Analysis of a 4-D Hyperchaotic Fractional-Order Memristive System with Hidden Attractors

Abstract This paper proposes new fractional-order (FO) models of seven nonequilibrium and stable equilibrium systems and investigates the existence of chaos and hyperchaos in them. It thereby challenges the conventional generation of chaos that involves starting the orbits from the vicinity of unstable manifold. This is followed by the discovery of coexisting hidden attractors in fractional dynamics. All the seven newly proposed fractional-order chaotic/hyperchaotic systems (FOCSs/FOHSs) ranging from minimum fractional dimension (nf) of 2.76 to 4.95, exhibit multiple hidden attractors, such as periodic orbits, stable foci, and strange attractors, often coexisting together. To the best of the our knowledge, this phenomenon of prevalence of FO coexisting hidden attractors in FOCSs is reported for the first time. These findings have significant practical relevance, because the attractors are discovered in real-life physical systems such as the FO homopolar disc dynamo, FO memristive system, FO model of the modulation instability in a dissipative medium, etc., as analyzed in this work. Numerical simulation results confirm the theoretical analyses and comply with the fact that multistability of hidden attractors does exist in the proposed FO models.

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Multiple coexisting analysis of a fractional-order coupled memristive system and its application in image encryption

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.111334 ◽

2021 ◽

Vol 152 ◽

pp. 111334

Author(s):

Yongbing Hu ◽

Qian Li ◽

Dawei Ding ◽

Li Jiang ◽

Zongli Yang ◽

...

Keyword(s):

Image Encryption ◽

Fractional Order ◽

Memristive System

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Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501675 ◽

2018 ◽

Vol 28 (13) ◽

pp. 1850167 ◽

Cited By ~ 18

Author(s):

Sen Zhang ◽

Yicheng Zeng ◽

Zhijun Li ◽

Chengyi Zhou

Keyword(s):

Fractional Order ◽

Initial Conditions ◽

State Feedback Control ◽

Hyperchaotic System ◽

Hidden Attractors ◽

System Parameters ◽

Offset Boosting ◽

Extreme Multistability ◽

Hidden Extreme Multistability ◽

New System

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.

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Fractional-order 4D hyperchaotic memristive system and application in color image encryption

EURASIP Journal on Image and Video Processing ◽

10.1186/s13640-018-0402-7 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 18

Author(s):

Peng Li ◽

Ji Xu ◽

Jun Mou ◽

Feifei Yang

Keyword(s):

Image Encryption ◽

Fractional Order ◽

Color Image ◽

Color Image Encryption ◽

Memristive System

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Finite Time Synchronization for Fractional Order Sprott C Systems with Hidden Attractors

Complexity ◽

10.1155/2019/1612752 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Cui Yan ◽

He Hongjun ◽

Lu Chenhui ◽

Sun Guan

Keyword(s):

Fractional Order ◽

Finite Time ◽

Chaotic Systems ◽

Time Synchronization ◽

Engineering Practice ◽

Fractional Order Systems ◽

Spectral Entropy ◽

Hidden Attractors ◽

Finite Time Stability ◽

Peculiar Phenomenon

Fractional order systems have a wider range of applications. Hidden attractors are a peculiar phenomenon in nonlinear systems. In this paper, we construct a fractional-order chaotic system with hidden attractors based on the Sprott C system. According to the Adomain decomposition method, we numerically simulate from several algorithms and study the dynamic characteristics of the system through bifurcation diagram, phase diagram, spectral entropy, and C0 complexity. The results of spectral entropy and C0 complexity simulations show that the system is highly complex. In order to apply such research results to engineering practice, for such fractional-order chaotic systems with hidden attractors, we design a controller to synchronize according to the finite-time stability theory. The simulation results show that the synchronization time is short and the robustness is stable. This paper lays the foundation for the study of fractional order systems with hidden attractors.

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A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽

10.3390/e20080564 ◽

2018 ◽

Vol 20 (8) ◽

pp. 564 ◽

Cited By ~ 36

Author(s):

Jesus Munoz-Pacheco ◽

Ernesto Zambrano-Serrano ◽

Christos Volos ◽

Sajad Jafari ◽

Jacques Kengne ◽

...

Keyword(s):

Fractional Order ◽

Chaotic System ◽

Chaotic Attractor ◽

Equilibrium Points ◽

Fractional Order System ◽

Order System ◽

Chaotic Attractors ◽

Equilibrium System ◽

Hidden Attractors ◽

Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

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Bursting, Dynamics, and Circuit Implementation of a New Fractional-Order Chaotic System With Coexisting Hidden Attractors

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4043003 ◽

2019 ◽

Vol 14 (7) ◽

Cited By ~ 3

Author(s):

Meng Jiao Wang ◽

Xiao Han Liao ◽

Yong Deng ◽

Zhi Jun Li ◽

Yi Ceng Zeng ◽

...

Keyword(s):

Numerical Simulation ◽

Fractional Order ◽

Chaotic System ◽

Chaotic Systems ◽

Equilibrium Points ◽

Neuroendocrine Cells ◽

Order System ◽

Hidden Attractors ◽

Main Mode ◽

Circuit Realization

Systems with hidden attractors have been the hot research topic of recent years because of their striking features. Fractional-order systems with hidden attractors are newly introduced and barely investigated. In this paper, a new 4D fractional-order chaotic system with hidden attractors is proposed. The abundant and complex hidden dynamical behaviors are studied by nonlinear theory, numerical simulation, and circuit realization. As the main mode of electrical behavior in many neuroendocrine cells, bursting oscillations (BOs) exist in this system. This complicated phenomenon is seldom found in the chaotic systems, especially in the fractional-order chaotic systems without equilibrium points. With the view of practical application, the spectral entropy (SE) algorithm is chosen to estimate the complexity of this fractional-order system for selecting more appropriate parameters. Interestingly, there is a state variable correlated with offset boosting that can adjust the amplitude of the variable conveniently. In addition, the circuit of this fractional-order chaotic system is designed and verified by analog as well as hardware circuit. All the results are very consistent with those of numerical simulation.

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