CHIN. PHYS. LETT. Vol. 26, No. 5 (2009) 050501
Increasing-order Projective Synchronization of Chaotic Systems with Time
Delay *
MIAO Qing-Ying(苗清影)1** , FANG Jian-An(方建安)2 , TANG Yang(唐漾)2 , DONG Ai-Hua(董爱华)2
2
1
Continuing Education School, Shanghai Jiaotong University, Shanghai 200036
School of Information Science and Technology, Donghua University, Shanghai 201620
(Received 18 November 2008)
This work is concerned with lag projective synchronization of chaotic systems with increasing order. The systems under consideration have unknown parameters and different structures. Combining the adaptive control
method and feedback control technique, we design a suitable controller and parameter update law to achieve
lag synchronization of chaotic systems with increasing order. The result is rigorously proved by the Lyapunov
stability theorem. Moreover, corresponding simulation results are given to verify the effectiveness of the proposed
methods.
PACS: 05. 45. +b
The past two decades have witnessed great development in the synchronization of chaotic systems
since the pioneering work of Pecora and Carroll in
1990.[1−4] There exist many types of synchronization
such as complete synchronization (CS),[6,7] phase synchronization (PS),[8] lag synchronization (LS),[9−12]
generalized synchronization (GS),[13] and projective
synchronization (PS),[14,15,16] Among all the kinds of
chaos synchronization, projective synchronization has
been extensively investigated in recent years. Mainieri
and Rehacek[14] first reported the concept of projective synchronization, in which the drive and the response systems synchronize up to a constant scaling
factor. Hu et al.[17] studied the hybrid projective synchronization of chaotic systems with fully unknown
parameters, in which the drive system and response
system can synchronize up to a constant scaling matrix. Tang et al.[15,16,18] employed some methods such
as nonlinear control method, active control and adaptive approach to achieve generalized projective synchronization. In engineering applications, time-delay
always exists and affects the dynamical behaviors of
chaotic systems. Strictly speaking, it is not reasonable
to require the response system to synchronize the drive
system at exactly the same time. In Refs. [9–13], the
authors discussed the lag synchronization of chaotic
systems.
There is increasing interest in the study of chaotic
synchronization with different structures and different
orders due to its wide existence in biological science
and social science.[19−22] For example, the order of the
thalamic neurons can be different from the hippocampal neurons. One more instance is the synchronization that occurs between heart and lungs, where one
can observe that circulatory and respiratory systems
synchronize with different orders. Hence, the investi-
gation of synchronization of different chaotic systems
with different orders is very important from the perspective of practical application and control theory.
Motivated by the above discussion, in this Letter we study the increasing-order projective synchronization of chaotic systems with time delay. Base on
the Lyapunov stability theorem, a general controller
and parameter update law are proposed to synchronize chaotic systems with increasing order. First, we
give the problem definition of increasing-order synchronization. Then we present the increasing-order
synchronization scheme of chaotic systems as well as
an application of the result.
Throughout this study, 𝑅𝑛 and 𝑅𝑛×𝑚 denote, respectively, the 𝑛-dimensional Euclidean space and the
set of all 𝑛 × 𝑚 real matrices. The superscript T denotes matrix transposition and the notation 𝑋 ≥ 𝑌
(respectively, 𝑋 > 𝑌 ), where 𝑋 and 𝑌 are symmetric
matrices, means that 𝑋 − 𝑌 is positive semi-definite
(respectively, positive definite). Let a vector norm
[︀ ∑︀𝑛
]︀
2 1/2
‖𝑥‖ on 𝑅𝑛 be defined as ‖𝑥‖ =
, so
𝑖=1 𝑥𝑖
‖𝑥‖2 = 𝑥𝑇 𝑥.
Consider that a chaotic system with unknown parameters can be given by
ˆ
𝑥(𝑡)
˙
= 𝑓 (𝑥(𝑡)) + 𝐶(𝑥(𝑡))𝜃,
(1)
where 𝑥(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡))𝑇 ∈ 𝑅𝑛 is the
state vector of the system, 𝑓 : 𝑅𝑛 → 𝑅𝑛 is a continuous vector function, 𝐶 : 𝑅𝑛 → 𝑅𝑛×𝑝 is a matrix
function, and 𝜃ˆ ∈ 𝑅𝑝 is a parameter vector.
Remark 1. Among existing nonlinear dynamical
systems,[23−26] many chaotic and hyper-chaotic systems belong to Eq. (1), such as the Lorenz system,
Genesio system, Chen system and Lü systems.
* Supported by the National Natural Science Foundation of China under Grant No 60874133, the Research Fund for the Doctoral
Program of Higher Education (No 200802550007), and the Key Creative Project of Shanghai Education Community (No 09ZZ66).
** Email: [email protected]
c 2009 Chinese Physical Society and IOP Publishing Ltd
○
050501-1
CHIN. PHYS. LETT. Vol. 26, No. 5 (2009) 050501
The response system is given in the form of
𝑦(𝑡)
˙ = 𝑔(𝑦(𝑡)) + 𝐷(𝑦(𝑡))𝛿ˆ + 𝑈 (𝑡),
(2)
where 𝑦(𝑡) = (𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑚 (𝑡))𝑇 ∈ 𝑅𝑛 is the
state vector, 𝑔 : 𝑅𝑛 → 𝑅𝑛 is a continuous vector function, 𝐷 : 𝑅𝑛 → 𝑅𝑛×𝑑 is a matrix function, 𝛿ˆ ∈ 𝑅𝑑 is
a parameter vector, and 𝑈 (𝑡) is the controller.
If the order of the drive system is lower than that
of the response system, i.e. 𝑛 < 𝑚, synchronization is
attained in increasing order. To achieve increasingorder synchronization, we must create order. One
feasible way is to construct an auxiliary state vector
which is the function of state 𝑥. Since 𝑟 = 𝑚 − 𝑛,
we construct the auxiliary state vector as 𝑥𝑛+1 (𝑡) =
𝜓1 (𝑥(𝑡)), . . . , 𝑥𝑛+𝑟 (𝑡) = 𝜓𝑟 (𝑥(𝑡)). In this way, we can
easily get a new 𝑚-dimensional state vector 𝑥(𝑡) =
(𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡), 𝑥𝑛+1 (𝑡), . . . , 𝑥𝑚 (𝑡))𝑇 . Therefore, the new drive system with time delay 𝑟(𝑡) can
be written as
𝑥˙ 𝑎 (𝑡 − 𝑟(𝑡)) = (1 − 𝑟(𝑡))𝑓
˙
˙
𝑎 (𝑥𝑎 (𝑡 − 𝑟(𝑡))) + (1 − 𝑟(𝑡))
ˆ
· 𝐶𝑎 (𝑥𝑎 (𝑡 − 𝑟(𝑡)))𝜃,
(3)
where 𝑥𝑎 ∈ 𝑅𝑚 , 𝑓𝑎 : 𝑅𝑚 → 𝑅𝑚 , 𝐶𝑎 : 𝑅𝑚 → 𝑅𝑚×𝑝 ,
𝜃ˆ ∈ 𝑅𝑝 , 𝑟(𝑡) is the function of 𝑡, and 𝑟(𝑡) ≥ 0. Hence,
the increasing-order synchronization of Eqs. (1) and
(2) becomes the synchronization of Eq. (3) and Eq. (2)
with the same order 𝑚.
Definition. If the trivial solution of the error system is asymptotically stable in the mean square with
the initial conditions 𝑥0 and 𝑦0 , respectively, i.e.
˙
𝛿˜ = 𝑆 −1 𝐷𝑇 (𝑦(𝑡))𝑒(𝑡),
(8)
ˆ
˜
ˆ
˜
where 𝜃 = 𝜃 − 𝜃, 𝛿 = 𝛿 − 𝛿 are the parameter
identification errors, 𝜉𝑖 > 0 is an arbitrary constant, which guarantees an effective feedback gain,
𝑇 = diag(𝑡1 , 𝑡2 , . . . , 𝑡𝑛 ) > 0 ∈ 𝑅𝑛×𝑛 and 𝑆 =
diag(𝑠1 , 𝑠2 , . . . , 𝑠𝑛 ) > 0 ∈ 𝑅𝑛×𝑛 are adaptive gain matrices.
Proof. The error system between the new drive
system Eq. (3) and response system Eq. (2) can be
written as
𝑒(𝑡)
˙ = 𝑔(𝑦(𝑡)) + 𝐷(𝑦(𝑡))𝛿ˆ − 𝐽(1 − 𝑟(𝑡)𝑓
˙
𝑎 (𝑥(𝑡 − 𝑟(𝑡)))
ˆ
− 𝐽(1 − 𝑟(𝑡)𝐶
˙
(9)
𝑎 (𝑥(𝑡 − 𝑟(𝑡)))𝜃 + 𝑈 (𝑡).
Substituting the controller Eq. (5) into Eq. (9), we can
obtain the new error system
𝑒(𝑡)
˙ = 𝐷(𝑦(𝑡))𝛿 − 𝐽(1 − 𝑟(𝑡)𝐶
˙
𝑎 (𝑥(𝑡 − 𝑟(𝑡)))𝜃 + 𝐾𝑒(𝑡).
(10)
Choose the following nonnegative function as
1
𝑉 (𝑡, 𝑒(𝑡)) = [𝑒𝑇 (𝑡)𝑒(𝑡) + 𝜃𝑇 𝑇 𝜃 + 𝛿 𝑇 𝑆𝛿]
2
𝑛
1 ∑︁ 1
+
(𝑘𝑖 + 𝑙𝑖 )2 ,
2 𝑖=1 𝜉𝑖
where 𝑙𝑖 is a constant to be determined.
[︀
𝑉˙ (𝑡, 𝑥) = 𝑒𝑇 (𝑡) 𝐷(𝑦(𝑡))𝛿−𝐽(1− 𝑟(𝑡))𝐶(𝑥(𝑡
˙
− 𝑟(𝑡)))𝜃
𝑛
∑︁
]︀
+ 𝐾𝑒(𝑡) + 𝜃˙𝑇 𝑇 𝜃 + 𝛿˙ 𝑇 𝑆𝛿 −
(𝑘𝑖 + 𝑙𝑖 )𝑒2𝑖 (𝑡)
𝑖=1
𝑇
= 𝑒 (𝑡)[𝐷(𝑦(𝑡))𝛿 − 𝐽(1 − 𝑟(𝑡))𝐶(𝑥(𝑡
˙
− 𝑟(𝑡)))𝜃]
𝑛
∑︁
−1 𝑇
−
𝑙𝑖 𝑒2𝑖 (𝑡) + [𝐽(1 − 𝑟(𝑡))𝑇
˙
𝐶 (𝑥(𝑡
2
lim ‖𝑒(𝑡)‖ = lim ‖𝑦(𝑡, 𝑦0 )
𝑡→∞
𝑡→∞
− 𝐽𝑥𝑎 (𝑡 − 𝑟(𝑡), 𝑥0 )‖2 ,
(4)
increasing-order synchronization is achieved. Here 𝐽
is a diagonal matrix meaning scaling factor.
Theorem. Equations (1) and (2) realize increasingorder synchronization if the controller is designed as
˜
𝑈 (𝑡) = − 𝑔(𝑦(𝑡))+𝐽(1 − 𝑟(𝑡))𝑓
˙
𝑎 (𝑥(𝑡 − 𝑟(𝑡)))−𝐷(𝑦(𝑡)𝛿
˜
+ 𝐽(1 − 𝑟(𝑡))𝐶
˙
(5)
𝑎 (𝑥(𝑡 − 𝑟(𝑡)))𝜃 + 𝐾𝑒(𝑡),
where 𝜃˜ and 𝛿˜ represent the estimate vectors of unknown parameter vectors 𝜃ˆ and 𝛿ˆ respectively; 𝑟(𝑡)
˙
is the derivate of the variant time delay 𝑟(𝑡), 𝐾 =
diag(𝑘1 , 𝑘2 , . . . , 𝑘𝑛 ) ∈ 𝑅𝑛×𝑛 is the feedback strength
and can be described according to the following update law
𝑘˙ 𝑖 = −𝜉𝑖 𝑒2𝑖 (𝑡),
𝑖 = (1, 2, . . . , 𝑛).
(6)
The parameter update law of unknown parameters is
chosen as
˙
−1 𝑇
𝜃˜ = −𝐽(1 − 𝑟(𝑡))𝑇
˙
𝐶 (𝑥(𝑡 − 𝑟(𝑡)))𝑒(𝑡),
(11)
(7)
𝑖=1
− 𝑟(𝑡)))𝑒(𝑡)]𝑇 𝑇 𝜃 + [−𝑆 −1 𝐷𝑇 (𝑦(𝑡))𝑒(𝑡)]𝑇 𝑆𝛿
= − 𝑒𝑇 (𝑡)𝐿𝑒(𝑡),
(12)
where 𝐿 = diag(𝑙1 , 𝑙2 , . . . , 𝑙𝑛 ) ∈ 𝑅𝑛×𝑛 is a constant diagonal matrix. Equations (1) and (2) attain
increasing-order synchronization, and the unknown
ˆ 𝛿˜ → 𝛿 when 𝑡 → ∞. This ends
parameters 𝜃˜ → 𝜃,
the proof.
Corollary 1. If the parameters 𝛿ˆ in the response
system are known a priori, the controller can be designed as follows:
𝑈 (𝑡) = − 𝑔(𝑦(𝑡)) + 𝐽(1 − 𝑟(𝑡))𝑓
˙
(𝑥(𝑡 − 𝑟(𝑡)))
− 𝐷(𝑦(𝑡))𝛿ˆ + 𝐽(1 − 𝑟(𝑡))𝐶(𝑥(𝑡
˙
− 𝑟(𝑡)))𝜃˜
+ 𝐾𝑒(𝑡).
(13)
Moreover, the coupling strength 𝐾 is updated according to Eq. (6) and the parameter is updated according
to Eq. (7). The drive system equation (1) and the corresponding system equation (2) can be synchronized
in the sense of increasing-order synchronization.
050501-2
CHIN. PHYS. LETT. Vol. 26, No. 5 (2009) 050501
Corollary 2. If the parameters 𝜃ˆ in the drive system are known a priori, the controller can be designed
as follows:
𝑈 (𝑡) = −𝑔(𝑦(𝑡))+𝐽(1− 𝑟(𝑡))𝑓
˙
(𝑥(𝑡 − 𝑟(𝑡)))−𝐷(𝑦(𝑡))𝛿˜
+ 𝐽(1 − 𝑟(𝑡))𝐶(𝑥(𝑡
˙
− 𝑟(𝑡)))𝜃ˆ + 𝐾𝑒(𝑡).
(14)
Moreover, the coupling strength 𝐾 is updated according to Eq. (6) and the parameter is updated to
Eq. (8). The drive system equation (1) and the system equation (2) can be synchronized in the sense of
increasing-order synchronization.
Remark 2. In Ref. [17], the authors study the projective synchronization of chaotic systems with different orders. However, in this study we consider time
delay in dynamical systems. This is more common in
real life.
Remark 3. In Ref. [15], the authors research lag
hybrid projective synchronization with unknown parameters and give a detailed discussion of a lag synchronization scheme. Here, we take into account dynamical systems with increasing order, which are more
general in nonlinear dynamical systems and chaos secure communication.
Next, we achieve increasing-order synchronization
of the Genesio system and generalized Lorenz system.
The drive system is the Genesio system and can be
written as
One can easily obtain 𝐺1 = diag(|𝑎1 |, |𝑎2 |, |𝑎3 |). Since
the order of the drive system is 3 less than that of the
response system, we need to construct a state variable
𝑥4 (𝑡). Here we choose two cases to study.
(1) If 𝑥4 = 𝑥1 + 𝑥2 + 𝑥3 , then 𝑥˙ 4 = 𝑥˙ 1 + 𝑥˙ 2 + 𝑥˙ 3 =
𝑥2 + 𝑥3 − 𝑎
ˆ𝑥1 − ˆ𝑏𝑥2 − 𝑐ˆ𝑥3 + 𝑥21 .
The drive system can be rewritten as
𝑥˙ 1 = 𝑥2 , 𝑥˙ 2 = 𝑥3 , 𝑥˙ 3 = −ˆ
𝑎𝑥1 − ˆ𝑏𝑥2 − 𝑐ˆ𝑥3 + 𝑥21 ,
𝑥˙ 4 = 𝑥2 + 𝑥3 − 𝑎
ˆ𝑥1 − ˆ𝑏𝑥2 − 𝑐ˆ𝑥3 + 𝑥2 ,
(21)
1
The Lyapunov exponents of Eq. (15) are 5.9204,
−3.1142, −4.0006. The new drive system Eq. (21)
has different Lyapunov exponents 0.7661, 0.2357,
−0.2489, −1.9486. With two positive Lyapunov exponents the Genesio system has changed into a hyperchaotic system. The increasing-order synchronization
of system Eq. (15) and system Eq. (16) has changed
into the same-order synchronization of hyper-chaotic
system Eq. (21) and chaotic system Eq. (16).
According to Eqs. (5)–(8), we obtain the controller
𝑢1 = 𝑗1 𝑥2 − 𝜎
˜ (𝑦2 − 𝑦1 ) + 𝛾˜ 𝑦4 + 𝑘1 𝑒1 ,
𝑢2 = 𝑦1 𝑦3 + 𝑦2 + 𝑗2 𝑥3 − 𝜌˜𝑦1 + 𝑘2 𝑒2 ,
˜ 3 − 𝑗3 𝑎
𝑢3 = − 𝑦1 𝑦2 + 𝑗3 𝑥2 + 𝛽𝑦
˜ 𝑥1
1
− 𝑗3˜𝑏𝑥2 − 𝑗3 𝑐˜𝑥3 + 𝑘3 𝑒3 ,
𝑥˙ 1 = 𝑥2 , 𝑥˙ 2 = 𝑥3 , 𝑥˙ 3 = −ˆ
𝑎𝑥1 − ˆ𝑏𝑥2 − 𝑐ˆ𝑥3 + 𝑥21 , (15)
where 𝑎
ˆ, ˆ𝑏, 𝑐ˆ are the system original parameters, and
the initial values are 𝑎
ˆ = 6, ˆ𝑏 = 2.92, 𝑐ˆ = 1.2.
The generalized Lorenz system is taken as the response system which can be described as follows:
𝑢4 = 𝑦1 + 𝑗(𝑥21 + 𝑥2 + 𝑥3 ) + 𝜎
˜ 𝑦4 − 𝑗˜
𝑎𝑥1
˜
− 𝑗 𝑏𝑥2 − 𝑗˜
𝑐𝑥3 + 𝑘4 𝑒4 ,
ˆ 𝛾ˆ obey the updating
and the estimates 𝑎
ˆ, ˆ𝑏, 𝑐ˆ, 𝜎
ˆ , 𝜌ˆ, 𝛽,
law
𝑦˙ 1 = 𝜎
ˆ (𝑦2 − 𝑦1 ) + 𝛾ˆ 𝑦4 + 𝑢1 ,
𝑎
˜˙ = 𝑡1 𝑗1 (𝑥1 𝑒3 + 𝑥1 𝑒4 ),
˜𝑏˙ = 𝑡2 𝑗2 (𝑥2 𝑒2 + 𝑥2 𝑒4 ),
𝑐˜˙ = 𝑡3 𝑗3 (𝑥3 𝑒3 + 𝑥3 𝑒4 ),
𝑦˙ 2 = 𝜌ˆ𝑦1 − 𝑦1 𝑦3 − 𝑦2 + 𝑢2 ,
ˆ 3 + 𝑢3 ,
𝑦˙ 3 = 𝑦1 𝑦2 − 𝛽𝑦
𝑦˙ 4 = − 𝑦1 − 𝜎
ˆ 𝑦4 + 𝑢4 ,
(16)
𝜎
˜˙ = 𝑠1 𝑒1 (𝑦2 − 𝑦1 ) − 𝑠1 𝑒4 𝑦4 ,
˙
𝛽˜ = −𝑠3 𝑦3 𝑒3 , 𝛾˜˙ = 𝑠4 𝑦4 𝑒1 .
ˆ 𝛾ˆ are the system original parameters.
where 𝜎
ˆ , 𝜌ˆ, 𝛽,
The system Eq. (16) can be written as
𝑦˙ = 𝑔(𝑦) + 𝐷(𝑦)ˆ(𝛿) + 𝑈,
where
⎞
0
⎜ −𝑦 𝑦 − 𝑦2 ⎟
𝑔(𝑦) = ⎝ 1 3
⎠,
𝑦1 𝑦2
−𝑦1
⎛𝑦 − 𝑦
0
𝑦
(17)
⎛
2
1
0
0
−𝑦4
⎛𝑢 ⎞
⎜
𝐷(𝑦) = ⎝
1
⎜𝑢 ⎟
𝑈 = ⎝ 2⎠,
𝑢3
𝑢4
4
𝑦1
0
0
(18)
⎞
0
0⎟
⎠,
−𝑦3 0
0
0
⎛𝜎
⎞
ˆ
⎜ 𝜌ˆ ⎟
𝛿ˆ = ⎝ ˆ ⎠ ,
𝛽
𝛾ˆ
(19)
(20)
(22)
(23)
𝜌˜˙ = 𝑠2 𝑦1 𝑒2 ,
(24)
In our simulations, the Euler-Maruyama numerical
scheme is used to solve the systems of differential
equations (15) and (16) with time step 0.01. The parameters of drive and response systems are selected
as 𝑎
ˆ = 10, ˆ𝑏 = 28, 𝑐ˆ = 8/3, 𝜎
ˆ = 1, 𝜌ˆ = 26,
ˆ
𝛽 = 0.7, 𝛾ˆ = 1.5, so the two systems display chaotic
behavior. The initial values of the drive and response
systems are 𝑥1 (0) = −1, 𝑥2 (0) = 2, 𝑥3 (0) = −1,
𝑥4 (0) = 2 and 𝑦1 (0) = −1, 𝑦2 (0) = 1, 𝑦3 (0) = −1,
𝑦4 (0) = −1, respectively. The initial states of feedback strength and unknown parameters are given as
𝑘1 (0) = −2, 𝑘2 (0) = −3, 𝑘3 (0) = −2, 𝑘4 (0) = −4 and
𝑎
˜(0) = 1, ˜𝑏(0) = 1, 𝑐˜(0) = 1, 𝜎
˜ (0) = 1, 𝜌˜(0) = 1,
˜
𝛽(0)
= 1, 𝛾˜ (0) = 1. Moreover, we choose propagation delay 𝑟(𝑡) = 10(1 + sin(𝑡)) and 𝑇 = diag(1, 1, 1),
050501-3
CHIN. PHYS. LETT. Vol. 26, No. 5 (2009) 050501
𝑆 = diag(1, 1, 1, 1). The scaling matrix is chosen as
𝐽 = diag(3, 3, 3, 3). Figure 1 shows the increasingorder synchronization between systems Eq. (21) and
Eq. (16) with the state variable 𝑥4 = 𝑥1 + 𝑥2 + 𝑥3 .
(2) If 𝑥4 = 𝑥21 , then 𝑥˙ 4 = 2𝑥1 𝑥˙ 1 = 2𝑥1 𝑥2 .
The drive system is
− 𝑗3˜𝑏𝑥2 − 𝑗3 𝑐˜𝑥3 + 𝑘3 𝑒3 ,
𝑢4 = 𝑦1 + 2𝑗𝑥1 𝑥2 + 𝜎
˜ 𝑦4 + 𝑘4 𝑒4 ,
˜ 𝛾˜ obey the updating
and the estimates 𝑎
˜, ˜𝑏, 𝑐˜, 𝜎
˜ , 𝜌˜, 𝛽,
law
˙
𝑎
˜˙ = 𝑡1 𝑗1 𝑥1 𝑒3 , ˜𝑏 = 𝑡2 𝑗2 𝑥2 𝑒3 , 𝑐˜˙ = 𝑡3 𝑗3 𝑥3 𝑒3 ,
𝜎
˜˙ = 𝑠1 (𝑦2 − 𝑦1 )𝑒1 − 𝑠1 𝑦4 𝑒4 , 𝜌˜˙ = 𝑠2 𝑦1 𝑒2 ,
˙
𝛽˜ = −𝑠3 𝑦3 𝑒3 , 𝛾˜˙ = 𝑠4 𝑦4 𝑒1 .
𝑥˙ 1 = 𝑥2 ,
𝑥˙ 2 = 𝑥3 ,
𝑥˙ 3 = −ˆ
𝑎𝑥1 − ˆ𝑏𝑥2 − 𝑐ˆ𝑥3 + 𝑥21 ,
𝑥˙ 4 = 2𝑥1 𝑥2 .
(25)
The Lyapunov exponents of system Eq. (25) are
0.6714, 0.0525, −0.0076, −1.9118. With two positive Lyapunov exponents, system Eq. (25) is a hyperchaotic system.
e3
e2
e1
20
10
0
-10
20
10
0
-10
20
e4
(28)
In the simulation, all the parameters and initial
values are the same as the above one. Figure 2
shows increasing-order synchronization between systems Eq. (25) and Eq. (16) with the state variable
𝑥4 = 𝑥21 .
In summary, we have studied lag projective synchronization of chaotic systems with increasing order.
By combining the adaptive control method and feedback control approach, a controller and parameter update law are designed to achieve synchronization. Corresponding simulation examples are given to illustrate
the effectiveness of the proposed method.
0
References
10
0
-10
0
10
20
ts
30
40
50
Fig. 1. Time evolution of increasing-order synchronization errors with the state variable 𝑥4 = 𝑥1 + 𝑥2 + 𝑥3 .
20
e2
10
0
10
20
10
0
10
20
e3
e1
(27)
10
-10
20
10
0
10
20
e4
(26)
10
0
10
0
10
20
ts
30
40
50
Fig. 2. Time evolution of increasing-order synchronization errors with the state variable 𝑥4 = 𝑥21 .
According to Eqs. (5)–(8), we obtain the controller
𝑢1 = 𝑗1 𝑥2 − 𝜎
˜ (𝑦2 − 𝑦1 ) − 𝛾˜ 𝑦4 + 𝑘1 𝑒1 ,
𝑢2 = 𝑦1 𝑦3 + 𝑦2 + 𝑗2 𝑥3 − 𝜌˜𝑦1 + 𝑘2 𝑒2 ,
˜ 3 − 𝑗3 𝑎
𝑢3 = − 𝑦1 𝑦2 + 𝑗3 𝑥2 + 𝛽𝑦
˜ 𝑥1
1
[1] Perora L M and Carroll T L 1990 Phys. Rev. Lett. 64 821
[2] Boccaletti S, Kurths J, Osipov G, Valladares D L and Zhou
C S 2002 Phys. Rep. 1 366
[3] Nijmeijer H 2001 Phys. D 154 219
[4] Maritan A and Banavar J 1994 Phys. Rev. Lett. 72 1451
[5] Wang X F, Wang Z Q and Chen G 1999 Int. J. Bifur.
Chaos 9 1169
[6] Kocarev L and Parlitz U 1996 Phys. Rev. Lett. 76 1816
[7] Yang X S and Chen G 2002 Chaos Solitons Fractals 13
1303
[8] Rosenblum M G, Pikovsky A S and Kurths J 1997 Phys.
Rev. Lett. 78 4193
[9] Li D M, Wang Z D, Zhou J, Fang J A and Ni J J 2008
Chaos Solitons Fractals 4 1217
[10] Yu W W and Cao J D 2007 Physica. A 375 467
[11] Zhang Q J and Lu J A 2008 Phys. Lett. A 372 1416
[12] Li C D, Liao X F and Wong K W 2004 Physica. D 194 187
[13] Senthilkumar D V and Lakshmanan M 2007 Chaos 17
[14] Mainieri R and Rehacek J 1999 Phys. Rev. Lett. 82 3042
[15] Tang Y and Fang J A 2008 Phys. Lett. A 372 1816
[16] Tang Y, Qiu R H, Fang J A and Miao Q Y, 2008 Phys.
Lett. A 372 4425
[17] Hu M F, Xu Z Y and Zhang R 2008 Nonlinear Sci. Numer.
Simulat. 13 456
[18] Tang Y, Fang J A and Miao Q Y 2009 Neurocomputing 72
1694
[19] Femat R and Solis-Perales G 2002 Phys. Rev. E. 65 036226
[20] Chen J and Liu Z R 2003 Chin. Phys. Lett. 20 1441
[21] Bowong S and McClintock PVE 2006 Phys. Lett. A 358
134
[22] Ho M C, Hung Y C, Liu Z Y and Jiang I M 2006 Phys.
Lett. A 348 251
[23] Genesio R and Tesi A 1992 Automatica 28 531
[24] Lorenz E N 1963 Atmos J. Sci. 20 130
[25] Bishop S R and Clifford M J 1996 J. Sound Vib. 189 142
[26] Chen G and Ueta T 1999 Int. J. Bifur. Chaos 9 1465
050501-4