Dynamic analysis of a fractional-order Lorenz chaotic system☆

Yu, Yongguang; Li, Han-Xiong; Wang, Sha; Yu, Junzhi

doi:10.1016/j.chaos.2009.03.016

Cited by 168 publications

(78 citation statements)

References 28 publications

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“…It has attracted many investigators [1][2]. Many famous fractional order systems, such as Rōssler system, Lorenz system, Chua's circuit, Duffing system and so on, have been studied [3][4][5][6][7][8][9]. In view of the fact that fractional calculus provides another good way to describe, predict and control physical systems accurately, it has been applied to control system, physics and system modeling [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of fractional order Duffing system

Chen

Zhu

et al. 2015

Chaos, Solitons & Fractals

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Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of fractional order Duffing system

Chen

Zhu

et al. 2015

Chaos, Solitons & Fractals

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“…It is worth mentioning that higher-dimensional chaotic systems with more than one positive Lyapunov exponent can indeed show more complex dynamics. It has been recently demonstrated that many fractional-order systems can display chaotic (or hyperchaotic) behaviours such as: fractional-order Duffing system [7], fractionalorder Chua's system [8], fractional-order Lorenz system [9], fractional-order Chen system [10], fractional-order Rössler system [11], fractional-order Liu system [12], fractional-order Arneodo system [13].…”

Section: Introductionmentioning

confidence: 99%

Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control

Bouzeriba

Boulkroune

Bouden

2015

Neural Comput & Applic

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In this paper, the projective synchronization problem of two fractional-order different chaotic (or hyperchaotic) systems with both uncertain dynamics and external disturbances is considered. More particularly, a fuzzy adaptive control system is investigated for achieving an appropriate projective synchronization of unknown fractional-order chaotic systems. The adaptive fuzzy logic systems are used to approximate some uncertain nonlinear functions appearing in the system model. These latter are augmented by a robust control term to compensate for the unavoidable fuzzy approximation errors and external disturbances as well as residual error due to the use of the socalled e-modification in the adaptive laws. A Lyapunov approach is adopted for the design of the parameter adaptation laws and the proof of the corresponding stability as well as the asymptotic convergence of the underlying synchronization errors towards zero. The effectiveness of the proposed synchronization system is illustrated through numerical experiment results.

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“…Systems with fractional-order dynamics were proven to exist in different disciplines [1], such as viscoelastic systems [2], electromagnetic waves [3], dielectric polarization [4], quantitative finance [5], and quantum evolution of complex systems [6]. Chaotic phenomena were found in several fractional-order systems, e.g., the fractional Lorenz system [7,8], the fractional Duffing system [9][10][11], the fractional Chua system [12,13], the fractional Chen system [14,15], the fractional van der Pol system [16,17], and the fractional Newton-Leipnik system [18,19].…”

Section: Introductionmentioning

confidence: 99%

Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems

Chang

Chen

2010

Nonlinear Dyn

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Recently, the fractional-order Chen-Lee system was proven to exhibit chaos by the presence of a positive Lyapunov exponent. However, the existence of chaos in fractional-order Chen-Lee systems has never been theoretically proven in the literature. Moreover, synchronization of chaotic fractional-order systems was extensively studied through numerical simulations in some of the literature, but a theoretical analysis is still lacking. Therefore, we devoted ourselves to investigating the theoretical basis of chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen-Lee systems in this paper. Based on the stability theorems of fractional-order systems, the necessary conditions for the existence of chaos and the controllers for hybrid projective synchronization were derived. The numerical simulations show coincidence with the theoretical results.

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Dynamic analysis of a fractional-order Lorenz chaotic system☆

Cited by 168 publications

References 28 publications

Nonlinear dynamics of fractional order Duffing system

Nonlinear dynamics of fractional order Duffing system

Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control

Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems

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