4
40
3
20
0
2
𝑦4
𝑦3
×104
−20
1
−40
−60
2000
0
𝑦2
−2000
−4000
0
−1
−2
2
1
0
100
3
×106
50
𝑦2
𝑦1
0
−50
−100
(a)
×104
4
3
3
2
2
1
1
0
100
0
100
50
𝑦3
0
−50
−100
−2
−1
0
2
3
×106
𝑦1
(b)
𝑦4
𝑦4
4
−2
1
−1
0
1
2
𝑦1
3
×106
×104
50
𝑦3
0
−50
−100 −3000
(c)
−2000
−1000
0
1000
2000
𝑦2
(d)
Figure 7: Projections of the phase portrait for chaotic system (13) of Case 3.
According to (10), we get the controller
𝑢1 = −2𝑦1 𝑥1 [𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
− 𝑥1̈ − 𝐾𝑦1̈ + 3𝑥42 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 )
− (1 +
𝑥12 ) [𝑎1
(𝑦2 − 𝑦1 ) + 𝑦4 ] + 𝑒1 ,
𝑢2 = −2𝑦2 𝑥2 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
− 𝑥2̈ − 𝐾𝑦2̈ +
3𝑥12
[𝑎 (𝑡) (𝑥2 − 𝑥1 ) + 𝑥4 ]
− (1 + 𝑥22 ) (𝑏1 𝑦1 − 𝑦2 − 𝑦1 𝑦3 ) + 𝑒2 ,
𝑢3 = −2𝑦3 𝑥3 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
− 𝐾𝑦4̈ + 3𝑥32 (−𝑏 (𝑡) 𝑥3 + 𝑥1 𝑥2 )
− (1 + 𝑥42 ) (𝑑1 𝑦4 − 𝑦1 𝑦3 ) + 𝑒4 .
(35)
Equation (34) becomes
𝑉̇ = − (𝑒12 + 𝑒22 + 𝑒32 + 𝑒42 ) < 0,
(36)
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The enhanced symplectic synchronization of the Chen system with uncertain parameters
and the Lorenz system is achieved. The numerical results
are shown in Figures 7 and 8. After 1 second, the motion
trajectories enter a chaotic attractor.
− 𝑥3̈ − 𝐾𝑦3̈ + 3𝑥22 (𝑑 (𝑡) 𝑥1 + 𝑐 (𝑡) 𝑥2 − 𝑥1 𝑥3 )
4. Conclusions
+ (1 + 𝑥32 ) (𝑐1 𝑦3 − 𝑦1 𝑦2 ) + 𝑒3 ,
We achieve the novel enhanced symplectic synchronization
of a Chen system with uncertain parameters, and a Lorenz
system is obtained by the Lyapunov asymptotical stability
𝑢4 = −2𝑦4 𝑥4 (𝑟 (𝑡) 𝑥4 + 𝑥2 𝑥3 ) − 𝑥4̈
Abstract and Applied Analysis
×106
2
1
1.5
0.5
1
𝑥2 , 𝑦2 , 𝐹2 , 𝐻2
𝑥1 , 𝑦1 , 𝐹1 , 𝐻1
1.5
11
0
−0.5
0.5
0
−0.5
−1
−1.5
×104
0
2
1
3
4
5
6
Time (s)
7
8
9
−1
10
0
1
𝐻1
𝐹1
𝑥1
𝑦1
2
3
×10
0.5
8
9
10
7
8
9
10
𝐻2
𝐹2
×10
5
0
𝑥4 , 𝑦4 , 𝐹4 , 𝐻4
−1
−2
−3
−4
−0.5
−1
−1.5
−2
−5
−2.5
−6
0
1
2
3
4
5
6
7
8
9
−3
10
0
1
2
3
4
Time (s)
𝐻3
𝐹3
𝑥3
𝑦3
5
6
Time (s)
𝐻4
𝐹4
𝑥4
𝑦4
(c)
(d)
4 ×10
5
3
2
𝑒1 , 𝑒2 , 𝑒3 , 𝑒4
𝑥3 , 𝑦3 , 𝐹3 , 𝐻3
7
(b)
4
0
−7
5
6
Time (s)
𝑥2
𝑦2
(a)
1
4
1
0
−1
−2
−3
0
1
2
3
4
5
6
Time (s)
𝑒1
𝑒2
7
8
9
10
𝑒3
𝑒4
(e)
Figure 8: Time histories of states, state errors, 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 for Case 3.
12
theorem. All the theoretical results are verified by numerical
simulations to demonstrate the effectiveness of the three
cases of proposed synchronization schemes. The enhanced
symplectic synchronization of chaotic systems with uncertain
parameters can be used to increase the security of secret
communication.
Acknowledgment
This research was supported by the National Science Council,
Taiwan, under Grant no. 98-2218-E-011-010.
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