Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 528325, 14 pages
http://dx.doi.org/10.1155/2013/528325
Research Article
Symplectic Synchronization of Lorenz-Stenflo System with
Uncertain Chaotic Parameters via Adaptive Control
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 24 October 2012; Accepted 30 December 2012
Academic Editor: Chuandong Li
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.
A new symplectic chaos synchronization of chaotic systems with uncertain chaotic parameters is studied. The traditional chaos
synchronizations are special cases of the symplectic chaos synchronization. A sufficient condition is given for the asymptotical
stability of the null solution of error dynamics and a parameter difference. The symplectic chaos synchronization with uncertain
chaotic parameters may be applied to the design of secure communication systems. Finally, numerical results are studied for
symplectic chaos synchronized from two identical Lorenz-Stenflo systems in three different cases.
1. Introduction
Chaos has been detected in a large number of nonlinear
dynamic systems of physical characteristics. In addition to
the control and stabilization of chaos, chaos synchronization systems are a fascinating concept which has received
considerable interest among nonlinear scientists in recent
times. However, chaos is desirable in some systems, such as
convective heat transfer, liquid mixing, encryption, power
converters, secure communications, biological systems and
chemical reactions. There are a chaotic master system (or
driver) and either an identical or a different slave system (or
responser). Our goal is the synchronization of the chaotic
master system and the chaotic slave system by coupling or by
other methods. In practice, some or all of the parameters of
chaotic system parameters are uncertain. A lot of works have
been preceded to solve this problem using adaptive control
concept.
Among many kinds of chaos synchronization, generalized chaos synchronization is investigated [1–6]. There exists
a functional relationship between the states of the master
system and those of the slave system. The symplectic chaos
synchronization concept [7]:
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, 𝐹(𝑡) is a given
function of time in different form. The 𝐹(𝑡) may be a regular
function or a chaotic function. When 𝐻(𝑡, 𝑥, 𝑦) + 𝐹(𝑡) =
𝑥 and 𝐻(𝑡, 𝑥, 𝑦) = 𝑥 of (1) reduces to the traditional
generalized chaos synchronization and the traditional chaos
synchronization given in [8–10], respectively.
As numerical examples, in 1996, Stenflo originally used a
four-dimensional autonomous chaotic system to describe the
low-frequency short-wavelength gravity wave disturbance in
the atmosphere [11]. This system is similar to the celebrated
Lorenz equation but more complex than it due to the introduction of a new feedback control and a new state variable
and thus is called a Lorenz-Stenflo system after the names of
Lorenz and Stenflo [12]. The nonlinear dynamical behaviors
of the Lorenz-Stenflo system have been investigated in [13–
16].
This paper is organized as follows. In Section 2, by
the Lyapunov asymptotic stability theorem, the symplectic
chaos synchronization with uncertain chaotic parameters
by adaptive control scheme is given. In Section 3, various
adaptive controllers and update laws are designed for the
symplectic chaos synchronization of the identical LorenzStenflo systems. Numerical simulations are also given in
Section 3. Finally, some concluding remarks are given in
Section 4.
2
Abstract and Applied Analysis
2. Symplectic Chaos Synchronization
with Chaotic Parameters by Adaptive
Control Scheme
Its derivative along any solution of (7) is
̃ (𝑡)) = 𝑒𝑇 { 𝜕𝐻 + 𝜕𝐻 𝑓 (𝑡, 𝑥, 𝐴 (𝑡))+ 𝜕𝐻 𝑓 (𝑡, 𝑦, 𝐴
̂ (𝑡))
𝑉 (𝑒, 𝐴
𝜕𝑡 𝜕𝑥
𝜕𝑦
.
There are two identical nonlinear dynamical systems, and the
partner 𝐴 controls the partner 𝐵. The partner 𝐴 is given by
.
𝑥 = 𝑓 (𝑡, 𝑥, 𝐴 (𝑡)) ,
(2)
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.
The partner 𝐵 is given by
.
̂ (𝑡)) ,
𝑦 = 𝑓 (𝑡, 𝑦, 𝐴
(3a)
̂ =
where 𝑦 = [𝑦1 , 𝑦2 , . . . , 𝑦𝑛 ]𝑇 ∈ 𝑅𝑛 is a state vector, 𝐴(𝑡)
𝑇
𝑚
̂2 (𝑡), . . . , 𝐴
̂𝑚 (𝑡)] ∈ 𝑅 is an estimated vector of
̂1 (𝑡), 𝐴
[𝐴
uncertain coefficients in 𝑓.
So a controller 𝑢(𝑡) is added on partner 𝐵, and the partner
𝐵 becomes
.
̂ (𝑡)) + 𝑢 (𝑡) ,
𝑦 = 𝑓 (𝑡, 𝑦, 𝐴
𝑇
(3b)
𝑛
where 𝑢(𝑡) = [𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡)] ∈ 𝑅 is the control
vector function.
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) + 𝐹(𝑡), a given function 𝐻(𝑡, 𝑥, 𝑦) plus a given
vector function 𝐹(𝑡) = [𝐹1 (𝑡), 𝐹2 (𝑡), . . . , 𝐹𝑛 (𝑡)]𝑇 which is the
regular function or the chaotic function.
To define error vector 𝑒(𝑡) = [𝑒1 , 𝑒2 , . . . , 𝑒𝑛 ]𝑇 :
.
lim 𝑒 = 0,
𝑡→∞
.
̃ (𝑡) = 0.
lim 𝐴
𝑡→∞
(10)
The symplectic chaos synchronization with uncertain
chaotic parameters is obtained [3–7, 11, 17–23].
3. Numerical Results for the Symplectic
Chaos Synchronization of Lorenz-Stenflo
System with Uncertain Chaotic
Parameters via Adaptive Control
The master Lorenz-Stenflo system can be described as [11]
.
𝑥1 = 𝑎 (𝑥2 − 𝑥1 ) + 𝑟𝑥4 ,
.
𝑥2 = 𝑑𝑥1 − 𝑥2 − 𝑥1 𝑥3 ,
lim 𝑒 = 0
(5)
𝑥4 = −𝑥4 − 𝑐𝑥1 .
.
(11)
.
And the slave Lorenz-Stenflo system can be described as
.
.
.
.
𝜕𝐻
+ ∇𝐻𝑇 Ψ − 𝑦 + 𝐹 (𝑡) ,
𝜕𝑡
𝑦1 = 𝑎̂ (𝑦2 − 𝑦1 ) + 𝑟̂𝑦4 ,
(6)
.
̂ −𝑦 −𝑦 𝑦 ,
𝑦2 = 𝑑𝑦
1
2
1 3
.
𝑦3 = −̂𝑏𝑦3 + 𝑦1 𝑦2 ,
.
where Ψ = [𝑥 𝑦].
By (2), (3a), and (3b), (6) can be rewritten as
𝑒=
.
̃ are designed so that 𝑉 = 𝑒𝑇 𝐶𝑛×𝑛 𝑒 +
In (9), the 𝑢(𝑡) and 𝐴(𝑡)
𝑇
̃ (𝑡)𝐷𝑚×𝑚 𝐴(𝑡)
̃ where 𝐶𝑛×𝑛 and 𝐷𝑚×𝑚 are two diagonal
𝐴
.
negative definite matrices. The 𝑉 is a negative definite
̃ By Lyapunov theorem of asymptotical
function of 𝑒 and 𝐴(𝑡).
stability
𝑥3 = −𝑏𝑥3 + 𝑥1 𝑥2 ,
𝑒=
.
(9)
(4)
is demanded.
From (4), it is obtained that
.𝑇
.
̃ (𝑡) .
̃𝑇 (𝑡) 𝐴
+𝐴
𝑒 = 𝐻 (𝑡, 𝑥, 𝑦) − 𝑦 + 𝐹 (𝑡) ,
𝑡→∞
.
.
̂ (𝑡)) + 𝐹 (𝑡) − 𝑢 (𝑡) }
−𝑓 (𝑡, 𝑦, 𝐴
𝜕𝐻 𝜕𝐻
𝜕𝐻
̂ (𝑡))
+
𝑓 (𝑡, 𝑥, 𝐴 (𝑡)) +
𝑓 (𝑡, 𝑦, 𝐴
𝜕𝑡
𝜕𝑥
𝜕𝑦
(12)
.
𝑦4 = −𝑦4 − 𝑐̂𝑦1 ,
̃
A positive definite Lyapunov function 𝑉(𝑒, 𝐴(𝑡))
is chosen:
where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑟 are called the first Prandtl number, geometric parameter, second Prandtl number, Rayleigh number,
and rotation number, respectively. The parameters of LorenzStenflo system are chosen as 𝑎 = 11, 𝑏 = 2.9, 𝑐 = 5, 𝑑 = 23,
and 𝑟 = 1.9.
The controllers, 𝑢1 , 𝑢2 , 𝑢3 , and 𝑢4 , are added to the four
equations of (12), respectively:
̃ (𝑡)) = 1 𝑒𝑇 𝑒 + 1 𝐴
̃𝑇 (𝑡) 𝐴
̃ (𝑡) ,
𝑉 (𝑒, 𝐴
2
2
𝑦1 = 𝑎̂ (𝑦2 − 𝑦1 ) + 𝑟̂𝑦4 + 𝑢1 ,
.
(7)
̂ (𝑡)) − 𝑢 (𝑡) + 𝐹 (𝑡) .
− 𝑓 (𝑡, 𝑦, 𝐴
̃ = 𝐴(𝑡) − 𝐴(𝑡).
̂
where 𝐴(𝑡)
(8)
.
.
̂ −𝑦 −𝑦 𝑦 +𝑢 ,
𝑦2 = 𝑑𝑦
1
2
1 3
2
Abstract and Applied Analysis
3
.
𝑦3 = −̂𝑏𝑦3 + 𝑦1 𝑦2 + 𝑢3 ,
.
𝑒4 = [2𝑥4 (−𝑥4 − 𝑐𝑥1 ) − 𝑦4 − 𝑐̂𝑦1 ] (sin 𝜔𝑡 + 3)
.
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐̂𝑦1 − 𝑢4
𝑦4 = −𝑦4 − 𝑐̂𝑦1 + 𝑢4 .
(13)
− 𝑥4 (𝑡 − 𝑇) − 𝑐𝑥1 (𝑡 − 𝑇) ,
(16)
The initial values of the states of the master system and
of the slave system are taken as 𝑥1 (0) = 10, 𝑥2 (0) = −13,
𝑥3 (0) = 12, 𝑥4 (0) = −5, 𝑦1 (0) = −3, 𝑦2 (0) = −4, 𝑦3 (0) = 3,
and 𝑦4 (0) = −5.
where
𝑒1 = (𝑥12 + 𝑦1 ) (sin 𝜔𝑡 + 3) − 𝑦1 + 𝑥1 (𝑡 − 𝑇) ,
Case 1. A time delay symplectic synchronization.
We take 𝐹1 (𝑡) = 𝑥1 (𝑡 − 𝑇), 𝐹2 (𝑡) = 𝑥2 (𝑡 − 𝑇), 𝐹3 (𝑡) =
𝑥3 (𝑡 − 𝑇), and 𝐹4 (𝑡) = 𝑥4 (𝑡 − 𝑇). They are chaotic functions
of time, where time delay 𝑇 = 0.5 sec. The 𝐻𝑖 (𝑥, 𝑦, 𝑡) = (𝑥𝑖2 +
𝑦𝑖 )(sin 𝑤𝑡 + 3) (𝑖 = 1, 2, 3, 4) are given. By (5) we have
lim 𝑒
𝑡→∞ 𝑖
=
lim ((𝑥𝑖2
𝑡→∞
𝑒2 = (𝑥22 + 𝑦2 ) (sin 𝜔𝑡 + 3) − 𝑦2 + 𝑥2 (𝑡 − 𝑇) ,
𝑒3 = (𝑥32 + 𝑦3 ) (sin 𝜔𝑡 + 3) − 𝑦3 + 𝑥3 (𝑡 − 𝑇) ,
𝑒4 = (𝑥42 + 𝑦4 ) (sin 𝜔𝑡 + 3) − 𝑦4 + 𝑥4 (𝑡 − 𝑇) .
+ 𝑦𝑖 ) (sin 𝜔𝑡 + 3)−𝑦𝑖 +𝑥𝑖 (𝑡 − 𝑇)) = 0,
(14)
𝑖 = 1, 2, 3, 4.
Choose a positive definite Lyapunov function:
From (14) we have
.
̃ 𝑟̃)
𝑉 (𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 , 𝑎̃, ̃𝑏, 𝑐̃, 𝑑,
.
.
𝑒𝑖 = (2𝑥𝑖 𝑥𝑖 + 𝑦𝑖 ) (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥𝑖2 + 𝑦𝑖 )
.
(17)
.
− 𝑦𝑖 + 𝑥𝑖 (𝑡 − 𝑇) ,
𝑖 = 1, 2, 3, 4,
=
(15)
where 𝜔 = 19.
Equation (15) can be expressed as
1 2 2 2 2
(𝑒 + 𝑒 + 𝑒 + 𝑒 + 𝑎̃2 + ̃𝑏2 + 𝑐̃2 + 𝑑̃2 + 𝑟̃2 ) ,
2 1 2 3 4
(18)
̂ 𝑟̃ = 𝑟 − 𝑟̂, and
where 𝑎̃ = 𝑎 − 𝑎̂, ̃𝑏 = 𝑏 − ̂𝑏, 𝑐̃ = 𝑐 − 𝑐̂, 𝑑̃ = 𝑑 − 𝑑,
̂ and 𝑟̂ are estimates of uncertain parameters a, b, c,
𝑎̂, ̂𝑏, 𝑐̂, 𝑑,
d, and r, respectively.
Its time derivative along any solution of (16) is
.
.
𝑒1 = [2𝑥1 (𝑎 (𝑥2 − 𝑥1 ) + 𝑟𝑥4 ) + (̂
𝑎 (𝑦2 − 𝑦1 ) + 𝑟̂𝑦4 )]
× (sin 𝜔𝑡 + 3) +
𝜔 cos 𝜔𝑡 (𝑥12
+ 𝑦1 ) − 𝑎̂ (𝑦2 − 𝑦1 )
− 𝑟̂𝑦4 − 𝑢1 + 𝑎 (𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇)) + 𝑟𝑥4 (𝑡 − 𝑇) ,
.
̂ − 𝑦 − 𝑦 𝑦 )]
𝑒2 = [2𝑥2 (𝑑𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦
1
2
1 3
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 )
̂ +𝑦 +𝑦 𝑦 +𝑢
− 𝑑𝑦
1
2
1 3
2
+ 𝑑𝑥1 (𝑡 − 𝑇) − 𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇) 𝑥3 (𝑡 − 𝑇) ,
.
𝑒3 = [2𝑥3 (−𝑏𝑥3 + 𝑥1 𝑥2 ) − ̂𝑏𝑦3 + 𝑦1 𝑦2 ] (sin 𝜔𝑡 + 3)
𝑉 = 𝑒1 { [2𝑥1 (𝑎 (𝑥2 − 𝑥1 ) + 𝑟𝑥4 ) + (̂
𝑎 (𝑦2 − 𝑦1 ) + 𝑟̂𝑦4 )]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎̂ (𝑦2 − 𝑦1 )
− 𝑟̂𝑦4 − 𝑢1 + 𝑎 (𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇)) + 𝑟𝑥4 (𝑡 − 𝑇)}
̂ − 𝑦 − 𝑦 𝑦 )]
+ 𝑒2 {[2𝑥2 (𝑑𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦
1
2
1 3
̂ +𝑦
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 ) − 𝑑𝑦
1
2
+ 𝑦1 𝑦3 − 𝑢2 + 𝑑𝑥1 (𝑡 − 𝑇) − 𝑥2 (𝑡 − 𝑇)
−𝑥1 (𝑡 − 𝑇) 𝑥3 (𝑡 − 𝑇)}
+ 𝑒3 {[2𝑥3 (−𝑏𝑥3 + 𝑥1 𝑥2 ) − ̂𝑏𝑦3 + 𝑦1 𝑦2 ]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + ̂𝑏𝑦3
+ 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + ̂𝑏𝑦3 − 𝑦1 𝑦2 − 𝑢3
− 𝑦1 𝑦2 − 𝑢3 − 𝑏𝑥3 (𝑡 − 𝑇)
− 𝑏𝑥3 (𝑡 − 𝑇) + 𝑥1 (𝑡 − 𝑇) 𝑥2 (𝑡 − 𝑇) ,
+ 𝑥1 (𝑡 − 𝑇) 𝑥2 (𝑡 − 𝑇)}
4
Abstract and Applied Analysis
+ 𝑒4 {[2𝑥4 (−𝑥4 − 𝑐𝑥1 ) − 𝑦4 − 𝑐̂𝑦1 ] (sin 𝜔𝑡 + 3)
The numerical results of the phase portrait of master system,
chaotic system (3), the time series of states, state errors, and
parameter differences are shown in Figures 1, 2, and 3, respectively. The symplectic chaos synchronization is accomplished
using adaptive control method.
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐̂𝑦1 − 𝑢4
.
.
− 𝑥4 (𝑡 − 𝑇) − 𝑐𝑥1 (𝑡 − 𝑇)} + 𝑎̃𝑎̃ + ̃𝑏̃𝑏
.
.
.
+ 𝑐̃𝑐̃ + 𝑑̃𝑑̃ + 𝑟̃𝑟̃.
(19)
Case 2. A time delay symplectic synchronization with uncertain chaotic parameters.
The Lorenz-Stenflo master system with uncertain chaotic
parameters is
The adaptive controllers are chosen as
.
𝑥1 = 𝐴 (𝑡) (𝑥2 − 𝑥1 ) + 𝑅 (𝑡) 𝑥4 ,
𝑢1 = [2𝑥1 (𝑎 (𝑥2 − 𝑥1 ) + 𝑟𝑥4 ) + (𝑎 (𝑦2 − 𝑦1 ) + 𝑟𝑦4 )]
.
𝑥2 = 𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ,
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎 (𝑦2 − 𝑦1 )
− 𝑟𝑦4 + 𝑎 (𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇)) + 𝑟𝑥4 (𝑡 − 𝑇) + 𝑒1 ,
.
𝑥4 = −𝑥4 − 𝐶 (𝑡) 𝑥1 ,
𝑢2 = [2𝑥2 (𝑑𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦1 − 𝑦2 − 𝑦1 𝑦3 )]
where 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡) are uncertain chaotic
parameters. In simulation, we take
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 )
− 𝑑𝑦1 + 𝑦2 + 𝑦1 𝑦3 + 𝑑𝑥1 (𝑡 − 𝑇)
− 𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇) 𝑥3 (𝑡 − 𝑇) + 𝑒2 ,
𝑢3 = [2𝑥3 (−𝑏𝑥3 + 𝑥1 𝑥2 ) − 𝑏𝑦3 + 𝑦1 𝑦2 ] (sin 𝜔𝑡 + 3)
𝐴 (𝑡) = 𝑎 (1 + 𝑘1 𝑧1 ) ,
𝐵 (𝑡) = 𝑏 (1 + 𝑘2 𝑧3 ) ,
𝐶 (𝑡) = 𝑐 (1 + 𝑘3 𝑧4 ) ,
𝐷 (𝑡) = 𝑑 (1 + 𝑘4 𝑧2 ) ,
where 𝑘1 , 𝑘2 , 𝑘3 , 𝑘4 , and 𝑘5 are positive constants. Chosen are
𝑘1 = 0.07, 𝑘2 = 0.08, 𝑘3 = 0.06, 𝑘4 = 0.07, and 𝑘5 = 0.08. The
system (23) is chaotic dynamic motion, shown in Figure 4.
− 𝑏𝑥3 (𝑡 − 𝑇) + 𝑥1 (𝑡 − 𝑇) 𝑥2 (𝑡 − 𝑇) + 𝑒3 ,
𝑢4 = [2𝑥4 (−𝑥4 − 𝑐𝑥1 ) − 𝑦4 − 𝑐𝑦1 ] (sin 𝜔𝑡 + 3)
The 𝐹(𝑡) is chaotic system, and the chaotic signal of goal
system can be described as
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐𝑦1
− 𝑥4 (𝑡 − 𝑇) − 𝑐𝑥1 (𝑡 − 𝑇) + 𝑒4 ,
.
𝑧1 = 𝑎 (𝑧2 − 𝑧1 ) + 𝑟𝑧4 ,
(20)
.
𝑧2 = 𝑑𝑧1 − 𝑧2 − 𝑧1 𝑧3 ,
and the update laws are chosen as
.
.
.
.
𝑧3 = −𝑏𝑧3 + 𝑧1 𝑧2 ,
.
̃𝑏 = −̂𝑏 = −2𝑥 𝑦 𝑒 (sin 𝜔𝑡 + 3) − 𝑦 𝑒 − ̃𝑏,
3 3 3
3 3
𝑧4 = −𝑧4 − 𝑐𝑧1 ,
𝑐̃ = −𝑐̂ = −2𝑥4 𝑦1 𝑒4 (sin 𝜔𝑡 + 3) − 𝑦1 𝑒4 − 𝑐̃,
.
.
.
.
(25)
.
𝑎̃ = −𝑎̂ = 2𝑥1 (𝑦2 − 𝑦1 ) (sin 𝜔𝑡 + 3) 𝑒1 + (𝑦2 − 𝑦1 ) 𝑒1 − 𝑎̃,
.
(24)
𝑅 (𝑡) = 𝑟 (1 + 𝑘5 𝑧1 ) ,
+ 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + 𝑏𝑦3 − 𝑦1 𝑦2
.
(23)
.
𝑥3 = −𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ,
where initial conditions of the chaotic signal of system are
𝑧1 (0) = 2, 𝑧2 (0) = 5, 𝑧3 (0) = −4, and 𝑧4 (0) = −6. The
𝐻𝑖 (𝑥, 𝑦, 𝑡) = (𝑥𝑖2 + 𝑦𝑖 )(sin 𝑤𝑡 + 3) (𝑖 = 1, 2, 3, 4) are given.
By (5) we have
̃
𝑑̃ = −𝑑̂ = 2𝑥2 𝑦1 𝑒2 (sin 𝜔𝑡 + 3) + 𝑦1 𝑒2 − 𝑑,
𝑟̃ = −𝑟̂ = 2𝑥1 𝑦4 𝑒1 (sin 𝜔𝑡 + 3) + 𝑦4 𝑒1 − 𝑟̃.
(21)
lim 𝑒
𝑡→∞ 𝑖
= lim ((𝑥𝑖2 + 𝑦𝑖 ) (sin 𝜔𝑡 + 3) − 𝑦𝑖 + 𝑥𝑖 (𝑡 − 𝑇)) = 0,
𝑡→∞
The initial values of estimate for uncertain parameters are
̂ = 𝑟̂(0) = 0.
𝑎̂(0) = ̂𝑏(0) = 𝑐̂(0) = 𝑑(0)
Equation (19) becomes
𝑖 = 1, 2, 3, 4.
(26)
.
𝑉 = − (𝑒12 + 𝑒22 + 𝑒32 + 𝑒42 + 𝑎̃2 + ̃𝑏2 + 𝑐̃2 + 𝑑̃2 + 𝑟̃2 ) < 0,
From (26) we have
(22)
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The time delay symplectic synchronization of the identical Lorenz-Stenflo systems is achieved.
.
.
.
𝑒𝑖 = (2𝑥𝑖 𝑥𝑖 + 𝑦𝑖 ) (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥𝑖2 + 𝑦𝑖 )
.
.
− 𝑦𝑖 + 𝑥𝑖 (𝑡 − 𝑇) ,
𝑖 = 1, 2, 3, 4.
(27)
Abstract and Applied Analysis
5
50
40
40
20
0
𝑥4
𝑥3
30
20
−20
10
0
40
20
0
𝑥2
−20
−40 −20
−10
0
10
20
−40
40
20
𝑥2
𝑥1
0
−20
40
40
20
20
0
0
−20
−20
−40
60
−40
60
40
𝑥3
−10
10
20
𝑥1
(b)
𝑥4
𝑥4
(a)
−40 −20
0
20
0 −20
−10
10
0
20
40
𝑥3
𝑥1
(c)
20
0 −40
−20
0
20
40
𝑥2
(d)
Figure 1: Projections of phase portrait for master system (11).
Equation (27) can be expressed as
where
.
𝑒1 = [2𝑥1 (𝐴 (𝑡) (𝑥2 − 𝑥1 ) + 𝑅 (𝑡) 𝑥4 ) + (̂
𝑎 (𝑦2 − 𝑦1 ) + 𝑟̂𝑦4 )]
𝑒1 = (𝑥12 + 𝑦1 ) (sin 𝜔𝑡 + 3) − 𝑦1 + 𝑥1 (𝑡 − 𝑇) ,
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎̂ (𝑦2 − 𝑦1 )
𝑒2 = (𝑥22 + 𝑦2 ) (sin 𝜔𝑡 + 3) − 𝑦2 + 𝑥2 (𝑡 − 𝑇) ,
− 𝑟̂𝑦4 − 𝑢1 + 𝐴 (𝑡) (𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇))
𝑒3 = (𝑥32 + 𝑦3 ) (sin 𝜔𝑡 + 3) − 𝑦3 + 𝑥3 (𝑡 − 𝑇) ,
+ 𝑅 (𝑡) 𝑥4 (𝑡 − 𝑇) ,
𝑒4 = (𝑥42 + 𝑦4 ) (sin 𝜔𝑡 + 3) − 𝑦4 + 𝑥4 (𝑡 − 𝑇) .
.
̂ − 𝑦 − 𝑦 𝑦 )]
𝑒2 = [2𝑥2 (𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦
1
2
1 3
(29)
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 )
̂ + 𝑦 + 𝑦 𝑦 + 𝑢 + 𝐷 (𝑡) 𝑥 (𝑡 − 𝑇)
− 𝑑𝑦
1
2
1 3
2
1
Choose a positive definite Lyapunov function:
− 𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇) 𝑥3 (𝑡 − 𝑇) ,
.
𝑒3 = [2𝑥3 (−𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) − ̂𝑏𝑦3 + 𝑦1 𝑦2 ] (sin 𝜔𝑡 + 3)
̃ 𝑟̃)
𝑉 (𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 , 𝑎̃, ̃𝑏, 𝑐̃, 𝑑,
+ 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + ̂𝑏𝑦3 − 𝑦1 𝑦2 − 𝑢3
.
− 𝐵 (𝑡) 𝑥3 (𝑡 − 𝑇) + 𝑥1 (𝑡 − 𝑇) 𝑥2 (𝑡 − 𝑇) ,
=
𝑒4 = [2𝑥4 (−𝑥4 − 𝐶 (𝑡) 𝑥1 ) − 𝑦4 − 𝑐̂𝑦1 ] (sin 𝜔𝑡 + 3)
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐̂𝑦1 − 𝑢4
− 𝑥4 (𝑡 − 𝑇) − 𝐶 (𝑡) 𝑥1 (𝑡 − 𝑇) ,
(28)
1 2 2 2 2
(𝑒 + 𝑒 + 𝑒 + 𝑒 + 𝑎̃2 + ̃𝑏2 + 𝑐̃2 + 𝑑̃2 + 𝑟̃2 ) ,
2 1 2 3 4
(30)
̂
where 𝑎̃ = 𝐴(𝑡) − 𝑎̂, ̃𝑏 = 𝐵(𝑡) − ̂𝑏, 𝑐̃ = 𝐶(𝑡) − 𝑐̂, 𝑑̃ = 𝐷(𝑡) − 𝑑,
̂
̂
𝑟̃ = 𝑅(𝑡) − 𝑟̂, and 𝑎̂, 𝑏, 𝑐̂, 𝑑, and 𝑟̂ are estimates of uncertain
parameters 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡), respectively.
Abstract and Applied Analysis
0
500
−1000
0
−500
−2000
𝑦4
𝑦3
6
−1000
−3000
−1500
−4000
500
−2000
500
0
200
0
−500
−200
𝑦2 −1000
−400
𝑦1
−1500 −600
−500
−200
𝑦2 −1000
−400
𝑦1
−1500 −600
0
200
(b)
500
500
0
0
−500
−500
𝑦4
𝑦4
(a)
0
−1000
−1000
−1500
−1500
−2000
0
−1000
−2000
−200
−3000
𝑦3
−4000 −600 −400
𝑦1
−2000
0
−1000
−2000
−3000
𝑦3
−4000
200
0
(c)
−1500
−1000
−500
𝑦2
0
500
(d)
Figure 2: Projections of the phase portrait for chaotic system (3) of Case 1.
Its time derivative along any solution of (28) is
The adaptive controllers are chosen as
.
𝑉 = 𝑒1 {[2𝑥1 (𝐴 (𝑡) (𝑥2 − 𝑥1 )+𝑅 (𝑡) 𝑥4 )+(̂
𝑎 (𝑦2 − 𝑦1 )+̂𝑟𝑦4 )]
𝑢1 = [2𝑥1 (𝐴 (𝑡) (𝑥2 − 𝑥1 ) + 𝑅 (𝑡) 𝑥4 ) + (𝑎 (𝑦2 − 𝑦1 ) + 𝑟𝑦4 )]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎̂ (𝑦2 − 𝑦1 )
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎 (𝑦2 − 𝑦1 )
− 𝑟̂𝑦4 − 𝑢1 + 𝐴 (𝑡) (𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇))
− 𝑟𝑦4 + 𝐴 (𝑡) (𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇))
+𝑅 (𝑡) 𝑥4 (𝑡 − 𝑇)}
+ 𝑅 (𝑡) 𝑥4 (𝑡 − 𝑇) + 𝑒1 ,
̂ − 𝑦 − 𝑦 𝑦 )]
+ 𝑒2 {[2𝑥2 (𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦
1
2
1 3
𝑢2 = [2𝑥2 (𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦1 − 𝑦2 − 𝑦1 𝑦3 )]
̂
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 ) − 𝑑𝑦
1
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 ) − 𝑑𝑦1
+ 𝑦2 + 𝑦1 𝑦3 − 𝑢2 + 𝐷 (𝑡) 𝑥1 (𝑡 − 𝑇)
+ 𝑦2 + 𝑦1 𝑦3 + 𝐷 (𝑡) 𝑥1 (𝑡 − 𝑇) − 𝑥2 (𝑡 − 𝑇)
− 𝑥1 (𝑡 − 𝑇) 𝑥3 (𝑡 − 𝑇) + 𝑒2 ,
− 𝑥2 (𝑡 − 𝑇) − 𝑥1 (𝑡 − 𝑇) 𝑥3 (𝑡 − 𝑇) }
𝑢3 = [2𝑥3 (−𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) − 𝑏𝑦3 + 𝑦1 𝑦2 ] (sin 𝜔𝑡 + 3)
+ 𝑒3 {[2𝑥3 (−𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) − ̂𝑏𝑦3 + 𝑦1 𝑦2 ]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + ̂𝑏𝑦3 − 𝑦1 𝑦2
− 𝑢3 − 𝐵 (𝑡) 𝑥3 (𝑡 − 𝑇) + 𝑥1 (𝑡 − 𝑇) 𝑥2 (𝑡 − 𝑇)}
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐̂𝑦1 − 𝑢4 − 𝑥4 (𝑡 − 𝑇)
.
.
.
− 𝐵 (𝑡) 𝑥3 (𝑡 − 𝑇) + 𝑥1 (𝑡 − 𝑇) 𝑥2 (𝑡 − 𝑇) + 𝑒3 ,
𝑢4 = [2𝑥4 (−𝑥4 − 𝐶 (𝑡) 𝑥1 ) − 𝑦4 − 𝑐𝑦1 ] (sin 𝜔𝑡 + 3)
+ 𝑒4 {[2𝑥4 (−𝑥4 − 𝐶 (𝑡) 𝑥1 ) − 𝑦4 − 𝑐̂𝑦1 ] (sin 𝜔𝑡 + 3)
.
+ 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + 𝑏𝑦3 − 𝑦1 𝑦2
.
− 𝐶 (𝑡) 𝑥1 (𝑡 − 𝑇)} + 𝑎̃𝑎̃ + ̃𝑏̃𝑏 + 𝑐̃𝑐̃ + 𝑑̃𝑑̃ + 𝑟̃𝑟̃.
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐𝑦1 − 𝑥4 (𝑡 − 𝑇)
− 𝐶 (𝑡) 𝑥1 (𝑡 − 𝑇) + 𝑒4 ,
(31)
(32)
Abstract and Applied Analysis
7
400
600
300
400
200
100
𝑥2 , 𝑦2 , 𝐹2 , 𝐻2
𝑥1 , 𝑦1 , 𝐹1 , 𝐻1
200
0
−100
−200
−300
0
−200
−400
−600
−400
−800
−500
−600
0
2
4
6
8
10 12
Time (s)
𝑥1
𝑦1
14
16
18
20
−1000
0
2
4
6
𝑥2
𝑦2
𝐻1
𝐹1
20
14
16
18
20
0
−200
𝑥4 , 𝑦4 , 𝐹4 , 𝐻4
𝑥3 , 𝑦3 , 𝐹3 , 𝐻3
18
(b)
−500
−1000
−1500
−2000
−400
−600
−800
−1000
−1200
−2500
−1400
0
2
4
6
8 10 12
Time (s)
𝑥3
𝑦3
14
16
18
−1600
20
0
2
𝐻3
𝐹3
4
6
8
10 12
Time (s)
𝑥4
𝑦4
𝐻4
𝐹4
(c)
(d)
×10
1
500
5
0.5
400
0
300
−0.5
̃ 𝑟̃
𝑎̃, ̃𝑏, 𝑐̃, 𝑑,
𝑒1 , 𝑒2 , 𝑒3 , 𝑒4
16
200
0
200
100
−1
−1.5
−2
−2.5
0
−100
14
𝐻2
𝐹2
(a)
500
−3000
8 10 12
Time (s)
−3
0
2
4
6
8
10 12
Time (s)
𝑒3
𝑒4
𝑒1
𝑒2
(e)
14
16
18
20
−3.5
0
2
4
6
8
10
12
14
16
18
20
Time (s)
𝑑̃
𝑟̃
𝑎̃
̃𝑏
𝑐̃
(f)
Figure 3: Time histories of states, state errors, parameter differences, 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 for Case 1.
Abstract and Applied Analysis
100
60
80
40
60
20
40
𝑥4
𝑥3
8
0
20
−20
0
−40
−20
100
−60
100
50
𝑥2
0
−50 −40
−20
0
20
40
60
50
0
𝑥2
𝑥1
−50 −40
60
60
40
40
20
20
0
−20
−40
−40
−60
100
−60
100
50
0
−50 −40
60
𝑥1
0
−20
𝑥3
0
40
(b)
𝑥4
𝑥4
(a)
−20
20
−20
0
20
40
60
50
0
𝑥3
𝑥1
−50 −50
(c)
0
50
100
𝑥2
(d)
Figure 4: Projections of the phase portrait for chaotic system (23).
and the update laws are chosen as
states, state errors, parameters, and parameter differences are
shown in Figures 5, 6, and 7, respectively. The symplectic
chaos synchronization is accomplished using adaptive control method.
.
𝑎̃ = 2𝑥1 (𝑦2 − 𝑦1 ) (sin 𝜔𝑡 + 3) 𝑒1 + (𝑦2 − 𝑦1 ) 𝑒1 − 𝑎̃,
.
̃𝑏 = −2𝑥 𝑦 𝑒 (sin 𝜔𝑡 + 3) − 𝑦 𝑒 − ̃𝑏,
3 3 3
3 3
.
𝑐̃ = −2𝑥4 𝑦1 𝑒4 (sin 𝜔𝑡 + 3) − 𝑦1 𝑒4 − 𝑐̃,
.
̃
𝑑̃ = 2𝑥2 𝑦1 𝑒2 (sin 𝜔𝑡 + 3) + 𝑦1 𝑒2 − 𝑑,
.
𝑟̃ = 2𝑥1 𝑦4 𝑒1 (sin 𝜔𝑡 + 3) + 𝑦4 𝑒1 − 𝑟̃.
(33)
The initial values of estimate for uncertain parameters are
̂ = 𝑟̂(0) = 0.
𝑎̂(0) = ̂𝑏(0) = 𝑐̂(0) = 𝑑(0)
Equation (31) becomes
Case 3. A multitime delay symplectic synchronization with
uncertain chaotic parameters.
We take 𝐹1 (𝑡) = 𝑥1 (𝑡 − 𝑇1 ), 𝐹2 (𝑡) = 𝑥2 (𝑡 − 𝑇2 ), 𝐹3 (𝑡) =
𝑥3 (𝑡 − 𝑇3 ), and 𝐹4 (𝑡) = 𝑥4 (𝑡 − 𝑇4 ). They are chaotic functions
of time, where multi time delay 𝑇1 , 𝑇2 , 𝑇3 , and 𝑇4 are positive
constants, 𝑇1 = 0.5 sec, 𝑇2 = 0.7 sec, 𝑇3 = 0.8 sec, and 𝑇4 =
0.9 sec. The 𝐻𝑖 (𝑥, 𝑦, 𝑡) = (𝑥𝑖2 + 𝑦𝑖 )(sin 𝑤𝑡 + 3) (𝑖 = 1, 2, 3, 4)
are given. By (5) we have
lim 𝑒
𝑡→∞ 𝑖
= lim ((𝑥𝑖2 + 𝑦𝑖 ) (sin 𝜔𝑡 + 3) − 𝑦𝑖 + 𝑥𝑖 (𝑡 − 𝑇𝑖 )) = 0,
𝑡→∞
𝑖 = 1, 2, 3, 4.
(35)
.
𝑉 = − (𝑒12 + 𝑒22 + 𝑒32 + 𝑒42 + 𝑎̃2 + ̃𝑏2 + 𝑐̃2 + 𝑑̃2 + 𝑟̃2 ) < 0,
(34)
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The time delay symplectic synchronization with uncertain chaotic parameters of the identical
Lorenz-Stenflo systems is achieved. The numerical results of
the phase portrait of chaotic system (3), the time series of
From (35) we have
.
.
.
𝑒𝑖 = (2𝑥𝑖 𝑥𝑖 + 𝑦𝑖 ) (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥𝑖2 + 𝑦𝑖 )
.
.
− 𝑦𝑖 + 𝑥𝑖 (𝑡 − 𝑇𝑖 ) ,
𝑖 = 1, 2, 3, 4.
(36)
Abstract and Applied Analysis
9
2000
0
−2000
0
−6000
𝑦4
𝑦3
−4000
−8000
−4000
−10000
−12000
2000
−2000
0
−2000
−4000
𝑦2
−6000 −3000
−1000
−2000
𝑦1
0
−6000
2000
1000
0
−2000
−4000
𝑦2
2000
2000
0
0
−2000
−2000
−4000
−4000
−6000
0
−6000
0
−5000
𝑦3
−10000
−15000 −3000
1000
0
(b)
𝑦4
𝑦4
(a)
−6000 −3000
−1000
−2000
𝑦1
−1000
−2000
𝑦1
0
1000
−5000
𝑦3
(c)
−10000
−15000 −6000
−2000
−4000 𝑦2
0
2000
(d)
Figure 5: Projections of the phase portrait for chaotic system (3) of Case 2.
Equation (36) can be expressed as
.
𝑒4 = [2𝑥4 (−𝑥4 − 𝐶 (𝑡) 𝑥1 ) − 𝑦4 − 𝑐̂𝑦1 ] (sin 𝜔𝑡 + 3)
.
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐̂𝑦1 − 𝑢4
𝑒1 = [2𝑥1 (𝐴 (𝑡) (𝑥2 − 𝑥1 ) + 𝑅 (𝑡) 𝑥4 ) + (̂
𝑎 (𝑦2 − 𝑦1 ) + 𝑟̂𝑦4 )]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎̂ (𝑦2 − 𝑦1 )
− 𝑥4 (𝑡 − 𝑇4 ) − 𝐶 (𝑡) 𝑥1 (𝑡 − 𝑇1 ) ,
(37)
− 𝑟̂𝑦4 − 𝑢1 + 𝐴 (𝑡) (𝑥2 (𝑡 − 𝑇2 ) − 𝑥1 (𝑡 − 𝑇1 ))
+ 𝑅 (𝑡) 𝑥4 (𝑡 − 𝑇4 ) ,
.
̂ − 𝑦 − 𝑦 𝑦 )]
𝑒2 = [2𝑥2 (𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦
1
2
1 3
where
𝑒1 = (𝑥12 + 𝑦1 ) (sin 𝜔𝑡 + 3) − 𝑦1 + 𝑥1 (𝑡 − 𝑇1 ) ,
̂
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 ) − 𝑑𝑦
1
𝑒2 = (𝑥22 + 𝑦2 ) (sin 𝜔𝑡 + 3) − 𝑦2 + 𝑥2 (𝑡 − 𝑇2 ) ,
+ 𝑦2 + 𝑦1 𝑦3 + 𝑢2 + 𝐷 (𝑡) 𝑥1 (𝑡 − 𝑇1 )
𝑒3 = (𝑥32 + 𝑦3 ) (sin 𝜔𝑡 + 3) − 𝑦3 + 𝑥3 (𝑡 − 𝑇3 ) ,
− 𝑥2 (𝑡 − 𝑇2 ) − 𝑥1 (𝑡 − 𝑇1 ) 𝑥3 (𝑡 − 𝑇3 ) ,
𝑒4 = (𝑥42 + 𝑦4 ) (sin 𝜔𝑡 + 3) − 𝑦4 + 𝑥4 (𝑡 − 𝑇4 ) .
.
𝑒3 = [2𝑥3 (−𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) − ̂𝑏𝑦3 + 𝑦1 𝑦2 ]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 )
+ ̂𝑏𝑦3 − 𝑦1 𝑦2 − 𝑢3 − 𝐵 (𝑡) 𝑥3 (𝑡 − 𝑇3 )
+ 𝑥1 (𝑡 − 𝑇1 ) 𝑥2 (𝑡 − 𝑇2 ) ,
(38)
Choose a positive definite Lyapunov function:
̃ 𝑟̃)
𝑉 (𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 , 𝑎̃, ̃𝑏, 𝑐̃, 𝑑,
=
1 2 2 2 2
(𝑒 + 𝑒 + 𝑒 + 𝑒 + 𝑎̃2 + ̃𝑏2 + 𝑐̃2 + 𝑑̃2 + 𝑟̃2 ) ,
2 1 2 3 4
(39)
10
Abstract and Applied Analysis
1000
0
0
−500
−1000
−1000
𝑥2 , 𝑦2 , 𝐹2 , 𝐻2
𝑥1 , 𝑦1 , 𝐹1 , 𝐻1
500
−1500
−2000
−2500
−2000
−3000
−4000
−3000
−5000
−3500
−6000
−4000
0
2
4
6
8
10
12
14
16
18
−7000
20
0
2
4
6
8
Time (s)
𝑥1
𝑦1
𝑥2
𝑦2
𝐻1
𝐹1
16
18
20
14
16
18
20
(b)
500
0
−500
𝑥4 , 𝑦4 , 𝐹4 , 𝐻4
−0.5
−1
−1.5
−1000
−1500
−2000
−2
−2500
0
2
4
6
8
10
12
14
16
18
−3000
20
0
2
4
6
8
Time (s)
𝑥3
𝑦3
10
12
Time (s)
𝑥4
𝑦4
𝐻3
𝐹3
𝐻4
𝐹4
(c)
(d)
500
400
𝑒1 , 𝑒2 , 𝑒3 , 𝑒4
𝑥3 , 𝑦3 , 𝐹3 , 𝐻3
14
4
0
−2.5
12
𝐻2
𝐹2
(a)
×10
0.5
10
Time (s)
300
200
100
0
−100
0
2
4
6
8
10
12
14
16
18
20
Time (s)
𝑒3
𝑒4
𝑒1
𝑒2
(e)
Figure 6: Time histories of states, state errors, 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , 𝐻1 ,𝐻2 , 𝐻3 , and 𝐻4 for Case 2.
11
80
×105
4
70
2
60
0
50
−2
40
̃ 𝑟̃
𝑎̃, ̃𝑏, 𝑐̃, 𝑑,
𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), 𝑅(𝑡)
Abstract and Applied Analysis
30
20
−4
−6
10
−8
0
−10
−10
−12
−20
0
2
4
6
𝐴(𝑡)
𝐵(𝑡)
𝐶(𝑡)
8
10 12
Time (s)
14
16
18
20
𝐷(𝑡)
𝑅(𝑡)
−14
0
2
𝑎̃
̃𝑏
𝑐̃
(a)
4
6
8
10
12
Time (s)
14
16
18
20
𝑑̃
𝑟̃
(b)
Figure 7: Time histories of 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡) and parameter differences for Case 2.
̂
where 𝑎̃ = 𝐴(𝑡) − 𝑎̂, ̃𝑏 = 𝐵(𝑡) − ̂𝑏, 𝑐̃ = 𝐶(𝑡) − 𝑐̂, 𝑑̃ = 𝐷(𝑡) − 𝑑,
̂ and 𝑟̂ are estimates of uncertain
𝑟̃ = 𝑅(𝑡) − 𝑟̂, and 𝑎̂, ̂𝑏, 𝑐̂, 𝑑,
parameters 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), 𝐷(𝑡), and 𝑅(𝑡), respectively.
Its time derivative along any solution of (37) is
The adaptive controllers are chosen as
𝑢1 = [2𝑥1 (𝐴 (𝑡) (𝑥2 − 𝑥1 ) + 𝑅 (𝑡) 𝑥4 ) + (𝑎 (𝑦2 − 𝑦1 ) + 𝑟𝑦4 )]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎 (𝑦2 − 𝑦1 )
.
𝑉 = 𝑒1 {[2𝑥1 (𝐴 (𝑡) (𝑥2 − 𝑥1 )+𝑅 (𝑡) 𝑥4 )+(̂
𝑎 (𝑦2 − 𝑦1 )+̂𝑟𝑦4 )]
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥12 + 𝑦1 ) − 𝑎̂ (𝑦2 − 𝑦1 )
− 𝑟̂𝑦4 − 𝑢1 + 𝐴 (𝑡) (𝑥2 (𝑡 − 𝑇2 ) − 𝑥1 (𝑡 − 𝑇1 ))
− 𝑟𝑦4 + 𝐴 (𝑡) (𝑥2 (𝑡 − 𝑇2 ) − 𝑥1 (𝑡 − 𝑇1 ))
+ 𝑅 (𝑡) 𝑥4 (𝑡 − 𝑇4 ) + 𝑒1 ,
𝑢2 = [2𝑥2 (𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦1 − 𝑦2 − 𝑦1 𝑦3 )]
+𝑅 (𝑡) 𝑥4 (𝑡 − 𝑇4 )}
̂ − 𝑦 − 𝑦 𝑦 )]
+ 𝑒2 {[2𝑥2 (𝐷 (𝑡) 𝑥1 − 𝑥2 − 𝑥1 𝑥3 ) + (𝑑𝑦
1
2
1 3
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 ) − 𝑑𝑦1 + 𝑦2
̂ +𝑦
× (sin 𝜔𝑡 + 3) + 𝜔 cos 𝜔𝑡 (𝑥22 + 𝑦2 ) − 𝑑𝑦
1
2
+ 𝑦1 𝑦3 + 𝐷 (𝑡) 𝑥1 (𝑡 − 𝑇1 ) − 𝑥2 (𝑡 − 𝑇2 )
+ 𝑦1 𝑦3 − 𝑢2 + 𝐷 (𝑡) 𝑥1 (𝑡 − 𝑇1 ) − 𝑥2 (𝑡 − 𝑇2 )
− 𝑥1 (𝑡 − 𝑇1 ) 𝑥3 (𝑡 − 𝑇3 ) + 𝑒2 ,
𝑢3 = [2𝑥3 (−𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) − 𝑏𝑦3 + 𝑦1 𝑦2 ] (sin 𝜔𝑡 + 3)
−𝑥1 (𝑡 − 𝑇1 ) 𝑥3 (𝑡 − 𝑇3 ) }
+ 𝑒3 {[2𝑥3 (−𝐵 (𝑡) 𝑥3 + 𝑥1 𝑥2 ) − ̂𝑏𝑦3 + 𝑦1 𝑦2 ] (sin 𝜔𝑡 + 3)
+ 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + ̂𝑏𝑦3 − 𝑦1 𝑦2 − 𝑢3
− 𝐵 (𝑡) 𝑥3 (𝑡 − 𝑇3 ) + 𝑥1 (𝑡 − 𝑇1 ) 𝑥2 (𝑡 − 𝑇2 ) + 𝑒3 ,
−𝐵 (𝑡) 𝑥3 (𝑡 − 𝑇3 ) + 𝑥1 (𝑡 − 𝑇1 ) 𝑥2 (𝑡 − 𝑇2 ) }
𝑢4 = [2𝑥4 (−𝑥4 − 𝐶 (𝑡) 𝑥1 ) − 𝑦4 − 𝑐𝑦1 ] (sin 𝜔𝑡 + 3)
+ 𝑒4 {[2𝑥4 (−𝑥4 − 𝐶 (𝑡) 𝑥1 ) − 𝑦4 − 𝑐̂𝑦1 ] (sin 𝜔𝑡 + 3)
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐̂𝑦1 − 𝑢4 − 𝑥4 (𝑡 − 𝑇4 )
.
.
.
.
+ 𝜔 cos 𝜔𝑡 (𝑥32 + 𝑦3 ) + 𝑏𝑦3 − 𝑦1 𝑦2
.
−𝐶 (𝑡) 𝑥1 (𝑡 − 𝑇1 )} + 𝑎̃𝑎̃ + ̃𝑏̃𝑏 + 𝑐̃𝑐̃ + 𝑑̃𝑑̃ + 𝑟̃𝑟̃.
+ 𝜔 cos 𝜔𝑡 (𝑥42 + 𝑦4 ) + 𝑦4 + 𝑐𝑦1
− 𝑥4 (𝑡 − 𝑇4 ) − 𝐶 (𝑡) 𝑥1 (𝑡 − 𝑇1 ) + 𝑒4 ,
(40)
(41)
Abstract and Applied Analysis
5000
2000
0
0
−5000
−2000
𝑦4
𝑦3
12
−10000
−4000
−15000
2000
−6000
2000
0
−2000
−1000
−2000
−4000
𝑦2
−3000
𝑦
1
−6000 −4000
1000
0
0
−2000
−1000
−2000
−4000
𝑦2
−3000
𝑦
1
−6000 −4000
2000
2000
0
0
−2000
−2000
−4000
−4000
−6000
5000
−6000
5000
0
−5000
−1000
−2000
−10000
𝑦3
−3000
𝑦1
−15000 −4000
1000
(b)
𝑦4
𝑦4
(a)
0
1000
0
0
−5000
−2000
−10000
−4000
𝑦3
𝑦2
−15000 −6000
(c)
0
2000
(d)
Figure 8: Projections of the phase portrait for chaotic system (3) of Case 3.
and the update laws are chosen as
.
𝑎̃ = 2𝑥1 (𝑦2 − 𝑦1 ) (sin 𝜔𝑡 + 3) 𝑒1 + (𝑦2 − 𝑦1 ) 𝑒1 − 𝑎̃,
.
̃𝑏 = −2𝑥 𝑦 𝑒 (sin 𝜔𝑡 + 3) − 𝑦 𝑒 − ̃𝑏,
3 3 3
3 3
.
𝑐̃ = −2𝑥4 𝑦1 𝑒4 (sin 𝜔𝑡 + 3) − 𝑦1 𝑒4 − 𝑐̃,
(42)
which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. The multi time delay symplectic
synchronization with uncertain chaotic parameters of the
identical Lorenz-Stenflo systems is achieved. The numerical
results of the phase portrait of chaotic system (3), the time
series of states, state errors, and parameter differences are
shown in Figures 8 and 9, respectively. The symplectic
chaos synchronization is accomplished using adaptive control method.
.
̃
𝑑̃ = 2𝑥2 𝑦1 𝑒2 (sin 𝜔𝑡 + 3) + 𝑦1 𝑒2 − 𝑑,
4. Conclusions
.
𝑟̃ = 2𝑥1 𝑦4 𝑒1 (sin 𝜔𝑡 + 3) + 𝑦4 𝑒1 − 𝑟̃.
The initial values of estimate for uncertain parameters are
̂ = 𝑟̂(0) = 0.
𝑎̂(0) = ̂𝑏(0) = 𝑐̂(0) = 𝑑(0)
Equation (40) becomes
.
𝑉 = − (𝑒12 + 𝑒22 + 𝑒32 + 𝑒42 + 𝑎̃2 + ̃𝑏2 + 𝑐̃2 + 𝑑̃2 + 𝑟̃2 ) < 0,
(43)
A novel symplectic synchronization of a Lorenz-Stenflo
system with uncertain chaotic parameters is obtained by
the Lyapunov asymptotical stability theorem. The simulation
results of three cases are shown in corresponding figures
which imply that the adaptive controllers and update laws we
designed are feasible and effective. The symplectic synchronization of chaotic systems with uncertain chaotic parameters
via adaptive control concept can be used to increase the
security of secret communication system.
13
500
1000
0
0
−500
−1000
−1000
𝑥2 , 𝑦2 , 𝐹2 , 𝐻2
𝑥1 , 𝑦1 , 𝐹1 , 𝐻1
Abstract and Applied Analysis
−1500
−2000
−2500
−2000
−3000
−4000
−3000
−5000
−3500
−6000
−4000
0
2
4
6
8
10
12
14
16
18
−7000
20
0
2
4
6
8
Time (s)
𝑥1
𝑦1
𝑥2
𝑦2
𝐻1
𝐹1
18
20
500
0
−500
𝑥4 , 𝑦4 , 𝐹4 , 𝐻4
𝑥3 , 𝑦3 , 𝐹3 , 𝐻3
16
(b)
−0.5
−1
−1.5
−1000
−1500
−2000
−2
−2500
0
2
4
6
8
10
12
14
16
18
20
−3000
0
2
4
6
8
𝑥3
𝑦3
𝑥4
𝑦4
𝐻3
𝐹3
12
14
16
18
20
𝐻4
𝐹4
(c)
(d)
×10
4
500
5
2
400
0
̃ 𝑟̃
𝑎̃, ̃𝑏, 𝑐̃, 𝑑,
300
200
100
−2
−4
−6
−8
−10
0
−100
10
Time (s)
Time (s)
𝑒1 , 𝑒2 , 𝑒3 , 𝑒4
14
4
0
−2.5
12
𝐻2
𝐹2
(a)
×10
0.5
10
Time (s)
−12
0
2
4
6
8
10
12
14
16
18
20
−14
0
2
4
Time (s)
𝑒3
𝑒4
𝑒1
𝑒2
(e)
6
8
10 12
Time (s)
14
16
18
20
𝑑̃
𝑟̃
𝑎̃
̃𝑏
𝑐̃
(f)
Figure 9: Time histories of states, state errors, parameter differences, 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 , 𝐻1 , 𝐻2 , 𝐻3 , and 𝐻4 for Case 3.
14
Abstract and Applied Analysis
Acknowledgment
This research was supported by the National Science Council,
Republic of China, under Grant no. 98-2218-E-011-010.
[19]
References
[20]
[1] R. Femat and G. Solı́s-Perales, “On the chaos synchronization
phenomena,” Physics Letters A, vol. 262, no. 1, pp. 50–60, 1999.
[2] Z.-M. Ge and C.-H. Yang, “Pragmatical generalized synchronization of chaotic systems with uncertain parameters by
adaptive control,” Physica D, vol. 231, no. 2, pp. 87–94, 2007.
[3] S. S. Yang and C. K. Duan, “Generalized synchronization in
chaotic systems,” Chaos, Solitons & Fractals, vol. 9, no. 10, pp.
1703–1707, 1998.
[4] A. Krawiecki and A. Sukiennicki, “Generalizations of the
concept of marginal synchronization of chaos,” Chaos, Solitons
& Fractals, vol. 11, no. 9, pp. 1445–1458, 2000.
[5] Z. M. Ge, C. H. Yang, H. H. Chen, and S. C. Lee, “Nonlinear dynamics and chaos control of a physical pendulum with
vibrating and rotating support,” Journal of Sound and Vibration,
vol. 242, no. 2, pp. 247–264, 2001.
[6] M. Y. Chen, Z. Z. Han, and Y. Shang, “General synchronization
of Genesio-Tesi systems,” International Journal of Bifurcation
and Chaos, vol. 14, no. 1, pp. 347–354, 2004.
[7] Z. M. Ge and C. H. Yang, “Symplectic synchronization of
different chaotic systems,” Chaos, Solitons & Fractals, vol. 40,
no. 5, pp. 2532–2543, 2009.
[8] Z. M. Ge and C. H. Yang, “The generalized synchronization of a
Quantum-CNN chaotic oscillator with different order systems,”
Chaos, Solitons & Fractals, vol. 35, no. 5, pp. 980–990, 2008.
[9] Z.-M. Ge and C.-H. Yang, “Synchronization of complex chaotic
systems in series expansion form,” Chaos, Solitons & Fractals,
vol. 34, no. 5, pp. 1649–1658, 2007.
[10] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic
systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824,
1990.
[11] L. Stenflo, “Generalized Lorenz equations for acoustic-gravity
waves in the atmosphere,” Physica Scripta, vol. 53, no. 1, pp. 83–
84, 1996.
[12] U. E. Vincent, “Synchronization of identical and non-identical
4-D chaotic systems using active control,” Chaos, Solitons &
Fractals, vol. 37, no. 4, pp. 1065–1075, 2008.
[13] Z. Liu, “The first integrals of nonlinear acoustic gravity wave
equations,” Physica Scripta, vol. 61, no. 5, p. 526, 2000.
[14] C. Zhou, C. H. Lai, and M. Y. Yu, “Bifurcation behavior of
the generalized Lorenz equations at large rotation numbers,”
Journal of Mathematical Physics, vol. 38, no. 10, pp. 5225–5239,
1997.
[15] S. Banerjee, P. Saha, and A. R. Chowdhury, “Chaotic scenario in
the Stenflo equations,” Physica Scripta, vol. 63, no. 3, pp. 177–180,
2001.
[16] Y. Chen, X. Wu, and Z. Gui, “Global synchronization criteria
for two Lorenz-Stenflo systems via single-variable substitution
control,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 361–369, 2010.
[17] Z.-M. Ge and Y.-S. Chen, “Synchronization of unidirectional
coupled chaotic systems via partial stability,” Chaos, Solitons &
Fractals, vol. 21, no. 1, pp. 101–111, 2004.
[18] C. H. Yang, S. Y. Li, and P. C. Tsen, “Synchronization of chaotic
system with uncertain variable parameters by linear coupling
[21]
[22]
[23]
and pragmatical adaptive tracking,” Nonlinear Dynamics, vol.
70, no. 3, pp. 2187–2202, 2012.
Z. M. Ge and C. C. Chen, “Phase synchronization of coupled
chaotic multiple time scales systems,” Chaos, Solitons & Fractals,
vol. 20, no. 3, pp. 639–647, 2004.
L. M. Tam, J. H. Chen, H. K. Chen, and W. M. Si Tou,
“Generation of hyperchaos from the Chen-Lee system via
sinusoidal perturbation,” Chaos, Solitons & Fractals, vol. 38, no.
3, pp. 826–839, 2008.
K. Sebastian Sudheer and M. Sabir, “Modified function projective synchronization of hyperchaotic systems through OpenPlus-Closed-Loop coupling,” Physics Letters A, vol. 374, no. 1920, pp. 2017–2023, 2010.
M. Sun, L. Tian, and Q. Jia, “Adaptive control and synchronization of a four-dimensional energy resources system with
unknown parameters,” Chaos, Solitons & Fractals, vol. 39, no.
4, pp. 1943–1949, 2009.
Z. Li and X. Zhao, “Generalized function projective synchronization of two different hyperchaotic systems with unknown
parameters,” Nonlinear Analysis: Real World Applications, vol.
12, no. 5, pp. 2607–2615, 2011.
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