Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 768587, 11 pages
doi:10.1155/2012/768587
Research Article
Hybrid Projective Synchronization for Two
Identical Fractional-Order Chaotic Systems
Ping Zhou,1, 2 Rui Ding,2 and Yu-xia Cao2
1
Research Center for System Theory and Applications, Chongqing University of Posts
and Telecommunications, Chongqing 400065, China
2
Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education,
Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Correspondence should be addressed to Ping Zhou, zhouping@cqupt.edu.cn
Received 23 May 2012; Accepted 27 June 2012
Academic Editor: Vimal Singh
Copyright q 2012 Ping Zhou et al. 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 hybrid projective synchronization scheme for two identical fractional-order chaotic systems
is proposed in this paper. Based on the stability theory of fractional-order systems, a controller
for the synchronization of two identical fractional-order chaotic systems is designed. This
synchronization scheme needs not to absorb all the nonlinear terms of response system. Hybrid
projective synchronization for the fractional-order Chen chaotic system and hybrid projective
synchronization for the fractional-order hyperchaotic Lu system are used to demonstrate the
validity and feasibility of the proposed scheme.
1. Introduction
Most recently, many authors begin to investigate the chaotic dynamics and synchronization
for fractional-order dynamical systems 1–6. Chaos synchronization of the fractional-order
systems is just beginning to attract some attention due to its potential applications in
secure communications and control processing 7–12. Several types of chaos synchronization
are well known, which include complete synchronization CS, antisynchronization AS,
phase synchronization, generalized synchronization GS, projective synchronization PS,
and modified projective synchronization MPS. Among all patterns of synchronization,
the most noticeable one may be projective synchronization PS, which was first studied
by Mainieri and Rehacek 13. Projective synchronization PS has been extensively
considered because it can obtain faster communication. The drive and response system could be synchronized up to a scaling factor in projective synchronization. In
application to secure communications, this proportional feature can be used to extend
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Discrete Dynamics in Nature and Society
binary digital to M-nary digital communication for getting faster communication 14,
15.
However, most of projective synchronizations for the fractional-order systems
have concentrated on studying the same scaling factor 16–19, and some projective
synchronization schemes 16, 17 are suitable for a class of fractional-order systems or for
some special fractional-order systems 19, and all the nonlinear terms of response system
was absorbed in some previous works. Moreover, in order to increase the degree of secrecy
for secure communications, the same scaling factor in PS can be replaced by vector function
factor. Motivated by the above discussions, we propose a hybrid projective synchronization
HPS scheme for two identical fractional-order chaotic systems in this paper. Hybrid
projective synchronization HPS is a more general definition of projective synchronization,
in which the drive system and response system could be synchronized up to a vector function
factor. HPS is different from the PS. Furthermore, HPS could be used to get more secure
communication than PS in application to secure communications, because it is obvious that
the unpredictability of the vector function factor in HPS is more than that of the same scaling
factor in PS. The main contribution of this paper is as follows: the HPS scheme in this paper
is suitable for a large number of fractional-order systems not for special fractional-order
system, and this HPS scheme needs not to absorb all the nonlinear terms of response system.
This is different from some previous works 16, 17, 19–21.
To illustrate the effectiveness of the proposed HPS scheme in this paper, the HPS for
the fractional-order Chen system and HPS for the fractional-order hyperchaotic Lu system
are investigated. Numerical simulations are used to demonstrate the effectiveness of the
proposed schemes. The organization of this paper is as follows. In Section 2, the definition
of HPS is given, and a HPS scheme for two identical fractional-order chaotic systems is
presented. In Section 3, two groups of examples are used to verify the effectiveness of the
proposed scheme. The conclusion is finally drawn in Section 4.
2. Hybrid Projective Synchronization Scheme
There are several definitions of fractional derivatives. In this paper, the Caputo-type fractional
derivative defined will be used. The Caputo definition of the fractional derivative, which is
sometimes called smooth fractional derivative, is described as
dq ft
1
q
dt
Γ m−q
t
0
f m τ
t − τq1−m
dτ,
m − 1 < q < m,
2.1
where 0 < q ≤ 1 is fractional order and dq /dtq denote the Caputo definition of the fractional
derivative. m is the smallest integer larger than q, and f m t is the m-order derivative in the
usual sense. Γ• is the gamma function.
The fractional-order chaotic drive and response systems can be written as follows,
respectively:
dq x
fx,
dtq
2.2
dq y
g y Ω x, y ,
q
dt
2.3
Discrete Dynamics in Nature and Society
3
where x ∈ Rn , y ∈ Rn are state vectors of the drive system 2.2 and the response system
2.3 and f, g : Rn → Rn are continuous vector functions, respectively. Ωx, y is a vector
controller.
Definition 2.1. For the drive system 2.2 and the response system 2.3, it is said to be
hybrid projective synchronization HPS if there exists an n × n reversible matrix A such
that
lim Ay − x 0,
t→∞
2.4
where · is the Euclidean norm.
Remark 2.2. If A I, and I is a unit matrix, then this synchronization is called complete
synchronization CS; if A −I, then this synchronization is called antisynchronization AS;
if A aI, and a / ± 1 is a nonzero real constant, then this synchronization is called projective
synchronization PS; if A diaga1 , a2 , . . . , an , and a1 , a2 , . . . , an are not the same nonzero
constant, then this synchronization is called modified projective synchronization MPS.
Therefore, CS, AS, PS, and MPS are the special cases of the hybrid projective synchronization
scheme HPS in this paper.
In order to realize HPS for the fractional-order chaotic system 2.2, we take the
fractional-order chaotic system 2.2 as drive system and construct a response system as
follows:
dq y
A−1 f Ay Ω x, y ,
q
dt
2.5
where A−1 is the reverse matrix of the reversible matrix A, y ∈ Rn are state vector of the
response system 2.5, and Ωx, y is a controller which will be designed.
Define the HPS errors between the response system 2.5 and the drive system 2.2 as
e Ay − x,
2.6
where
e e1 , e2 , . . . , en T ,
⎞
n
ei ⎝ aij yj ⎠ − xi
⎛
i, j 1, 2, . . . n .
2.7
j1
Let
f Ay − fx Fx, e.
2.8
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Discrete Dynamics in Nature and Society
Now, we assume that the errors vector e can be divided into eα es1 , . . . , esm T and
eβ esm1 , . . . , esn T , such that Fx, e has the form of
Fx, e Bα eα h1 x, eα , eβ
,
Bβ eβ h21 x, eα , eβ h22 x, eα , eβ
2.9
where h1 x, eα , eβ ∈ Rm , h21 x, eα , eβ ∈ Rn−m , h22 x, eα , eβ ∈ Rn−m and limeα → 0 h21 x, eα , eβ 0, respectively. Bα ∈ Rm×m and Bβ ∈ Rn−m×n−m are real constant matrix.
Rewrite the controller Ωx, y as follows:
μα x, e
,
Ω x, y μx, e μβ x, e
2.10
where μα x, e ∈ Rm and μβ x, e ∈ Rn−m , respectively.
Now, Theorem 2.3 is given based on the previously mentioned conditions in order to
achieve the HPS between the drive system 2.2 and the response system 2.5.
Theorem 2.3. Choose the following controller:
Ω x, y μx, e μα x, e
μβ x, e
Λα eα − h1 x, eα , eβ
,
Λβ eβ − h22 x, eα , eβ
2.11
where Λα ∈ Rm×m and Λβ ∈ Rn−m×n−m are suitable constant matrix, respectively.
If all the eigenvalues of Bα Λα satisfy | arg λi | > 0.5πq i 1, 2, . . . , m and all
the eigenvalues of Bβ Λβ satisfy | arg λj | > 0.5πq j 1, 2, . . . , n − m, then hybrid
projective synchronization between the drive system 2.2 and the response system 2.5 can
be achieved.
Proof. According to the drive system 2.2 and the response system 2.5, the error dynamic
system of hybrid projective synchronization can be obtained as follows:
dq e Adq y dq x
− q f Ay − fx Ω x, y Fx, e μx, e.
q
q
dt
dt
dt
2.12
According to 2.9 and 2.10, the error dynamic system 2.12 can be rewritten as
d q eα
Bα eα h1 x, eα , eβ μα x, e,
dtq
d q eβ
Bβ eβ h21 x, eα , eβ h22 x, eα , eβ μβ x, e,
q
dt
2.13
Discrete Dynamics in Nature and Society
5
because
μα x, e
μβ x, e
Λα eα − h1 x, eα , eβ
.
Λβ eβ − h22 x, eα , eβ
2.14
So,
d q eα
Bα Λα eα ,
dtq
d q eβ Bβ Λβ eβ h21 x, eα , eβ .
q
dt
2.15
Because all the eigenvalues of Bα Λα satisfy | arg λi | > 0.5πq i 1, 2, . . . , m,
according to the stability theory of linear fractional-order systems 22, the equilibrium point
ei 0 i 1, 2, . . . , m in the first equation of system 2.15 is asymptotically stable, which
indicates limt → ∞ eα 0.
Since limt → ∞ eα 0 and limeα → 0 h21 x, eα , eβ 0, therefore when time t → ∞, the
second equation of system 2.15 can be changed as
d q eβ Bβ Λβ eβ ,
q
dt
2.16
because all the eigenvalues of Bβ Λβ satisfy | arg λj | > 0.5πq j 1, 2, . . . , n − m. According
to the stability theory of linear fractional-order systems 22, the equilibrium point ei 0 i 1, 2, . . . , n − m of system 2.16 is asymptotically stable, which indicates limt → ∞ eβ 0.
According to limt → ∞ eα 0 and limt → ∞ eβ 0, the hybrid projective synchronization
between the drive system 2.2 and the response system 2.5 can be achieved. This finishes
the proof.
Remark 2.4. In order to use the stability theory
fractional-order
systems 22, the
Λ eof−hlinear
α α
1 x,eα ,eβ controller Ωx, y or μx, e are chosen as Λβ eβ −h22 x,eα ,eβ . Moreover, the nonlinear term
h21 x, eα , eβ ∈ Rn−m in the error dynamic system 2.13 or response system 2.5 is preserved;
this is different from some previous works 17, 19–21 which need to absorb all the nonlinear
terms of response system or error dynamic system.
Remark 2.5. For the complex fractional-order multiscroll chaotic systems 23–25 and the
complex dynamical network or the small-world dynamical networks 26–28, the hybrid
projective synchronization would be much more complex. Further work on this issue is an
ongoing research topic in our group.
3. Applications
In order to illustrate the effectiveness of the proposed hybrid projective synchronization
scheme obtained in Section 2, two examples are considered in this section, which are HPS for
the fractional-order Chen system and HPS for the fractional-order hyperchaotic Lu system.
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Discrete Dynamics in Nature and Society
20
x2
x3
40
20
0
20
0
−20
x2
20
0
−20
−20
0
−20
x1
0
20
x1
a
b
Figure 1: Chaotic attractor of fractional-order Chen chaotic system 3.1 for q 0.9.
3.1. HPS for the Fractional-Order Chen System
The fractional-order Chen system 23 is described as follows:
dq x1
35x2 − x1 ,
dtq
dq x2
−7x1 28x2 − x1 x3 ,
dtq
3.1
dq x3
x1 x2 − 3x3 .
dtq
Tavazoei and Haeri pointed out that fractional-order Chen system 3.1 exhibits
chaotic behavior for q ≥ 0.83 29. The chaotic attractor of fractional-order Chen system for
q 0.9, is depicted in Figure 1.
According to the HPS scheme presented in the previous section, the response system
is described by
⎞
⎞
⎤
⎡⎛
3
3
dq y1
35
a2j yj − a1j yj
⎟
⎜ dtq ⎟
⎥
⎢⎜
⎟
⎟
⎜
⎥
⎢⎜
j1
j1
⎟
⎜ q ⎟
⎥
⎢⎜
3
3
3
3
⎜
⎟
⎜ d y2 ⎟
⎥
⎢
⎟ A−1 ⎢⎜−7 × a1j yj 28 × a2j yj − a1j yj × a3j yj ⎟ Ω x, y ⎥,
⎜
⎟
⎜ dtq ⎟
⎥
⎢⎜
j1
j1
j1
j1
⎟
⎟
⎜
⎥
⎢⎜
⎟
⎜ q ⎟
⎥
⎢⎜
3
3
3
⎠
⎝ d y3 ⎠
⎦
⎣⎝
a1j yj × a2j yj − 3 × a3j yj
dtq
j1
j1
j1
⎛
3.2
a11 a12 a13 where A aa21 aa22 aa23 is a reversible matrix and A−1 is its reverse matrix.
31 32 33
Now, the term of fAy − fx Fx, e is yielded firstly:
⎞
35e2 − 35e1
f Ay − fx Fx, e ⎝−7e1 28e2 − x3 e1 − x1 e3 − e1 e3 ⎠.
x2 e1 x1 e2 e1 e2 − 3e3
⎛
3.3
Discrete Dynamics in Nature and Society
7
6
4
ε
2
0
0
5
10
15
20
t
Figure 2: The HPS result between the drive system 3.1 and its response system 3.2 for q 0.9.
So, we can choose eα e1 and eβ e2 , e3 T . Therefore, h1 x, eα , eβ −35e2 ,
1 e3 28 0 3 e1 −e1 e3
h21 x, eα , eβ −7ex12−x
, h22 x, eα , eβ −x
x1 e2 , Bα −35, and Bβ e1 e1 e2
0 −3 , respectively.
Obviously, limeα → 0 h21 x, eα , eβ 0.
According to Theorem 2.3 in Section 2, the controller Ωx, y can be chosen as follows:
Ω x, y μα x, e
μβ x, e
Λα eα − h1 x, eα , eβ
,
Λβ eβ − h22 x, eα , eβ
3.4
1 e3 1×1
where μα x, e Λα e1 − 35e2 , μβ x, e Λβ ee23 − −x
, and Λβ ∈ R2 × 2 ,
x1 e2 , Λα ∈ R
1×1
2×2
and Λβ ∈ R
. If the eigenvalues
respectively. Therefore, choose suitable matrix Λα ∈ R
of Bα Λα satisfy | arg λ| > 0.5πq and if all the eigenvalues of Bβ Λβ satisfy | arg λj | > 0.5πq
j 1, 2, then hybrid projective synchronization between the drive system 3.1 and its
response system 3.2 can be achieved.
For example, choose reversible matrix
⎛
−2 2
0
⎞
⎟
⎜
⎟
A⎜
⎝−1 1 1 ⎠,
0 1 −0.5
Λα 30,
Λβ −24 10
−5
0
.
3.5
So, the eigenvalues of Bα Λα are −5, and all the eigenvalues of Bβ Λβ are 0.5 ±
6.1441 j and | arg λBβ Λβ | 0.9483 × π/2 > 0.5πq q 0.9, respectively. Therefore, the
hybrid projective synchronization between drive system 3.1 and its response system 3.2
can be achieved. The corresponding numerical result is shown in Figure 2, in which the initial
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Discrete Dynamics in Nature and Society
60
100
x4
x3
40
20
0
0
−100
20
0
x2 −20
−40 −30 −20
10
−10 0
x1
20
40
x3
a
−10 0
0 −30 −20
x2
20
10
20
b
Figure 3: Chaotic attractor of fractional-order hyperchaotic Lu system 3.6 for q 0.95.
conditions are x10 , x20 , x30 3, 4, 5 for the drive system 3.1, and y10 , y20 , y30 8, 10,
4 for the response system 3.2, respectively, and ε 3i1 ei2 1/2 .
3.2. HPS for Fractional-Order Hyperchaotic Lǔ System
Min et al. reported a fractional-order hyperchaotic Lu system 30, 31 based on the
hyperchaotic Lu system, which is described as follows:
dq x1
36x2 − x1 x4 ,
dtq
dq x2
20x2 − x1 x3 ,
dtq
dq x3
x1 x2 − 3x3 ,
dtq
3.6
dq x4
1.3x4 x1 x3 .
dtq
The chaotic attractor of fractional-order hyperchaotic Lu system for q 0.95, is shown
in Figure 3.
Taking system 3.6 as the drive system, according to the HPS scheme presented in
Section 2, the response system is described by
⎞
⎞
⎤
⎡⎛
dq y1
4
4
4
⎜ dtq ⎟
a2j yj − a1j yj a4j yj ⎟
⎥
⎢⎜36 ×
⎟
⎜
⎥
⎟
⎢⎜
j1
j1
j1
⎟
⎜
⎥
⎟
⎢⎜
⎜ dq y2 ⎟
⎥
⎜
⎟
⎢
4
4
4
⎟
⎜
⎥
⎟
⎢⎜
⎜ q ⎟
20
×
a
y
−
a
y
×
a
y
⎥
⎜
⎟
⎢
2j
j
1j
j
3j
j
⎜ dt ⎟
⎜
⎟
⎥
⎢
j1
j1
j1
⎟
⎜
⎜
⎟ Ω x, y ⎥,
⎜ q ⎟ A−1 ⎢
⎜
⎟
⎥
⎢
4
4
4
⎜ d y3 ⎟
⎟
⎥
⎢⎜ ⎟
⎜
a
y
×
a
y
−
3
×
a
y
⎜
⎟
⎥
⎢
1j
j
2j
j
3j
j
⎜ dtq ⎟
⎟
⎥
⎢⎜ j1
⎟
⎜
j1
j1
⎟
⎥
⎢⎜
⎟
⎜
⎜
⎟
⎥
⎢
4
4
4
⎜ dq y ⎟
⎠
⎦
⎣⎝
⎝
4⎠
1.3 × a4j yj a1j yj × a3j yj
q
j1
j1
j1
dt
⎛
3.7
Discrete Dynamics in Nature and Society
9
a11 a12 a13 a14 a22 a23 a24
where A aa21
is a reversible matrix and A−1 is its reverse matrix. As the same as
31 a32 a33 a34
a41 a42 a43 a44
what is mentioned previously, we can obtain
⎛
⎞
36e2 − 36e1 e4
⎜
⎟
⎜ 20e2 − x3 e1 − x1 e3 − e1 e3 ⎟
⎜
⎟
f Ay − fx Fx, e ⎜
⎟.
⎜ x2 e1 x1 e2 e1 e2 − 3e3 ⎟
⎝
⎠
3.8
1.3e4 x3 e1 x1 e3 e1 e3
T
Now, we
eβ e3 , e4 T . Therefore,
h1x, eα ,eβ 1 , e2 and
can choose eα x2 ee
e4
0
−36 36 , and
1 x1 e2 e1 e2
x,
e
,
e
x,
e
,
e
,
h
,
h
,
B
21
α β
22
α β
α
−x3 e1−x1 e3 −e1e3
x3 e1 e1 e3
x1 e3
0 20
0
Bβ −3
0 1.3 , respectively. Obviously, limeα → 0 h21 x, eα , eβ 0.
According to Theorem 2.3 in Section 2, the controller Ωx, y can be chosen as follows:
Ω x, y μα x, e
μβ x, e
Λα eα − h1 x, eα , eβ
,
Λβ eβ − h22 x, eα , eβ
3.9
where μα x, e Λα ee12 − −x3 e1 −xe14e3 −e1 e3 , μβ x, e Λβ ee23 − x10e3 , Λα ∈ R2 × 2 and
Λβ ∈ R2 × 2 , respectively. Therefore, choose suitable matrix Λα ∈ R2 × 2 , and Λβ ∈ R2 × 2 . If
the eigenvalues of Bα Λα satisfy | arg λi | > 0.5πq i 1, 2 and if all the eigenvalues of
Bβ Λβ satisfy | arg λj | > 0.5πq j 1, 2, then the hybrid projective synchronization between
the drive system 3.6 and its response system 3.7 can be achieved.
For example, choose reversible matrix
⎛
0.5 0 0
⎜
⎜ 0 −2 2
⎜
A⎜
⎜ 0 −1 1
⎝
0
0
⎞
⎟
0 ⎟
⎟
⎟,
1 ⎟
⎠
0 1 −0.5
Λα 0
0
0 −30
,
Λβ 0
0
0 −2.3
.
3.10
So, the eigenvalues of Bα Λα are −36 and −10, and the eigenvalues of Bβ Λβ are −3 and
−1, respectively. Therefore, the hybrid projective synchronization between drive system 3.6
and its response system 3.7 can be achieved. The corresponding numerical result is shown
in Figure 4, in which the initial conditions are x10 , x20 , x30 , x40 3,4,5,6 for the drive
system 3.6 and y10 , y20 , y30 , y40 16,7,12,4 for the response system 3.7, respectively,
and ε 4i1 ei2 1/2 .
4. Conclusions
We proposed a new synchronization scheme to achieve hybrid projective synchronization for
two identical fractional-order chaotic systems in this paper. The drive system and response
system could be synchronized up to a vector function factor, and this synchronization
scheme needs not to absorb all the nonlinear terms of response system. The synchronization
technique, based on stability theory of fractional-order systems, is simple and theoretically
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Discrete Dynamics in Nature and Society
20
15
ε
10
5
0
0
10
20
30
t
Figure 4: The HPS result between the drive system 3.6 and its response system 3.7 for q 0.95.
rigorous. Numerical simulations are used to illustrate the effectiveness of the proposed
synchronization method.
Acknowledgments
The authors are very grateful to the reviewers for their valuable comments and suggestions,
which have led to the improved presentation of this paper. This work is supported by
Foundation of Science and Technology project of Chongqing Education Commission under
Grant KJ110525.
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