Although a large number of hidden chaotic attractors have been studied in recent years, most studies only refer to integer-order chaotic systems and neglect the relationships among chaotic attractors. In this paper, we first extend LE1 of sprott from integer-order chaotic systems to fractional-order chaotic systems, and we add two constant controllers which could produce a novel fractional-order chaotic system with hidden chaotic attractors. Second, we discuss its complicated dynamic characteristics with the help of projection pictures and bifurcation diagrams. The new fractional-order chaotic system can exhibit self-excited attractor and three different types of hidden attractors. Moreover, based on fractional-order finite time stability theory, we design finite time synchronization scheme of this new system. And combination synchronization of three fractional-order chaotic systems with hidden chaotic attractors is also derived. Finally, numerical simulations demonstrate the effectiveness of the proposed synchronization methods.

References

References
1.
Leonov
,
G. A.
, and
Kuznetsov
,
N. V.
,
2013
, “
Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits
,”
Int. J. Bifurcation Chaos
,
23
(
1
), p.
1330002
.
2.
Munozpacheco
,
J.
,
Zambranoserrano
,
E.
,
Volos
,
C.
,
Jafari
,
S.
,
Kengne
,
J.
, and
Rajagopal
,
K.
,
2018
, “
A New Fractional-Order Chaotic System With Different Families of Hidden and Self-Excited Attractors
,”
Entropy
,
20
(8), p. 564.
3.
Sprott
,
J. C.
,
1994
, “
Some Simple Chaotic Flows
,”
Phys. Rev. E
,
50
(
2
), pp.
R647
R650
.
4.
Julien Clinton
,
S.
,
William Graham
,
H.
, and
Carol Griswold
,
H.
,
2014
, “
Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators With a Temperature Gradient
,”
Phys. Rev. E
,
89
(
4
), p.
042914
.
5.
Leonov
,
G.
,
Kuznetsov
,
N.
,
Kiseleva
,
M.
,
Solovyeva
,
E.
, and
Zaretskiy
,
A.
,
2014
, “
Hidden Oscillations in Mathematical Model of Drilling System Actuated by Induction Motor With a Wound Rotor
,”
Nonlinear Dyn.
,
77
(
1–2
), pp.
277
288
.
6.
Zhang
,
F.
, and
Liu
,
S.
,
2014
, “
Self-Time-Delay Synchronization of Time-Delay Coupled Complex Chaotic System and Its Applications to Communication
,”
Int. J. Mod. Phys. C
,
25
(
3
), p.
1350102
.
7.
Li, C., Liao, X., and Yu, J., 2003, “Synchronization of Fractional Order Chaotic Systems,”
Phys. Rev. E
,
68
(6), p. 067203.
8.
Dudkowski
,
D.
,
Jafari
,
S.
,
Kapitaniak
,
T.
,
Kuznetsov
,
N. V.
,
Leonov
,
G. A.
, and
Prasad
,
A.
,
2016
, “
Hidden Attractors in Dynamical Systems
,”
Phys. Rep.
,
637
, pp.
1
50
.
9.
Jafari
,
S.
,
Sprott
,
J. C.
,
Pham
,
V. T.
,
Volos
,
C.
, and
Li
,
C.
,
2016
, “
Simple Chaotic 3D Flows With Surfaces of Equilibria
,”
Nonlinear Dyn.
,
86
(
2
), pp.
1
10
.
10.
Singh
,
J. P.
,
Roy
,
B. K.
, and
Jafari
,
S.
,
2018
, “
New Family of 4-D Hyperchaotic and Chaotic Systems With Quadric Surfaces of Equilibria
,”
Chaos, Solitons Fractals
,
106
, pp.
243
257
.
11.
Panahi
,
S.
,
Aram
,
Z.
,
Jafari
,
S.
,
Pham
,
V.-T.
,
Volos
,
C.
, and
Rajagopal
,
K.
,
2018
, “
A New Transiently Chaotic Flow With Ellipsoid Equilibria
,”
Pramana
,
90
(
3
), p.
31
.
12.
Singh
,
J. P.
, and
Roy
,
B.
,
2017
, “
The Simplest 4-D Chaotic System With Line of Equilibria, Chaotic 2-Torus and 3-Torus Behaviour
,”
Nonlinear Dyn.
,
89
(
3
), pp.
1845
1862
.
13.
Mobayen
,
S.
,
Kingni
,
S. T.
,
Pham
,
V.-T.
,
Nazarimehr
,
F.
, and
Jafari
,
S.
,
2018
, “
Analysis, Synchronisation and Circuit Design of a New Highly Nonlinear Chaotic System
,”
Int. J. Syst. Sci.
,
49
(
3
), pp.
617
630
.
14.
Nazarimehr
,
F.
,
Rajagopal
,
K.
,
Kengne
,
J.
,
Jafari
,
S.
, and
Pham
,
V.-T.
,
2018
, “
A New Four-Dimensional System Containing Chaotic or Hyper-Chaotic Attractors With No Equilibrium, a Line of Equilibria and Unstable Equilibria
,”
Chaos, Solitons Fractals
,
111
, pp.
108
118
.
15.
Messias
,
M.
, and
Reinol
,
A. C.
,
2017
, “
On the Formation of Hidden Chaotic Attractors and Nested Invariant Tori in the Sprott a System
,”
Nonlinear Dyn.
,
88
(
2
), pp.
807
821
.
16.
Zhang
,
S.
,
Zeng
,
Y.
,
Li
,
Z.
,
Wang
,
M.
,
Zhang
,
X.
, and
Chang
,
D.
,
2018
, “
A Novel Simple No-Equilibrium Chaotic System With Complex Hidden Dynamics
,”
Int. J. Dyn. Control
,
6
(4), pp. 1465–1476.https://link.springer.com/article/10.1007/s40435-018-0413-3
17.
Jafari
,
S.
,
Sprott
,
J.
, and
Golpayegani
,
S. M. R. H.
,
2013
, “
Elementary Quadratic Chaotic Flows With No Equilibria
,”
Phys. Lett. A
,
377
(
9
), pp.
699
702
.
18.
Wei
,
Z.
,
2011
, “
Dynamical Behaviors of a Chaotic System With No Equilibria
,”
Phys. Lett. A
,
376
(
2
), pp.
102
108
.
19.
Pham
,
V.-T.
,
Jafari
,
S.
,
Volos
,
C.
,
Wang
,
X.
, and
Golpayegani
,
S. M. R. H.
,
2014
, “
Is That Really Hidden? The Presence of Complex Fixed-Points in Chaotic Flows With No Equilibria
,”
Int. J. Bifurcation Chaos
,
24
(
11
), p.
1450146
.
20.
Wei
,
Z.
,
Wang
,
R.
, and
Liu
,
A.
,
2014
, “
A New Finding of the Existence of Hidden Hyperchaotic Attractors With No Equilibria
,”
Math. Comput. Simul.
,
100
, pp.
13
23
.
21.
Kuznetsov
,
N.
, and
Leonov
,
G.
,
2014
, “
Hidden Attractors in Dynamical Systems: Systems With No Equilibria, Multistability and Coexisting Attractors
,”
IFAC Proc. Vol.
,
47
(
3
), pp.
5445
5454
.
22.
Tahir
,
F. R.
,
Jafari
,
S.
,
Pham
,
V.-T.
,
Volos
,
C.
, and
Wang
,
X.
,
2015
, “
A Novel No-Equilibrium Chaotic System With Multiwing Butterfly Attractors
,”
Int. J. Bifurcation Chaos
,
25
(
4
), p.
1550056
.
23.
Pham
,
V.-T.
,
Volos
,
C.
,
Jafari
,
S.
, and
Kapitaniak
,
T.
,
2017
, “
Coexistence of Hidden Chaotic Attractors in a Novel No-Equilibrium System
,”
Nonlinear Dyn.
,
87
(
3
), pp.
2001
2010
.
24.
Wei
,
Z.
,
Sprott
,
J.
, and
Chen
,
H.
,
2015
, “
Elementary Quadratic Chaotic Flows With a Single Non-Hyperbolic Equilibrium
,”
Phys. Lett. A
,
379
(
37
), pp.
2184
2187
.
25.
Pham
,
V.-T.
,
Vaidyanathan
,
S.
,
Volos
,
C.
,
Jafari
,
S.
,
Kuznetsov
,
N.
, and
Hoang
,
T.
,
2016
, “
A Novel Memristive Time–Delay Chaotic System Without Equilibrium Points
,”
Eur. Phys. J. Spec. Top.
,
225
(
1
), pp.
127
136
.
26.
Pham
,
V.-T.
,
Vaidyanathan
,
S.
,
Volos
,
C.
,
Jafari
,
S.
, and
Kingni
,
S. T.
,
2016
, “
A No-Equilibrium Hyperchaotic System With a Cubic Nonlinear Term
,”
Optik-Int. J. Light Electron Opt.
,
127
(
6
), pp.
3259
3265
.
27.
Lin
,
Y.
,
Wang
,
C.
,
He
,
H.
, and
Zhou
,
L. L.
,
2016
, “
A Novel Four-Wing Non-Equilibrium Chaotic System and Its Circuit Implementation
,”
Pramana
,
86
(
4
), pp.
801
807
.
28.
Pham
,
V.-T.
,
Akgul
,
A.
,
Volos
,
C.
,
Jafari
,
S.
, and
Kapitaniak
,
T.
,
2017
, “
Dynamics and Circuit Realization of a No-Equilibrium Chaotic System With a Boostable Variable
,”
AEU-Int. J. Electron. Commun.
,
78
, pp.
134
140
.
29.
Pham
,
V.-T.
,
Vaidyanathan
,
S.
,
Volos
,
C. K.
,
Azar
,
A. T.
,
Hoang
,
T. M.
, and
Van Yem
,
V.
,
2017
, “
A Three-Dimensional No-Equilibrium Chaotic System: Analysis, Synchronization and Its Fractional Order Form
,”
Fractional Order Control and Synchronization of Chaotic Systems
,
Springer
, Berlin, pp.
449
470
.
30.
Pham
,
V.-T.
,
Kingni
,
S. T.
,
Volos
,
C.
,
Jafari
,
S.
, and
Kapitaniak
,
T.
,
2017
, “
A Simple Three-Dimensional Fractional-Order Chaotic System Without Equilibrium: Dynamics, Circuitry Implementation, Chaos Control and Synchronization
,”
AEU-Int. J. Electron. Commun.
,
78
, pp.
220
227
.
31.
Pham
,
V.-T.
,
Wang
,
X.
,
Jafari
,
S.
,
Volos
,
C.
, and
Kapitaniak
,
T.
,
2017
, “
From Wang–Chen System With Only One Stable Equilibrium to a New Chaotic System Without Equilibrium
,”
Int. J. Bifurcation Chaos
,
27
(
6
), p.
1750097
.
32.
Wang
,
X.
, and
Chen
,
G.
,
2012
, “
A Chaotic System With Only One Stable Equilibrium
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
3
), pp.
1264
1272
.
33.
Molaie
,
M.
,
Jafari
,
S.
,
Sprott
,
J. C.
, and
Golpayegani
,
S. M. R. H.
,
2013
, “
Simple Chaotic Flows With One Stable Equilibrium
,”
Int. J. Bifurcation Chaos
,
23
(
11
), p.
1350188
.
34.
Wang
,
X.
, and
Chen
,
G.
,
2013
, “
Constructing a Chaotic System With Any Number of Equilibria
,”
Nonlinear Dyn.
,
71
(
3
), pp.
429
436
.
35.
Wei
,
Z.
, and
Yang
,
Q.
,
2012
, “
Dynamical Analysis of the Generalized Sprott C System With Only Two Stable Equilibria
,”
Nonlinear Dyn.
,
68
(
4
), pp.
543
554
.
36.
Wei
,
Z.
, and
Pehlivan
,
I.
,
2012
, “
Chaos, Coexisting Attractors, and Circuit Design of the Generalized Sprott C System With Only Two Stable Equilibria
,”
Optoelectron. Adv. Mater. Rapid Commun
,
6
(
7–8
), pp.
742
745
.https://www.researchgate.net/publication/236651172
37.
Li
,
Q.
,
Zeng
,
H.
, and
Li
,
J.
,
2015
, “
Hyperchaos in a 4D Memristive Circuit With Infinitely Many Stable Equilibria
,”
Nonlinear Dyn.
,
79
(
4
), pp.
2295
2308
.
38.
Yang
,
Q.
,
Wei
,
Z.
, and
Chen
,
G.
,
2010
, “
An Unusual 3D Autonomous Quadratic Chaotic System With Two Stable Node-Foci
,”
Int. J. Bifurcation Chaos
,
20
(
4
), pp.
1061
1083
.
39.
Wei
,
Z.
,
Moroz
,
I.
, and
Liu
,
A.
,
2014
, “
Degenerate Hopf Bifurcations, Hidden Attractors, and Control in the Extended Sprott E System With Only One Stable Equilibrium
,”
Turk. J. Math.
,
38
(
4
), pp.
672
687
.
40.
Barati
,
K.
,
Jafari
,
S.
,
Sprott
,
J. C.
, and
Pham
,
V. T.
,
2016
, “
Simple Chaotic Flows With a Curve of Equilibria
,”
Int. J. Bifurcation Chaos
,
26
(
12
), pp.
511
543
.
41.
Pham
,
V.-T.
,
Jafari
,
S.
,
Volos
,
C.
,
Vaidyanathan
,
S.
, and
Kapitaniak
,
T.
,
2016
, “
A Chaotic System With Infinite Equilibria Located on a Piecewise Linear Curve
,”
Optik-Int. J. Light Electron Opt.
,
127
(
20
), pp.
9111
9117
.
42.
Chen
,
G.
, and
Ueta
,
T.
,
1999
, “
Yet Another Chaotic Attractor
,”
Int. J. Bifurcation Chaos
,
9
(
7
), pp.
1465
1466
.
43.
Jafari
,
S.
, and
Sprott
,
J.
,
2013
, “
Simple Chaotic Flows With a Line Equilibrium
,”
Chaos, Solitons Fractals
,
57
, pp.
79
84
.
44.
Li
,
Q.
,
Hu
,
S.
,
Tang
,
S.
, and
Zeng
,
G.
,
2014
, “
Hyperchaos and Horseshoe in a 4D Memristive System With a Line of Equilibria and Its Implementation
,”
Int. J. Circuit Theory Appl.
,
42
(
11
), pp.
1172
1188
.
45.
Jafari
,
S.
,
Sprott
,
J.
, and
Nazarimehr
,
F.
,
2015
, “
Recent New Examples of Hidden Attractors
,”
Eur. Phys. J. Spec. Top.
,
224
(
8
), pp.
1469
1476
.
46.
Semenov
,
V.
,
Korneev
,
I.
,
Arinushkin
,
P.
,
Strelkova
,
G.
,
Vadivasova
,
T.
, and
Anishchenko
,
V.
,
2015
, “
Numerical and Experimental Studies of Attractors in Memristor-Based Chua's Oscillator With a Line of Equilibria: Noise-Induced Effects
,”
Eur. Phys. J. Spec. Top.
,
224
(
8
), pp.
1553
1561
.
47.
Ma
,
J.
,
Chen
,
Z.
,
Wang
,
Z.
, and
Zhang
,
Q.
,
2015
, “
A Four-Wing Hyper-Chaotic Attractor Generated From a 4-D Memristive System With a Line Equilibrium
,”
Nonlinear Dyn.
,
81
(
3
), pp.
1275
1288
.
48.
Li
,
C.
,
Sprott
,
J.
, and
Thio
,
W.
,
2014
, “
Bistability in a Hyperchaotic System With a Line Equilibrium
,”
J. Exp. Theor. Phys.
,
118
(
3
), pp.
494
500
.
49.
Bao
,
H.
,
Wang
,
N.
,
Bao
,
B.
,
Chen
,
M.
,
Jin
,
P.
, and
Wang
,
G.
,
2018
, “
Initial Condition-Dependent Dynamics and Transient Period in Memristor-Based Hypogenetic Jerk System With Four Line Equilibria
,”
Commun. Nonlinear Sci. Numer. Simul.
,
57
, pp.
264
275
.
50.
Chen
,
E.
,
Min
,
L.
, and
Chen
,
G.
,
2017
, “
Discrete Chaotic Systems With One-Line Equilibria and Their Application to Image Encryption
,”
Int. J. Bifurcation Chaos
,
27
(
3
), p.
1750046
.
51.
Korneev
,
I. A.
, and
Semenov
,
V. V.
,
2017
, “
Andronov–Hopf Bifurcation With and Without Parameter in a Cubic Memristor Oscillator With a Line of Equilibria
,”
Chaos: An Interdiscip. J. Nonlinear Sci.
,
27
(
8
), p.
081104
.
52.
Korneev
,
I. A.
,
Vadivasova
,
T. E.
, and
Semenov
,
V. V.
,
2017
, “
Hard and Soft Excitation of Oscillations in Memristor-Based Oscillators With a Line of Equilibria
,”
Nonlinear Dyn.
,
89
(
4
), pp.
2829
2843
.
53.
Kamal
,
N.
,
Varshney
,
V.
,
Shrimali
,
M.
,
Prasad
,
A.
,
Kuznetsov
,
N.
, and
Leonov
,
G.
,
2018
, “
Shadowing in Hidden Attractors
,”
Nonlinear Dyn.
,
91
(
4
), pp.
2429
2434
.
54.
Oumbé Tékam
,
G. T.
,
Kwuimy
,
C. A.
, and
Woafo
,
P.
,
2015
, “
Analysis of Tristable Energy Harvesting System Having Fractional Order Viscoelastic Material
,”
Chaos
,
25
(
1
), pp.
191
206
.
55.
Laskin
,
N.
,
2000
, “
Fractional Market Dynamics
,”
Phys. A Stat. Mech. Appl.
,
287
(
3–4
), pp.
482
492
.
56.
Bagley
,
R. L.
, and
Calico
,
R. A.
,
1991
, “
Fractional Order State Equations for the Control of Viscoelastically damped Structures
,”
Proc. Damping
,
1
(
4
), pp.
431
436
.
57.
Efe
,
M. N.
,
2011
, “
Fractional Order Systems in Industrial Automation—A Survey
,”
IEEE Trans. Ind. Inf.
,
7
(
4
), pp.
582
591
.
58.
Hosseinnia
,
S. H.
,
Magin
,
R. L.
, and
Vinagre
,
B. M.
,
2015
, “
Chaos in Fractional and Integer Order Nsg Systems
,”
Signal Process.
,
107
(
C
), pp.
302
311
.
59.
Ding
,
Y.
,
Wang
,
Z.
, and
Ye
,
H.
,
2012
, “
Optimal Control of a Fractional-Order Hiv-Immune System With Memory
,”
IEEE Trans. Control Syst. Technol.
,
20
(
3
), pp.
763
769
.
60.
Wang
,
X.
,
Ouannas
,
A.
,
Pham
,
V. T.
, and
Abdolmohammadi
,
H. R.
,
2018
, “
A Fractional-Order Form of a System With Stable Equilibria and Its Synchronization
,”
Adv. Diff. Equations
,
2018
(
1
), p.
20
.
61.
Kingni
,
S. T.
,
Jafari
,
S.
,
Simo
,
H.
, and
Woafo
,
P.
,
2014
, “
Three-Dimensional Chaotic Autonomous System With Only One Stable Equilibrium: Analysis, Circuit Design, Parameter Estimation, Control, Synchronization and Its Fractional-Order Form
,”
Eur. Phys. J. Plus
,
129
(
5
), pp.
1
16
.
62.
Kingni
,
S. T.
,
Pham
,
V.-T.
,
Jafari
,
S.
, and
Woafo
,
P.
,
2017
, “
A Chaotic System With an Infinite Number of Equilibrium Points Located on a Line and on a Hyperbola and Its Fractional-Order Form
,”
Chaos, Solitons Fractals
,
99
, pp.
209
218
.
63.
Zhang
,
F.
, and
Liu
,
S.
,
2015
, “
Adaptive Complex Function Projective Synchronization of Uncertain Complex Chaotic Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
1
), p.
011013
.
64.
Jiang
,
C.
,
Zhang
,
F.
, and
Li
,
T.
,
2018
, “
Synchronization and Antisynchronization of n-Coupled Fractional-Order Complex Chaotic Systems With Ring Connection
,”
Math. Methods Appl. Sci.
,
41
(
7
), pp.
2625
2638
.
65.
Zhang
,
F.
,
2015
, “
Complete Synchronization of Coupled Multiple-Time-Delay Complex Chaotic System With Applications to Secure Communication
,”
Acta Phys. Polonica B
,
46
(
8
), pp.
1473
1486
.
66.
Jafari
,
M. A.
,
Mliki
,
E.
,
Akgul
,
A.
,
Pham
,
V.-T.
,
Kingni
,
S. T.
,
Wang
,
X.
, and
Jafari
,
S.
,
2017
, “
Chameleon: The Most Hidden Chaotic Flow
,”
Nonlinear Dyn.
,
88
(
3
), pp.
2303
2317
.
67.
Runzi
,
L.
,
Yinglan
,
W.
, and
Shucheng
,
D.
,
2011
, “
Combination Synchronization of Three Classic Chaotic Systems Using Active Backstepping Design
,”
Chaos
,
21
(
4
), p.
821
.
68.
Zhao
,
L. D.
,
2011
, “
A Finite-Time Stable Theorem About Fractional Systems and Finite-Time Synchronizing Fractional Super Chaotic Lorenz Systems
,”
Acta Phys. Sin.
,
60
(
10
), pp.
687
709
.
You do not currently have access to this content.