On the dynamics, control and synchronization of fractional-order Ikeda map

Ouannas, Adel; Khennaoui, Amina–Aicha; Odibat, Zaid; Pham, Viet–Thanh; Grassi, Giuseppe

doi:10.1016/j.chaos.2019.04.002

Cited by 74 publications

(34 citation statements)

References 46 publications

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“…Referring to fractional-order chaotic discrete-time systems (i.e., systems outlined by difference equations of fractional order), many scholars have mainly focused on the system's dynamics characterized by the presence of "selfexcited attractors" [7,8]. For example, the so-called generalized Hénon map of three dimensions has been studied in [9], while some dynamics of the fractionalized logistic map were examined in [10].…”

Section: Introductionmentioning

confidence: 99%

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Khennaoui

Almatroud

Ouannas

et al. 2021

Mathematical Problems in Engineering

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Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.

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Section: Introductionmentioning

confidence: 99%

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Khennaoui

Almatroud

Ouannas

et al. 2021

Mathematical Problems in Engineering

Self Cite

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“…Modeling chaotic attractors have been a great interest of humankind in the last decades 1–6 . Some interested models have been suggested, and many more have been modified with the aim to capture more attractors.…”

Section: Introductionmentioning

confidence: 99%

“…Modeling chaotic attractors have been a great interest of humankind in the last decades. [1][2][3][4][5][6] Some interested models have been suggested, and many more have been modified with the aim to capture more attractors. However, there exist in nature attractors with self-similarities; this class cannot be captured neither with classical differentiation nor with classical fractional and new trends in fractional differentiation and integration.…”

Section: Introductionmentioning

confidence: 99%

New chaotic attractors: Application of fractal‐fractional differentiation and integration

Gómez‐Aguilar

Atangana

2020

Math Methods in App Sciences

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Very recently, the concept of fractal differentiation and fractional differentiation has been combined to produce new differentiation operators. The new operators were constructed using three different kernels, namely, power law, exponential decay, and the generalized Mittag‐Leffler function. The new operators have two parameters: the first is considered as fractional order and the second as fractal dimension. In this work, we applied these new operators to model some chaotic attractors, and the models were solved numerically using a new and very efficient numerical scheme. We presented numerical simulations for some specific fractional order and fractal dimension. The classical fractional differential models could be recovered when the fractal dimension is equal to 1; in these cases, the obtained attractors with power law presented no similarities. Nevertheless, those obtained via Caputo‐Fabrizio and the Atangana‐Baleanu derivative show some crossover effects, which is due to non‐index law property. However, those obtained from fractal‐fractional, in particular, those with the Mittag‐Leffler kernel, show very strange and new attractors with self‐similarities; these results are obtained for the first time. We conclude that this new concept is the future to modelling complexities with self‐similarities.

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“…In the past three centuries, the study of fractional calculus theory has been carried out mainly in the purely theoretical field of mathematics, but in recent decades, fractional differential equations and fractional difference equations have been used more and more to describe optical and thermal systems, mechanics systems, signal processing, system identification, robotics and other applications [9][10][11][12][13][14][15]. A great number of FO chaotic maps in continuous-time and discrete-time were investigated in recent years, including FO Chen map [16], FO Rossler map [17], FO Lorenz map [18], FO Lu map [19], FO Ikeda map [20], FO Sine map [21], FO cubic Logistic map [22]. In recent years, many scholars have also studied the sliding mode control and circuit implementation of FO chaotic systems [39][40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

A New Matrix Projective Synchronization of Fractional-Order Discrete-Time Systems and its Application in Secure Communication

Yan

Ding

2020

IEEE Access

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In this study, it is proposed that a new matrix projective synchronization of fractional-order (FO) chaotic maps in discrete-time. A new synchronization error is introduced and a control law is constructed, which makes the synchronization error converge towards zero in sufficient time under the stability theory of linearization method of FO systems. Numerical simulation results are presented to illustrate the feasibility of the scheme. Finally, a secure communication scheme based on FO discrete-time (FODT) systems was proposed.

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On the dynamics, control and synchronization of fractional-order Ikeda map

Cited by 74 publications

References 46 publications

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

New chaotic attractors: Application of fractal‐fractional differentiation and integration

A New Matrix Projective Synchronization of Fractional-Order Discrete-Time Systems and its Application in Secure Communication

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