Chaos in a new system with fractional order

Sheu, Long-Jye; Chen, Hsien-Keng; Chen, Juhn-Horng; Tam, Lap Mou

doi:10.1016/j.chaos.2005.10.073

Cited by 53 publications

(11 citation statements)

References 29 publications

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“…Since then, the Chen-Lee system has garnered much attention [34][35][36]. The study of Chen-Lee systems was extended from integer orders to fractional orders by Sheu et al [23] in 2007. In their study, the existence of chaos in fractional-order ChenLee systems was numerically proven by the presence of a positive Lyapunov exponent, but the existence of chaos in fractional-order Chen-Lee system has never been theoretically proven in the literature.…”

Section: Chaos In Fractional-order Chen-lee Systemsmentioning

confidence: 99%

“…The Chen-Lee system relates to gyro motion originating from the anticontrol of chaos in a rigid body, and it can be implemented in electronic circuits [22]. Recently, Sheu et al studied chaos of fractionalorder Chen-Lee systems [23]. In their study, the existence of chaos in fractional-order Chen-Lee systems was proven numerically by the presence of a positive Lyapunov exponent, but the existence of chaos in fractional-order Chen-Lee systems has never been theoretically proven in the literature.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Chaos In Fractional-order Chen-lee Systemsmentioning

confidence: 99%

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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“…Recent studies show that chaotic fractional order systems can also be synchronized [19]- [32]. In many literatures, synchronization among fractional order systems is only investigated through numerical simulations that are based on the stability criteria of linear fractional order systems, such as the work presented in [24]- [27], or based on Laplace transform theory, such as the work presented in [28]- [32].…”

Section: Introductionmentioning

confidence: 99%

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

Odibat

2009

Nonlinear Dyn

150

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To cite this version:Zaid M. Odibat. Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dynamics, Springer Verlag, 2009, 60 (4) Abstract This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have nonidentical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to asses the performance of the proposed adaptive controller in synchronizing chaotic systems.

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“…This notion of the Lyapunov spectrum is used to investigate the chaotic behavior in a class of fractional differential systems, see e.g. [5,13,20].…”

Section: Introductionmentioning

confidence: 99%

On fractional lyapunov exponent for solutions of linear fractional differential equations

Cong

Doàn

2014

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Our aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.MSC 2010 : Primary 34A08; Secondary 34D08, 34D20

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Chaos in a new system with fractional order

Cited by 53 publications

References 29 publications

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

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

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

On fractional lyapunov exponent for solutions of linear fractional differential equations

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