A fractional-order form of a system with stable equilibria and its synchronization

Wang, Xiong; Ouannas, Adel; Pham, Viet–Thanh; Abdolmohammadi, Hamid Reza

doi:10.1186/s13662-018-1479-0

Cited by 18 publications

(7 citation statements)

References 70 publications

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“…e parameter A was calculated using (17) and the degrees of freedom for the first approximation were R g � 100 Ω, C x � 0.1 mF, Ω � 1 E 5 , C � 1 nF, and R x � 5 kΩ. For the second approximation, we used R y � 1 kΩ, C x � 0.1 μF, Ω � 1 E 5 , and C � 1 pF.…”

Section: Fractional-order Multiscroll Lü Chaotic Systemmentioning

confidence: 99%

“…Regarding chaos, there has been recently an increased number of works reporting fractional-order chaotic systems because fractional order provides an extra degree of freedom, which can be useful for generating diverse dynamics such as self-excited attractors, hidden attractors, multistability, and extreme stability [16][17][18][19][20]. In particular, the multiscroll chaotic attractors present plenty of complex topological structures contrary to single-or double-scroll attractors.…”

Section: Introductionmentioning

confidence: 99%

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Active Realization of Fractional‐Order Integrators and Their Application in Multiscroll Chaotic Systems

et al. 2021

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This paper presents the design, simulation, and experimental verification of the fractional-order multiscroll Lü chaotic system. We base them on op-amp-based approximations of fractional-order integrators and saturated series of nonlinear functions. The integrators are first-order active realizations tuned to reduce the inaccuracy of the frequency response. By an exponential curve fitting, we got a convenient design equation for realizing fractional-order integrators of orders from 0.1 to 0.95. The results include simulations in SPICE of the mathematical description and the electronic implementation and experimental measurements that confirm them. Monte Carlo and sensitivity tests revealed a robust realization. Contrary to its passive counterparts, the suggested realizations significantly reduce design and implementation efforts by favoring resistors and capacitors with commercial values and reducing hardware requirements.

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Section: Fractional-order Multiscroll Lü Chaotic Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Active Realization of Fractional‐Order Integrators and Their Application in Multiscroll Chaotic Systems

et al. 2021

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“…There is a wide variety of literature [24,[33][34][35][36][37] on. Research about finite-time synchronization in allusion to fractional-order chaotic system with hidden attractors, are quite limited.…”

Section: Finite-time Synchronization Of the Fractional-order System Wmentioning

confidence: 99%

Hidden Coexisting Attractors in a Fractional‐Order System without Equilibrium: Analysis, Circuit Implementation, and Finite‐Time Synchronization

Zheng

Liu

2019

Mathematical Problems in Engineering

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In this paper, a novel three-dimensional fractional-order chaotic system without equilibrium, which can present symmetric hidden coexisting chaotic attractors, is proposed. Dynamical characteristics of the fractional-order system are analyzed fully through numerical simulations, mainly including finite-time local Lyapunov exponents, bifurcation diagram, and the basins of attraction. In particular, the system can generate diverse coexisting attractors varying with different orders, which presents ample and complex dynamic characteristics. And there is great potential for secure communication. Then electronic circuit of the fractional-order system is designed to help verify its effectiveness. What is more, taking the disturbances into account, a finite-time synchronization of the fractional-order chaotic system without equilibrium is achieved and the improved controller is proven strictly by applying finite-time stable theorem. Eventually, simulation results verify the validity and rapidness of the proposed method. Therefore, the fractional-order chaotic system with hidden attractors can present better performance for practical applications, such as secure communication and image encryption, which deserve further investigation.

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“…With equilibrium, we are stating the equilibrium points of the state variables. Definition 2 also includes fractional-order dynamical systems with no-equilibria, line and surfaces of equilibria, and stable equilibria [35][36][37][38][39][40][41][42].…”

Section: Hidden Chaotic Attractor Localization In the Fractional-ordementioning

confidence: 99%

“…Therefore, the research effort oriented to hidden attractors in fractional-order dynamical systems is vital to understand this exciting and still less-explored subject of importance. In the literature, few works have reported hidden attractors in fractional-order dynamical systems with one stable equilibrium [35,36], with no-equilibria [37][38][39][40], with a line or surfaces of equilibria [41,42], or even in fractional-order hyperchaotic systems [43,44]. However, those fractional order systems generate only one family of hidden attractors, i.e., line, surface, stable, and without equilibrium.…”

Section: Introductionmentioning

confidence: 99%

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

Muñoz‐Pacheco

Zambrano-Serrano

Volos

et al. 2018

Entropy

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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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A fractional-order form of a system with stable equilibria and its synchronization

Cited by 18 publications

References 70 publications

Active Realization of Fractional‐Order Integrators and Their Application in Multiscroll Chaotic Systems

Active Realization of Fractional‐Order Integrators and Their Application in Multiscroll Chaotic Systems

Hidden Coexisting Attractors in a Fractional‐Order System without Equilibrium: Analysis, Circuit Implementation, and Finite‐Time Synchronization

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

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