Based on the pure mathematical model of the memristor, this paper proposes a novel memristor-based chaotic system without equilibrium points. By selecting different parameters and initial conditions, the system shows extremely diverse forms of winglike attractors, such as period-1 to period-12 wings, chaotic single-wing, and chaotic double-wing attractors. It was found that the attractor basins with three different sets of parameters are interwoven in a complex manner within the relatively large (but not the entire) initial phase plane. This means that small perturbations in the initial conditions in the mixing region will lead to the production of hidden extreme multistability. At the same time, these sieve-shaped basins are confirmed by the uncertainty exponent. Additionally, in the case of fixed parameters, when different initial values are chosen, the system exhibits a variety of coexisting transient transition behaviors. These 14 were also where the same state transition from period 18 to period 18 was first discovered. The above dynamical behavior is analyzed in detail through time-domain waveforms, phase diagrams, attraction basin, bifurcation diagrams, and Lyapunov exponent spectrum . Finally, the circuit implementation based on the digital signal processor verifies the numerical simulation and theoretical analysis.
REFERENCES
1
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 2
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
–398
(1976
). 3
G.
Chen
and T.
Ueta
, “Yet another chaotic attractor
,” Int. J. Bifurcat. Chaos
09
, 1465
–1466
(1999
). 4
T.
Matsumoto
, “A chaotic attractor from Chua's circuit
,” IEEE Trans. Circuits Syst.
31
, 1055
–1058
(1984
). 5
J. C.
Sprott
, “Some simple chaotic flows
,” Phys. Rev. E
50
(2
), 647
–650
(1994
). 6
J.
Lü
and G.
Chen
, “A new chaotic attractor coined
,” Int. J. Bifurcat. Chaos
12
, 659
–661
(2002
). 7
Q.
Xie
and Y. C.
Zeng
, “Generating different types of multi-double-scroll and multi-double-wing hidden attractors
,” Eur. Phys. J. Spec. Top.
229
, 1361
–1371
(2020
). 8
Q. J.
Wu
, Q. H.
Hong
, X. Y.
Liu
, X. P.
Wang
, and Z. G.
Zeng
, “Constructing multi-butterfly attractors based on Sprott C system via non-autonomous approaches
,” Chaos
29
, 043112
(2019
). 9
L.
Chua
, “Memristor—The missing circuit element
,” IEEE Trans. Circuit Theory
18
, 507
–519
(1971
). 10
J.
Ma
, Z. Q.
Chen
, Z. L.
Wang
, and Q.
Zhang
, “A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium
,” Nonlinear Dyn.
81
, 1275
–1288
(2015
). 11
X. Y.
Zhong
, M. F.
Peng
, and M.
Shahidehpousr
, “Creation and circuit implementation of two-to-eight-wing chaotic attractors using a 3D memristor-based system
,” Int. J. Circuit Theory Appl.
47
, 686
–701
(2019
). 12
M. J.
Wang
, Y.
Deng
, X. H.
Liao
, Z. J.
Li
, M. L.
Ma
, and Y. C.
Zeng
, “Dynamics and circuit implementation of a four-wing memristive chaotic system with attractor rotation
,” Int. J. Non-Linear Mech.
111
, 149
–159
(2019
). 13
K.
Rajagopal
, A.
Bayani
, A. J. M.
Khalaf
, H.
Namazi
, S.
Jafari
, and V. T.
Pham
, “A no-equilibrium memristive system with four-wing hyperchaotic attractor
,” Int. J. Electron. Commun.
95
, 207
–215
(2018
). 14
G. Y.
Peng
, F. H.
Min
, and E. R.
Wang
, “Circuit implementation, synchronization of multistability, and image encryption of a four-wing memristive chaotic system
,” J. Electr. Comput. Eng.
2018
, 8649294
(2018
). 15
L.
Zhou
, C.
Wang
, and L.
Zhou
, “Generating four-wing hyperchaotic attractor and two-wing, three-wing, and four-wing chaotic attractors in 4D memristive system
,” Int. J. Bifurcat. Chaos
27
, 1750027
(2017
). 16
G. Q.
Min
, S. K.
Duan
, and L. D.
Wang
, “A double-wing chaotic system based on ion migration memristor and its sliding mode control
,” Int. J. Bifurcat. Chaos
26
, 1650129
(2016
). 17
S. J.
Cang
, A. G.
Wu
, Z. H.
Wang
, W.
Xue
, and Z. Q.
Chen
, “Birth of one-to-four-wing chaotic attractors in a class of simplest three-dimensional continuous memristive systems
,” Nonlinear Dyn.
83
, 1987
–2001
(2016
). 18
L.
Zhou
, C. H.
Wang
, and L. L.
Zhou
, “Generating hyperchaotic multi-wing attractor in a 4D memristive circuit
,” Nonlinear Dyn.
85
, 2653
–2663
(2016
). 19
L. L.
Huang
, Z. F.
Zhang
, J. H.
Xiang
, and S. M.
Wang
, “A new 4D chaotic system with two-wing, four-wing, and coexisting attractors and its circuit simulation
,” Complexity
2019
, 5803506
(2019
). 20
H.
Natiq
, M. R. M.
Said
, M. R. K.
Ariffin
, S.
He
, L.
Rondoni
, and S.
Banerjee
, “Self-excited and hidden attractors in a novel chaotic system with complicated multistability
,” Eur. Phys. J. Plus
133
, 557
(2018
). 21
D.
Dudkowski
, S.
Jafari
, T.
Kapitaniak
, N. V.
Kuznetsov
, G. A.
Leonov
, and A.
Prasad
, “Hidden attractors in dynamical systems
,” Phys. Rep.
637
, 1
–50
(2016
). 22
H.
Chang
, Y. X.
Li
, and G. R.
Chen
, “A novel memristor-based dynamical system with multi-wing attractors and symmetric periodic bursting
,” Chaos
30
, 043110
(2020
). 23
C.
Zhou
, Z.
Li
, and F.
Xie
, “Coexisting attractors, crisis route to chaos in a novel 4D fractional-order system and variable-order circuit implementation
,” Eur. Phys. J. Plus
134
, 73
(2019
). 24
S.
Zhang
, Y. C.
Zeng
, Z. J.
Li
, M. J.
Wang
, and L.
Xiong
, “Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability
,” Chaos
28
, 013113
(2018
). 25
S.
Zhang
, Y. C.
Zeng
, and Z. J.
Li
, “One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics
,” Chin. J. Phys.
56
, 793
–806
(2018
). 26
V. R. F.
Signing
, J.
Kengne
, and L. K.
Kana
, “Dynamic analysis and multistability of a novel four-wing chaotic system with smooth piecewise quadratic nonlinearity
,” Chaos Solitons Fractals
113
, 263
–274
(2018
). 27
M.
Borah
and B. K.
Roy
, “An enhanced multi-wing fractional-order chaotic system with coexisting attractors and switching hybrid synchronisation with its nonautonomous counterpart
,” Chaos Solitons Fractals
102
, 372
–386
(2017
). 28
G. A.
Leonov
, N. V.
Kuznetsov
, M. A.
Kiseleva
, E. P.
Solovyeva
, and A. M.
Zaretskiy
, “Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor
,” Nonlinear Dyn.
77
, 277
–288
(2014
). 29
H.
Bao
, N.
Wang
, B. C.
Bao
, M.
Chen
, P. P.
Jin
, and G. Y.
Wang
, “Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria
,” Commun. Nonlinear. Sci. Numer. Simul.
57
, 264
–275
(2018
). 30
G. Y.
Wang
, F.
Yuan
, G. R.
Chen
, and Y.
Zhang
, “Coexisting multiple attractors and riddled basins of a memristive system
,” Chaos
28
, 013125
(2018
). 31
F.
Yuan
, G. Y.
Wang
, and X. W.
Wang
, “Extreme multistability in a memristor-based multi-scroll hyper-chaotic system
,” Chaos
26
, 073107
(2016
). 32
X.
Zhang
, C. B.
Li
, T. F.
Lei
, Z. H.
Liu
, and C. Y.
Tao
, “A symmetric controllable hyperchaotic hidden attractor
,” Symmetry
12
, 550
(2020
). 33
A.
Saha
and U.
Feudel
, “Riddled basins of attraction in systems exhibiting extreme events
,” Chaos
28
, 033610
(2018
). 34
K.
Sathiyadevi
, S.
Karthiga
, V. K.
Chandrasekar
, D. V.
Senthilkumar
, and M.
Lakshmanan
, “Frustration induced transient chaos, fractal and riddled basins in coupled limit cycle oscillators
,” Commun. Nonlinear. Sci. Numer. Simul.
72
, 586
–599
(2019
). 35
S. R.
Ujjwal
, N.
Punetha
, R.
Ramaswamy
, M.
Agrawal
, and A.
Prasad
, “Driving-induced multistability in coupled chaotic oscillators: Symmetries and riddled basins
,” Chaos
26
, 063111
(2016
). 36
H. R.
Lin
, C. H.
Wang
, and Y. M.
Tan
, “Hidden extreme multistability with hyperchaos and transient chaos in a Hopfield neural network affected by electromagnetic radiation
,” Nonlinear Dyn.
99
, 2369
–2386
(2020
). 37
B. A.
Mezatio
, M. T.
Motchongom
, B. R. W.
Tekam
, R.
Kengne
, R.
Tchitnga
, and A.
Fomethe
, “A novel memristive 6D hyperchaotic autonomous system with hidden extreme multistability
,” Chaos Solitons Fractals
120
, 100
–115
(2019
). 38
M.-F.
Danca
, P.
Bourke
, and N.
Kuznetsov
, “Graphical structure of attraction basins of hidden chaotic attractors: The Rabinovich-Fabrikant system
,” Int. J. Bifurcat. Chaos
29
, 1930001
(2019
). 39
M.
Chen
, X.
Ren
, H. G.
Wu
, Q.
Xu
, and B. C.
Bao
, “Interpreting initial offset boosting via reconstitution in integral domain
,” Chaos Solitons Fractals
131
, 109544
(2020
). 40
C.
Grebogi
, S. W.
McDonald
, E.
Ott
, and J. A.
Yorke
, “Final state sensitivity: An obstruction to predictability
,” Phys. Lett. A
99
, 415
–418
(1983
). 41
C. H.
Du
, L. C.
Liu
, S. S.
Shi
, and Y.
Wei
, “Multiple transient transitions behavior analysis of a double memristor's hidden system and its circuit
,” IEEE Access
8
, 76642
–76656
(2020
). © 2021 Author(s).
2021
Author(s)
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