In this article, the authors have proposed a novel scheme for the dual combination synchronization among four master systems and two slave systems for the fractional order complex chaotic systems. Dual combination synchronization for the integer order has already been investigated in real space; but for the case of fractional order in complex space, it is the first of its kind. Due to complexity and presence of additional variable, it will be more secure and interesting to transmit and receive signals in communication theory. Based on the Lyapunov stability theory, six complex chaotic systems are considered and corresponding controllers are designed to achieve synchronization. The special cases, such as combination synchronization, projective synchronization, complete synchronization, and many more, can be derived from the proposed scheme. The corresponding theoretical analysis and numerical simulations are shown to verify the feasibility and effectiveness of the proposed dual combination synchronization scheme.

References

References
1.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier, Amsterdam
,
The Netherlands
.
2.
Xi
,
H.
,
Lia
,
Y.
, and
Huanga
,
X.
,
2015
, “
Adaptive Function Projective Combination Synchronization of Three Different Fractional-Order Chaotic Systems
,”
Optik
,
126
(
24
), pp.
5346
5349
.
3.
Dadras
,
S.
,
Momani
,
H. R.
,
Qi
,
G.
, and
Wang
,
Z. L.
,
2012
, “
Four-Wing Hyperchaotic Attractor Generated From a New 4D System With One Equilibrium and Its Fractional-Order Form
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1161
1173
.
4.
Zeng
,
C.
,
Yang
,
Q.
, and
Wang
,
J.
,
2011
, “
Chaos and Mixed Synchronization of a New Fractional-Order System With One Saddle and Two Stable Node-Foci
,”
Nonlinear Dyn.
,
65
(
4
), pp.
457
466
.
5.
Pinto
,
C. M. A.
, and
Machado
,
J. A. T.
,
2011
, “
Complex Order Van Der Pol Oscillator
,”
Nonlinear Dyn.
,
65
(
3
), pp.
247
254
.
6.
Podlubny
,
I.
,
1999
, “
Fractional-Order Systems and PIλDμ Controllers
,”
IEEE Trans. Autom. Control
,
44
(
1
), pp.
208
214
.
7.
Arena
,
P.
,
Caponetto
,
R.
,
Fortuna
,
L.
, and
Porto
,
D.
,
2000
,
Nonlinear Non-Integer Order Circuits and Systems—An Introduction
,
World Scientific
,
Singapore
.
8.
Tseng
,
C. C.
,
2007
, “
Design of FIR and IIR Fractional Order Simpson Digital Integrators
,”
Signal Process.
,
87
(
5
), pp.
1045
1057
.
9.
Sheu
,
L. J.
,
2011
, “
A Speech Encryption Using Fractional Chaotic Systems
,”
Nonlinear Dyn.
,
65
(
1
), pp.
103
108
.
10.
Golmankhaneh
,
A. K.
,
Arefi
,
R.
, and
Baleanu
,
D.
,
2013
, “
The Proposed Modified Liu System With Fractional Order
,”
Adv. Math. Phys.
,
2013
, p.
186037
.
11.
Grogirenko
,
I.
, and
Grogirenko
,
E.
,
2003
, “
Chaotic Dynamics of the Fractional Order Lorenz System
,”
Phys. Rev. Lett.
,
91
(
3
), p.
034101
.
12.
Sheu
,
L. J.
,
Chen
,
H. K.
,
Chen
,
J. H.
,
Tam
,
L. M.
,
Chen
,
W. C.
,
Lin
,
K. T.
, and
Kang
,
Y.
,
2008
, “
Chaos in the Newton-Leipnik System With Fractional Order
,”
Chaos, Solitons Fractals
,
36
, pp.
98
103
.
13.
Pecora
,
L. M.
, and
Carroll
,
T. L.
,
1990
, “
Synchronization in Chaotic Systems
,”
Phys. Rev. Lett.
,
64
(
8
), pp.
821
824
.
14.
Lakshmanan
,
M.
, and
Murali
,
K.
,
1996
,
Chaos in Nonlinear Oscillators: Controlling and Synchronization
,
World Scientific
,
Singapore
.
15.
Blasius
,
B.
,
Huppert
,
A.
, and
Stone
,
L.
,
1999
, “
Complex Dynamics and Phase Synchronization in Spatially Extended Ecological System
,”
Nature
,
399
(
6734
), pp.
354
359
.
16.
Han
,
S. K.
,
Kerrer
,
C.
, and
Kuramoto
,
Y.
,
1995
, “
Dephasing and Bursting in Coupled Neural Oscillators
,”
Phys. Rev. Lett.
,
75
(
17
), pp.
3190
3193
.
17.
Cuomo
,
K. M.
, and
Oppenheim
,
A. V.
,
1993
, “
Circuit Implementation of Synchronized Chaos With Application to Communication
,”
Phys. Rev. Lett.
,
71
(
1
), pp.
65
68
.
18.
Murali
,
K.
, and
Lakshmanan
,
M.
,
2003
, “
Secure Communication Using a Compound Signal Using Sampled-Data Feedback
,”
Appl. Math. Mech.
,
11
, pp.
1309
1315
.
19.
Liu
,
Y.
,
Li
,
L.
, and
Feng
,
Y.
,
2015
, “
Finite Time Synchronization for High-Dimensional Chaotic Systems and Its Application in Secure Communication
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
5
), p.
0510281
.
20.
Wu
,
G. C.
,
Baleanu
,
D.
,
Xie
,
H. P.
, and
Chen
,
F. L.
,
2016
, “
Chaos Synchronization of Fractional Chaotic Maps Based on the Stability Condition
,”
Phys. A
,
460
, pp.
374
383
.
21.
Golmankhaneh
,
A. K.
,
Arefi
,
R.
, and
Baleanu
,
D.
,
2015
, “
Synchronization in a Non-Identical Fractional Order of a Proposed Modified System
,”
J. Vib. Control
,
21
(
6
), pp.
1154
1161
.
22.
Luo
,
C.
, and
Wang
,
X.
,
2013
, “
Chaos in the Fractional-Order Complex Lorenz System and Its Synchronization
,”
Nonlinear Dyn.
,
71
(
1
), pp.
241
257
.
23.
Jiang
,
C.
,
Liu
,
S.
, and
Luo
,
C.
,
2014
, “
A New Fractional-Order Chaotic Complex System and Its Antisynchronization
,”
Abstr. Appl. Anal.
,
2014
, p.
326354
.
24.
Liu
,
X.
,
Hong
,
L.
, and
Yang
,
L.
,
2014
, “
Fractional-Order Complex T System: Bifurcations, Chaos Control, and Synchronization
,”
Nonlinear Dyn.
,
75
(
3
), pp.
589
602
.
25.
Singh
,
A. K.
,
Yadav
,
V. K.
, and
Das
,
S.
,
2016
, “
Synchronization Between Fractional Order Complex Chaotic Systems
,”
Int. J. Dyn. Control
(online).
26.
Luo
,
C.
, and
Wang
,
X.
,
2013
, “
Chaos Generated From the Fractional-Order Complex Chen System and Its Application to Digital Secure Communication
,”
Int. J. Mod. Phys. C
,
24
(
04
), p.
1350025
.
27.
Liu
,
J.
,
2014
, “
Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional Order Complex Chaos and Real Hyper-Chaos
,”
Entropy
,
16
(
12
), pp.
6195
6211
.
28.
Velmurugan
,
G.
, and
Rakkiyapan
,
R.
,
2016
, “
Hybrid Projective Synchronization of Fractional Order Chaotic Complex Systems With Time Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
3
), p.
031016
.
29.
Liu
,
Y.
, and
Davis
,
P.
,
2000
, “
Dual Synchronization of Chaos
,”
Phys. Rev. E
,
61
, pp.
2176
2184
.
30.
Xiao
,
J.
,
Ma
,
Z. Z.
, and
Yang
,
Y. H.
,
2013
, “
Dual Synchronization of Fractional Order Chaotic Systems Via Linear Controller
,”
Sci. World J.
,
2013
, p.
159194
.
31.
Jiang
,
C.
,
Liu
,
S.
, and
Wang
,
D.
,
2015
, “
Generalised Combination Complex Synchronization for Fractional Order Chaotic Complex Systems
,”
Entropy
,
17
(
8
), pp.
5199
5217
.
32.
Runzi
,
L.
,
Yinglan
,
W.
, and
Shucheng
,
D.
,
2011
, “
Combination Synchronization of Three Classical Chaotic Systems Using Active Backstepping Design
,”
Chaos
,
21
(
4
), p.
043114
.
33.
Sun
,
J.
,
Jiang
,
S.
,
Cui
,
G.
, and
Wang
,
Y.
,
2016
, “
Dual Combination Synchronization of Six Chaotic Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
11
, p. 034501-5.
34.
Aguila-Camacho
,
N.
,
Duarte-Mermoud
,
M. A.
, and
Gallegos
,
J. A.
,
2014
, “
Lyapunov Functions for Fractional Order Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
19
(
9
), pp.
2951
2957
.
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