This paper addresses the projective synchronization (PS) of the complex modified Van der Pol-Duffing (MVDPD for short) chaotic oscillator by using the nonlinear observer control and also discusses its applications to secure communication in theory. First, we construct the complex MVDPD oscillator and analysis its chaotic behavior. Moreover, an observer design method is applied to achieve PS of two identical MVDPD chaotic oscillators with complex offset terms, which are synchronized to the desired scale factor. The unpredictability of the scaling factor could further enhance the security of the communication. Finally, numerical simulations are given to demonstrate the effectiveness and feasibility of the proposed synchronization approach and also verify the success application to the proposed scheme’s in the secure communication.

References

References
1.
Liao
,
T. L.
, and
Huang
,
N. S.
,
1999
, “
An Observer Based Approach for Chaotic Synchronization and Secure Communication
,”
IEEE Trans. Circuits Syst. I
,
46
(
9
), pp.
1144
1149
.10.1109/81.788817
2.
Luo
,
A. C. J.
,
2009
, “
A Theory for Synchronization of Dynamical Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
5
), pp.
1901
1951
.10.1016/j.cnsns.2008.07.002
3.
Parlitz
,
U.
,
Kocarev
,
L.
,
Stojanovski
,
T.
, and
Preckel
,
H.
,
1996
, “
Encoding Messages Using Chaotic Synchronization
,”
Phys. Rev. E
,
53
(
5
), pp.
4351
4361
.10.1103/PhysRevE.53.4351
4.
Chen
,
G.
, and
Dong
,
X.
,
1998
,
From Chaos to Order: Methodologies, Perspectives, and Applications
,
World Scientific
,
Singapore
.10.1142/9789812798640
5.
Alvarez
,
G.
,
Li
,
S.
,
Montoya
,
F.
,
Pastor
,
G.
, and
Romera
,
M.
,
2004
, “
Breaking a Secure Communication Scheme Based on the Phase Synchronization of Chaotic Systems
,”
Chaos
,
14
(
2
), pp.
274
278
.10.1063/1.1688092
6.
Mahmoud
,
G.
,
Kashif
,
M.
, and
Farghaly
,
A.
,
2008
, “
Chaotic and Hyperchaotic Attractors of a Complex Nonlinear System
,”
J. Phys. A
,
41
(
5
), p.
055104
.10.1088/1751-8113/41/5/055104
7.
Mahmoud
,
G.
, and
Aly
,
S. A.
,
2000
, “
Periodic Attractors of Complex Damped Non-Linear Systems
,”
Int. J. Nonlinear Mech.
,
35
(
2
), pp.
309
323
.10.1016/S0020-7462(99)00016-5
8.
Mahmoud
,
G. M.
, and
Mahmoud
,
E. E.
,
2010
, “
Complete Synchronization of Chaotic Complex Nonlinear Systems With Uncertain Parameters
,”
Nonlinear Dyn.
,
62
(
4
), pp.
875
882
.10.1007/s11071-010-9770-y
9.
Abarbanel
,
H. D. I.
,
Rulkov
,
N. F.
, and
Sushchik
,
M. M.
,
1996
, “
Generalized Synchronization of Chaos: The Auxiliary System Approach
,”
Phys. Rev. E
,
53
(
5
), pp.
4528
4535
.10.1103/PhysRevE.53.4528
10.
Mainieri
,
R.
, and
Rehacek
,
J.
,
1999
, “
Projective Synchronization in Three-Dimensional Chaotic Systems
,”
Phys. Rev. Lett.
,
82
(
15
), pp.
3042
3045
.10.1103/PhysRevLett.82.3042
11.
Xu
,
D.
,
2001
, “
Control of Projective Synchronization in Chaotic Systems
,”
Phys. Rev. E
,
63
(
2
), p.
027201
.10.1103/PhysRevE.63.027201
12.
Xu
,
D.
,
Li
,
Z.
, and
Bishop
,
S.
,
2001
, “
Manipulating the Scaling Factor of Projective Synchronization in Three-Dimensional Chaotic Systems
,”
Chaos
,
11
(
3
), pp.
439
442
.10.1063/1.1380370
13.
Li
,
G. H.
,
2006
, “
Generalized Projective Synchronization of Two Chaotic Systems by Using Active Control
,”
Chaos Solitons Fract.
,
30
(
1
), pp.
77
82
.10.1016/j.chaos.2005.08.130
14.
Shi
,
X.
, and
Wang
,
Z.
,
2009
, “
Projective Synchronization of Chaotic Systems With Different Dimensions Via Backstepping Design
,”
Int. J. Nonlinear Sci.
,
7
, pp.
301
306
.
15.
Ghosh
,
D.
, and
Bhattacharya
,
S.
,
2010
, “
Projective Synchronization of New Hyperchaotic System With Fully Unknown Parameters
,”
Nonlinear Dyn.
,
61
(
1–2
), pp.
11
21, 2010
.10.1007/s11071-009-9627-4
16.
Hua
,
M.
,
Yang
,
Y.
,
Xua
,
Z.
, and
Guo
,
L.
,
2008
, “
Hybrid Projective Synchronization in a Chaotic Complex Nonlinear System
,”
Math. Comput. Simul.
,
79
(
3
), pp.
449
457
.10.1016/j.matcom.2008.01.047
17.
Callier
,
F. M.
, and
Desoer
,
C. A.
,
1991
,
Linear System Theory
,
Springer-Verlag
,
New York
.10.1007/978-1-4612-0957-7
18.
Vidyasagar
,
M.
,
1993
,
Nonlinear Systems Analysis
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
19.
Darouach
,
M.
, and
Boutayeb
,
M.
,
1995
, “
Design of Observers for Descriptor Systems
,”
IEEE Trans. Autom. Control
,
40
(
7
), pp.
1323
1327
.10.1109/9.400467
20.
Nijmeijer
,
H.
, and
Mareels
,
I.
,
1997
, “
An Observer Looks at Synchronization
,”
IEEE Trans. Circuits Syst. I
,
44
(
10
), pp.
882
890
.10.1109/81.633877
21.
Grassi
,
G.
, and
Mascolo
,
S.
,
1997
, “
Nonlinear Observer Design to Synchronize Hyperchaotic Systems Via a Scalar Signal
,”
IEEE Trans. Circuits Syst. I
,
44
(
10
), pp.
1011
1014
.10.1109/81.633891
22.
Liu
,
B.
, and
Peng
,
J.
,
2004
,
Nonlinear Dynamics
,
Higher Education Press
,
Beijing, China
.
23.
Qiu
,
S.
, and
Filanovsky
, I
. M.
,
1987
, “
Periodic Solution of Van Der Pol Equation With Moderate Values of Damping Coefficient
,”
IEEE Trans. Circuits Syst. I
,
34
(
8
), pp.
913
918
.10.1109/TCS.1987.1086241
24.
Sira-Ramirez
,
H.
,
2002
, “
Harmonic Response of Variable-Structure-Controlled Van der Pol Oscillators
,”
IEEE Trans. Circuits Syst. I
,
34
(
1
), pp.
103
106
.10.1109/TCS.1987.1086026
25.
Li
,
X.
,
Ji
,
J. C.
,
Hansen
,
C. H.
, and
Tan
,
C.
,
2006
, “
The Response of a Duffing-Van Der Pol Oscillator Under Delayed Feedback Control
,”
J. Sound Vib.
,
291
(
3–5
), pp.
644
655
.10.1016/j.jsv.2005.06.033
26.
Mahmoud
,
G. M.
, and
Farghaly
,
A. A.
,
2004
, “
Chaos Control of Chaotic Limit Cycles of Real and Complex Van Der Pol Oscillators
,”
Chaos Solitons Fract.
,
21
(
4
), pp.
915
924
.10.1016/j.chaos.2003.12.039
27.
Mahmoud
,
G. M.
,
Mahmoud
,
E. E.
,
Farghaly
,
A. A.
, and
Aly
,
S. A.
,
2009
, “
Chaotic Synchronization of Two Complex Nonlinear Oscillators
,”
Chaos Solitons Fract.
,
42
(
5
), pp.
2858
2864
.10.1016/j.chaos.2009.04.027
28.
Xu
,
Y.
,
Xu
,
W.
, and
Mahmoud
,
G. M.
,
2004
, “
On a Complex Beam–Beam Interaction Model With Random Forcing
,”
Physica A
,
336
(
3–4
), pp.
347
360
.10.1016/j.physa.2003.12.030
29.
Liao
,
T.
, and
Tsai
,
S.
,
2000
, “
Adaptive Synchronization of Chaotic Systems and Its Application to Secure Communications
,”
Chaos Solitons Fract.
,
11
(
9
), pp.
1387
1396
.10.1016/S0960-0779(99)00051-X
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