Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York.
2.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Application of Fractional Differential Equations
,
Elsevier
,
New York
.
3.
Koeller
,
R. C.
,
1984
, “
Applications of Fractional Calculus to the Theory of Viscoelasticity
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
299
307
.
4.
Sabatier
,
J.
,
Agrawal
,
O. P.
, and
Machado
,
J. T.
,
2007
,
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
,
Springer
,
Dordrecht, The Netherlands
.
5.
Machado
,
J. T.
,
Kiryakova
,
V.
, and
Mainardi
,
F.
,
2011
, “
Recent History of Fractional Calculus
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
3
), pp.
1140
1153
.
6.
Hartley
,
T. T.
,
Lorenzo
,
C. F.
, and
Killory Qammer
,
H.
,
1995
, “
Chaos in a Fractional Order Chua's System
,”
IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.
,
42
(
8
), pp.
485
490
.
7.
Razminia
,
A.
, and
Baleanu
,
D.
,
2013
, “
Fractional Hyperchaotic Telecommunication Systems: A New Paradigm
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
3
), p.
031012
.
8.
Wang
,
X. Y.
, and
Zhang
,
H.
,
2013
, “
Bivariate Module-Phase Synchronization of a Fractional-Order Lorenz System in Different Dimensions
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
3
), p.
031017
.
9.
Li
,
C.
, and
Chen
,
G.
,
2004
, “
Chaos and Hyperchaos in the Fractional-Order Rossler Equations
,”
Phys. A
,
341
, pp.
55
61
.
10.
Grigorenko
,
I.
, and
Grigorenko
,
E.
,
2003
, “
Chaotic Dynamics of the Fractional Lorenz System
,”
Phys. Rev. Lett.
,
91
(
3
), p.
034101
.
11.
Li
,
C.
, and
Chen
,
G.
,
2004
, “
Chaos in the Fractional Order Chen System and Its Control
,”
Chaos, Solitons Fract.
,
22
(
3
), pp.
549
554
.
12.
Lu
,
J. G.
,
2006
, “
Chaotic Dynamics of the Fractional-Order Lu System and Its Synchronization
,”
Phys. Lett. A
,
354
(
4
), pp.
305
311
.
13.
Gibbon
,
J. D.
, and
McGuinness
,
M. J.
,
1982
, “
The Real and Complex Lorenz Equations in Rotating Fluids and Lasers
,”
Phys. D
,
5
(
1
), pp.
108
122
.
14.
Fowler
,
A. C.
,
Gibbon
,
J. D.
, and
McGuinness
,
M. J.
,
1982
, “
The Complex Lorenz Equations
,”
Phys. D
,
4
(
2
), pp.
139
163
.
15.
Fowler
,
A. C.
,
Gibbon
,
J. D.
, and
McGuinness
,
M. J.
,
1983
, “
The Real and Complex Lorenz Equations and Their Relevance to Physical Systems
,”
Phys. D
,
7
(
1
), pp.
126
134
.
16.
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
.
17.
Li
,
C.
,
Liao
,
X.
, and
Yu
,
J.
,
2003
, “
Synchronization of Fractional Order Chaotic Systems
,”
Phys. Rev. E
,
68
(
6
), p.
067203
.
18.
Zhou
,
P.
, and
Zhu
,
W.
,
2011
, “
Function Projective Synchronization for Fractional-Order Chaotic Systems
,”
Nonlinear Anal.: Real World Appl.
,
12
(
2
), pp.
811
816
.
19.
Bhalekar
,
S.
, and
Daftardar-Gejji
,
V.
,
2010
, “
Synchronization of Different Fractional Order Chaotic Systems Using Active Control
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
11
), pp.
3536
3546
.
20.
Aghababa
,
M. P.
,
2014
, “
Synchronization and Stabilization of Fractional Second-Order Nonlinear Complex Systems
,”
Nonlinear Dyn.
,
80
(
4
), pp.
1731
1744
.
21.
Sun
,
J.
, and
Shen
,
Y.
,
2014
, “
Adaptive Anti-Synchronization of Chaotic Complex Systems and Chaotic Real Systems With Unknown Parameters
,”
J. Vib. Control
(in press).
22.
Ma
,
T.
, and
Zhang
,
J.
,
2015
, “
Hybrid Synchronization of Coupled Fractional-Order Complex Networks
,”
Neurocomputing
,
157
, pp.
166
172
.
23.
Mahmoud
,
G. M.
, and
Mahmoud
,
E. E.
,
2010
, “
Synchronization and Control of Hyperchaotic Complex Lorenz System
,”
Math. Comput. Simul.
,
80
(
12
), pp.
2286
2296
.
24.
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
.
25.
Mahmoud
,
G. M.
,
Aly
,
S. A.
, and
Al-Kashif
,
M. A.
,
2008
, “
Dynamical Properties and Chaos Synchronization of a New Chaotic Complex Nonlinear System
,”
Nonlinear Dyn.
,
51
(
1–2
), pp.
171
181
.
26.
Wang
,
W. X.
,
Huang
,
L.
,
Lai
,
Y. C.
, and
Chen
,
G.
,
2009
, “
Onset of Synchronization in Weighted Scale-Free Networks
,”
Chaos
,
19
(
1
), p.
013134
.
27.
Xie
,
Y.
,
Gong
,
Y.
,
Hao
,
Y.
, and
Ma
,
X.
,
2010
, “
Synchronization Transitions on Complex Thermo-Sensitive Neuron Networks With Time Delays
,”
Biophys. chem.
,
146
(
2
), pp.
126
132
.
28.
Yang
,
Y.
,
Duan
,
X.
, and
Li
,
L.
,
2011
, “
Wireless Sensor Network Time Synchronization Design for Large Generator On-Line Monitoring
,”
Sensor Lett.
,
9
(
4
), pp.
1467
1471
.
29.
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
.
30.
Chen
,
D.
,
Zhao
,
W.
,
Liu
,
X.
, and
Ma
,
X.
,
2015
, “
Synchronization and Antisynchronization of a Class of Chaotic Systems With Nonidentical Orders and Uncertain Parameters
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
1
), p.
011003
.
31.
Kim
,
C. M.
,
Rim
,
S.
,
Kye
,
W. H.
,
Ryu
,
J. W.
, and
Park
,
Y. J.
,
2003
, “
Anti-Synchronization of Chaotic Oscillators
,”
Phys. Lett. A
,
320
(
1
), pp.
39
46
.
32.
Kocarev
,
L.
, and
Parlitz
,
U.
,
1996
, “
Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems
,”
Phys. Rev. Lett.
,
76
(
11
), pp.
1816
1819
.
33.
Mainieri
,
R.
, and
Rehacek
,
J.
,
1999
, “
Projective Synchronization in Three-Dimensional Chaotic Systems
,”
Phys. Rev. Lett.
,
82
(
15
), pp.
3042
3045
.
34.
Wang
,
S.
,
Yu
,
Y.
, and
Wen
,
G.
,
2014
, “
Hybrid Projective Synchronization of Time-Delayed Fractional Order Chaotic Systems
,”
Nonlinear Anal. Hybrid Syst.
,
11
, pp.
129
138
.
35.
Hu
,
M.
,
Yang
,
Y.
,
Xu
,
Z.
, and
Guo
,
L.
,
2008
, “
Hybrid Projective Synchronization in a Chaotic Complex Nonlinear System
,”
Math. Comput. Simul.
,
79
(
3
), pp.
449
457
.
36.
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
.
37.
Sun
,
J.
,
Shen
,
Y.
, and
Zhang
,
X.
,
2014
, “
Modified Projective and Modified Function Projective Synchronization of a Class of Real Nonlinear Systems and a Class of Complex Nonlinear Systems
,”
Nonlinear Dyn.
,
78
(
3
), pp.
1755
1764
.
38.
Liu
,
J.
,
Liu
,
S.
, and
Yuan
,
C.
,
2014
, “
Adaptive Complex Modified Projective Synchronization of Complex Chaotic (Hyperchaotic) Systems With Uncertain Complex Parameters
,”
Nonlinear Dyn.
,
79
(
2
), pp.
1035
1047
.
39.
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
.
40.
Chee
,
C. Y.
, and
Xu
,
D.
,
2006
, “
Chaos-Based M-Nary Digital Communication Technique Using Controller Projective Synchronization
,”
IEE Proc-G Circ. Dev. Syst.
,
153
(
4
), pp.
357
360
.
41.
Asl
,
F. M.
, and
Ulsoy
,
A. G.
,
2003
, “
Analysis of a System of Linear Delay Differential Equations
,”
ASME J. Dyn. Syst., Meas., Control
,
125
(
2
), pp.
215
223
.
42.
Chen
,
Y.
, and
Moore
,
K. L.
,
2002
, “
Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems
,”
Nonlinear Dyn.
,
29
(
1
), pp.
191
200
.
43.
Deng
,
W.
,
Li
,
C.
, and
Lu
,
J.
,
2007
, “
Stability Analysis of Linear Fractional Differential System With Multiple Time Delays
,”
Nonlinear Dyn.
,
48
(
4
), pp.
409
416
.
44.
Li
,
L.
,
Peng
,
H.
,
Yang
,
Y.
, and
Wang
,
X.
,
2009
, “
On the Chaotic Synchronization of Lorenz Systems With Time-Varying Lags
,”
Chaos, Solitons Fract.
,
41
(
2
), pp.
783
794
.
45.
Zhang
,
F.
,
2015
, “
Lag Synchronization of Complex Lorenz System With Applications to Communication
,”
Entropy
,
17
(
7
), pp.
4974
4985
.
46.
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
(
4
), p.
1350025
.
47.
Mahmoud
,
G. M.
,
Mahmoud
,
E. E.
, and
Arafa
,
A. A.
,
2015
, “
On Modified Time Delay Hyperchaotic Complex Lu System
,”
Nonlinear Dyn.
,
80
(
1–2
), pp.
855
869
.
48.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
,
2002
, “
A Predictor–Corrector Approach for the Numerical Solution of Fractional Differential Equations
,”
Nonlinear Dyn.
,
29
(
1–4
), pp.
3
22
.
You do not currently have access to this content.