Lyapunov functions for Riemann–Liouville-like fractional difference equations

Wu, Guo–Cheng; Băleanu, Dumitru; Luo, Wei-Hua

doi:10.1016/j.amc.2017.06.019

Cited by 115 publications

(76 citation statements)

References 36 publications

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“…Researchers have shown an increased interest in control and synchronization of fractional-order systems [55][56][57][68][69][70]. We have studied the synchronization of fractional systems with different orders.…”

Section: Resultsmentioning

confidence: 99%

“…The complexity of proposed synchronization schemes can be used in secure communication and chaotic encryption schemes. In our future works, we will use the recent stability results of fractional systems [55][56][57] for discrete fractional systems and their synchronization.…”

Section: Resultsmentioning

confidence: 99%

“…The fractional derivatives play important roles in the field of mathematical modeling of numerous models such as fractional model of regularized long-wave equation [51], Lienard's equation [52], fractional model of convective radial fins [53], modified Kawahara equation [54], etc. In recent years, there has been an increasing interest in the stability of fractional systems [55][56][57]. Moreover, it is worth noting that local fractional derivatives with special functions have received significant attention in different areas [58][59][60].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2018

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There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2018

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“…Recently, there have been many papers concerning on the stability of fractional-order systems [2,3,7,22,23,36,39]. Next, we give the subsequent lemmas which are helpful in proving our results.…”

Section: Definition 21 ([5]mentioning

confidence: 99%

Anti-synchronization of fractional-order chaotic complex systems with unknown parameters via adaptive control

Jiang¹,

Zhang²,

Qin³

et al. 2017

J. Nonlinear Sci. Appl.

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This paper is concerned with adaptive control for anti-synchronization of a class of uncertain fractional-order chaotic complex systems described by a unified mathematical expression. By utilizing the recently established result for the Caputo fractional derivative of a quadratic function and employing the adaptive control technique, we design controllers and some fractional-order parameter update laws to anti-synchronize two fractional-order chaotic complex systems with unknown parameters. The proposed method has generality, simplicity, and feasibility. Moreover, anti-synchronization between uncertain fractional-order complex Lorenz system and fractional-order complex Lü system is implemented as an example to demonstrate the effectiveness and feasibility of the proposed scheme.

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“…The numerical approach to fractional differential equations (FDEs) has been discussed by numerous researchers [2-5, 7, 10-14, 17, 19, 26]. Recently, in [23][24][25], some new finite memory fractional differences have been proposed. And these definitions can be considered for discrete fractional modeling.…”

Section: Introductionmentioning

confidence: 99%

Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term

Yang¹

2017

J. Nonlinear Sci. Appl.

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In this paper, we propose a finite difference method for the Riesz space fractional diffusion equations with delay and a nonlinear source term on a finite domain. The proposed method combines a time scheme based on the predictor-corrector method and the Crank-Nicolson scheme for the spatial discretization. The corresponding theoretical results including stability and convergence are provided. Some numerical examples are presented to validate the proposed method.

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Lyapunov functions for Riemann–Liouville-like fractional difference equations

Cited by 115 publications

References 36 publications

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

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

Anti-synchronization of fractional-order chaotic complex systems with unknown parameters via adaptive control

Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term

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