Compared with chaotic systems over the real number field, complex chaotic dynamics have some unique properties. In this paper, a kind of novel hybrid synchronizations of complex chaotic systems is discussed analytically and numerically. Between two nonidentical complex chaotic systems, modified projective synchronization (MPS) in the modulus space and complete synchronization in the phase space are simultaneously achieved by means of active control. Based on the Lyapunov stability theory, a controller is developed, in which time delay as an important consideration is involved. Furthermore, a switch-modulated digital secure communication system based on the proposed synchronization scheme is carried out. Different from the previous works, only one set of drive-response chaotic systems can implement switch-modulated secure communication, which could simplify the complexity of design. Furthermore, the latency of a signal transmitted between transmitter and receiver is simulated by channel delay. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.

References

References
1.
Lang
,
R. L.
,
2010
, “
A Stochastic Complex Model With Random Imaginary Noise
,”
Nonlinear Dyn.
,
62
(
3
), pp.
561
565
.
2.
Ning
,
C. Z.
, and
Haken
,
H.
,
1990
, “
Detuned Lasers and the Complex Lorenz Equations–Subcritical and Supercritical Hopf Bifurcations
,”
Phys. Rev. A
,
41
(
7
), pp.
3826
3837
.
3.
Vladimirov
,
A. G.
,
Toronov
,
V. Y.
, and
Derbov
,
V. L.
,
1998
, “
Properties of the Phase Space and Bifurcations in the Complex Lorenz Model
,”
Tech. Phys.
,
43
(
8
), pp.
877
884
.
4.
Xu
,
Y.
,
Xu
,
W.
, and
Mahmoud
,
G. M.
,
2004
, “
On a Complex Beam–Beam Interaction Model With Random Forcing
,”
Phys. A
,
336
(
3
), pp.
347
360
.
5.
Xu
,
Y.
,
Mahmoud
,
G. M.
,
Xu
,
W.
, and
Lei
,
Y.
,
2005
, “
Suppressing Chaos of a Complex Duffing's System Using a Random Phase
,”
Chaos, Solitons Fractals
,
23
(
1
), pp.
265
273
.
6.
Mahmoud
,
G. M.
, and
Aly
,
S. A.
,
2000
, “
Periodic Attractors of Complex Damped Non-Linear Systems
,”
Int. J. Nonlinear Mech.
,
35
(
2
), pp.
309
323
.
7.
Mahmoud
,
G. M.
,
Mahmoud
,
E. E.
, and
Arafa
,
A. A.
,
2013
, “
On Projective Synchronization of Hyperchaotic Complex Nonlinear Systems Based on Passive Theory for Secure Communications
,”
Phys. Scr.
,
87
(
5
), p.
055002
.
8.
Fowler
,
A. C.
,
Gibbon
,
J. D.
, and
McGuinness
,
M. J.
,
1982
, “
The Complex Lorenz Equations
,”
Phys. D
,
4
(
2
), pp.
139
163
.
9.
Mahmoud
,
G. M.
,
Al-Kashif
,
M. A.
, and
Aly
,
S. A.
,
2007
, “
Basic Properties and Chaotic Synchronization of Complex Lorenz System
,”
Int. J. Mod. Phys. C
,
18
(
2
), pp.
253
265
.
10.
Mahmoud
,
G. M.
,
Mahmoud
,
E. E.
, and
Ahmed
,
M. E.
,
2009
, “
On the Hyperchaotic Complex Lü System
,”
Nonlinear Dyn.
,
58
(
4
), pp.
725
738
.
11.
Mahmoud
,
G. M.
,
Mahmoud
,
E. E.
, and
Arafa
,
A. A.
,
2015
, “
On Modified Time Delay Hyperchaotic Complex Lü System
,”
Nonlinear Dyn.
,
80
(
1–2
), pp.
855
869
.
12.
Mahmoud
,
G. M.
,
Ahmed
,
M. E.
, and
Mahmoud
,
E. E.
,
2008
, “
Analysis of Hyperchaotic Complex Lorenz Systems
,”
Int. J. Mod. Phys. C
,
19
(
10
), pp.
1477
1494
.
13.
Mahmoud
,
E. E.
,
2012
, “
Dynamics and Synchronization of New Hyperchaotic Complex Lorenz System
,”
Math. Comput. Modell.
,
55
(
7
), pp.
1951
1962
.
14.
Luo
,
C.
, and
Wang
,
X.
,
2013
, “
Chaos in the Fractional-Order Complex Lorenz System and Its Synchronization
,”
Nonlinear Dyn.
,
71
(
1–2
), pp.
241
257
.
15.
Peng
,
J. H.
,
Ding
,
E. J.
,
Ding
,
M.
, and
Yang
,
W.
,
1996
, “
Synchronizing Hyperchaos With a Scalar Transmitted Signal
,”
Phys. Rev. Lett.
,
76
(
6
), pp.
904
907
.
16.
Liu
,
P.
,
Song
,
H.
, and
Li
,
X.
,
2015
, “
Observe-Based Projective Synchronization of Chaotic Complex Modified Van der Pol-Duffing Oscillator With Application to Secure Communication
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p.
051015
.
17.
Li
,
C. L.
,
2012
, “
Tracking Control and Generalized Projective Synchronization of a Class of Hyperchaotic System With Unknown Parameter and Disturbance
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
1
), pp.
405
413
.
18.
Liu
,
P.
,
2015
, “
Adaptive Hybrid Function Projective Synchronization of General Chaotic Complex Systems With Different Orders
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
021018
.
19.
Zhang
,
F.
, and
Liu
,
S.
,
2014
, “
Full State Hybrid Projective Synchronization and Parameters Identification for Uncertain Chaotic (Hyperchaotic) Complex Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
2
), p.
021009
.
20.
Chen
,
D. Y.
,
Zhang
,
R. F.
,
Sprott
,
J. C.
,
Chen
,
H. T.
, and
Ma
,
X. Y.
,
2012
, “
Synchronization Between Integer-Order Chaotic Systems and a Class of Fractional-Order Chaotic Systems Via Sliding Mode Control
,”
Chaos
,
22
(
2
), p.
023109
.
21.
Yang
,
J.
,
Chen
,
Y.
, and
Zhu
,
F.
,
2014
, “
Singular Reduced-Order Observer-Based Synchronization for Uncertain Chaotic Systems Subject to Channel Disturbance and Chaos-Based Secure Communication
,”
Appl. Math. Comput.
,
229
(
6
), pp.
227
238
.
22.
Pai
,
M. C.
,
2014
, “
Global Synchronization of Uncertain Chaotic Systems Via Discrete-Time Sliding Mode Control
,”
Appl. Math. Comput.
,
227
(
2
), pp.
663
671
.
23.
Liu
,
S. T.
, and
Liu
,
P.
,
2011
, “
Adaptive Anti-Synchronization of Chaotic Complex Nonlinear Systems With Unknown Parameters
,”
Nonlinear Anal.: Real World Appl.
,
12
(
6
), pp.
3046
3055
.
24.
Liu
,
P.
,
Liu
,
S. T.
, and
Li
,
X.
,
2012
, “
Adaptive Modified Function Projective Synchronization of General Uncertain Chaotic Complex Systems
,”
Phys. Scr.
,
85
(
3
), p.
035005
.
25.
Mahmoud
,
G. M.
, and
Mahmoud
,
E. E.
,
2010
, “
Phase and Antiphase Synchronization of Two Identical Hyperchaotic Complex Nonlinear Systems
,”
Nonlinear Dyn.
,
61
(
1–2
), pp.
141
152
.
26.
Nian
,
F. Z.
, and
Wang
,
X. Y.
,
2010
, “
Module-Phase Synchronization in Complex Dynamic System
,”
Appl. Math. Comput.
,
217
(
6
), pp.
2481
2489
.
27.
Wang
,
X. Y.
, and
Luo
,
C.
,
2013
, “
Hybrid Modulus-Phase Synchronization of Hyperchaotic Complex Systems and Its Application to Secure Communication
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
14
(
7–8
), pp.
533
542
.
28.
Pourdehi
,
S.
,
Karimaghaee
,
P.
, and
Karimipour
,
D.
,
2011
, “
Adaptive Controller Design for Lag-Synchronization of Two Non-Identical Time-Delayed Chaotic Systems With Unknown Parameters
,”
Phys. Lett. A
,
375
(
17
), pp.
1769
1778
.
29.
Mahmoud
,
G. M.
,
Mahmoud
,
E. E.
, and
Ahmed
,
M. E.
,
2007
, “
A Hyperchaotic Complex Chen System and Its Dynamics
,”
Int. J. Appl. Math. Stat.
,
12
(
D07
), pp.
90
100
.
30.
Yang
,
J.
, and
Zhu
,
F.
,
2013
, “
Synchronization for Chaotic Systems and Chaos-Based Secure Communications Via Both Reduced-Order and Step-by-Step Sliding Mode Observers
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
4
), pp.
926
937
.
31.
Pan
,
J.
,
Ding
,
Q.
, and
Du
,
B. X.
,
2012
, “
A New Improved Scheme of Chaotic Masking Secure Communication Based on Lorenz System
,”
Int. J. Bifurcation Chaos
,
22
(
5
), p.
1250125
.
32.
Sheikhan
,
M.
,
Shahnazi
,
R.
, and
Garoucy
,
S.
,
2013
, “
Synchronization of General Chaotic Systems Using Neural Controllers With Application to Secure Communication
,”
Neural Comput. Appl.
,
22
(
2
), pp.
361
373
.
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