On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Khennaoui, Amina–Aicha; Ouannas, Adel; Momani, Shaher; Batiha, Iqbal M.; Dibi, Z.; Grassi, Giuseppe

doi:10.3390/electronics9122179

Cited by 11 publications

(3 citation statements)

References 33 publications

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“…Several complex phenomena can be modeled with the help of using these equations [10,11]. As a result, many applications of these equations can be found in the study of viscoelasticity, electrochemistry, signal processing, control theory, porous media, fluid mechanics, rheology, transport by diffusion, electrical networks, electromagnetic, probability distributions, and many other processes [5,6,12]. In fact, the existence and uniqueness results of the weak solutions were widely investigated for several fractional-order partial differential equations using Lax Milgram theorem, see [13,14] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On a Weak Solution of a Fractional-order Temporal Equation

Batiha¹,

Chebana²,

Oussaeif³

et al. 2022

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Several real-world phenomena emerging in engineering and science fields can be described successfully by developing certain models using fractional-order partial differential equations. The exact, analytical, semi-analytical or even numerical solutions for these models should be examined and investigated by distinguishing between their solvablities and non-solvabilities. In this paper, we aim to establish some sufficient conditions for exploring the existence and uniqueness of solution for a class of initial-boundary value problems with Dirichlet condition. The gained results from this research paper are established for the class of fractional-order partial differential equations by a method based on Lax Milgram theorem, which relies in its construction on properties of the symmetric part of the bilinear form. Lax Milgram theorem is deemed as a mathematical scheme that can be used to examine the existence and uniqueness of weak solutions for fractional-order partial differential equations. These equations are formulated here in view of the Caputo fractional-order derivative operator, which its inverse operator is the Riemann-Louville fractional-order integral one. The results of this paper will be supportive for mathematical analyzers and researchers when a fractional-order partial differential equation is handled in terms of finding its exact, analytical, semi-analytical or numerical solution.

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Section: Introductionmentioning

confidence: 99%

On a Weak Solution of a Fractional-order Temporal Equation

Batiha¹,

Chebana²,

Oussaeif³

et al. 2022

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“…In [16], some synchronization and control schemes of FoDSs formulated by the Caputo h-difference operator were investigated and evolved. In [17,18], several dynamics of the FoDSs were explored and investigated in terms of their chaotic phenomena. In [19], a specific stabilization of the chaotic dynamics of the FoDSs formulated by the h-fractional-order difference operator was performed.…”

Section: Introductionmentioning

confidence: 99%

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

et al. 2022

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This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand.

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“…With the help of using the Poincar'e section of the Lorenz system, the first discrete-time chaotic system (map) was established by Hénon [2]. Further consideration increased in dealing with chaotic maps of discrete-time were then established over the years including [3][4][5][6][7][8]. These chaotic maps might be categorized into certain classes in accordance with their nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

A new two-dimensional fractional discrete rational map: chaos and complexity

Shatnawi¹,

Abbes²,

Ouannas³

et al. 2022

Phys. Scr.

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In this paper, a new two-dimensional fractional discrete rational map with γth-Caputo fractional difference operator is introduced. The study of the presence and stability of the equilibrium points shows that there are four types of equilibrium points; no equilibrium point, a line of equilibrium points, one equilibrium point and two equilibrium points. In addition, in the context of the commensurate and incommensurate instances, the nonlinear dynamics of the suggested fractional discrete map in different cases of equilibrium points are investigated through several numerical techniques including Lyapunov exponents, phase attractors and bifurcation diagrams. These dynamic behaviors suggest that the fractional discrete rational map has both hidden and self-excited attractors, which have rarely been described in the literature. Finally, to validate the presence of chaos, a complexity analysis is carried out using approximation entropy (ApEn) and the C0-measure.

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On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Cited by 11 publications

References 33 publications

On a Weak Solution of a Fractional-order Temporal Equation

On a Weak Solution of a Fractional-order Temporal Equation

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

A new two-dimensional fractional discrete rational map: chaos and complexity

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