On chaos in the fractional-order Grassi–Miller map and its control

Ouannas, Adel; Khennaoui, Amina–Aicha; Grassi, Giuseppe; Bendoukha, Samir

doi:10.1016/j.cam.2019.03.031

Cited by 39 publications

(41 citation statements)

References 20 publications

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“…In [11], three different fractional-order discrete-time systems (FoDs) have been investigated, i.e., Wang's, Rossler's, and Stefanski's maps. In [12], the chaotic behaviors of the fractional-order sine and standard maps were analyzed, whereas in [13], the dynamic properties of the fractional-order Grassi-Miller map have been illustrated in detail. Additionally, the presence of chaos in the fractionalorder discrete double scroll map has been investigated in [14], whereas in [12], the fractional-order delayed logistic map was analyzed regarding to its chaotic behavior.…”

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

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“…However, some research workers have recently concentrated their considerations on studying the chaotic behaviours associated with the dynamics of the Fractional-order Discrete Systems (FoDSs), i.e., maps outlined by Fractional-order Difference Equations (FoDEs) [11,12]. To this purpose, the chaotic dynamics of some FoDSs have been recently explored and analyzed in [13][14][15][16][17][18][19]. In particular, a generalized version of a 3D fractional-order Hénon system has been investigated in [14], while the fractional-order Wang, Rossler, and Stefanski discrete systems have been demonstrated in [13].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, a generalized version of a 3D fractional-order Hénon system has been investigated in [14], while the fractional-order Wang, Rossler, and Stefanski discrete systems have been demonstrated in [13]. Additionally, the fractional discrete double scroll has been introduced in [16], whereas the chaotic dynamics of the fractional Grassi-Miller map have been studied in [15]. Referring to very recent results in literature for fractional discrete systems, a fractional logistic map characterized by two-parameters has been illustrated in [20].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.

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“…Controlling these systems consists in proposing a suitable adaptive controller for their chaotic modes, so that their states are forced to be asymptotically stable, or are stabilized at zero [18,41,42]. Control issues are, for instance, of great importance in several industrial processes, like in robotics where chaotic motions of a rigid body need to be controlled [43][44][45]. On the other hand, synchronization, which has been considered a key concept in chaos theory over the last three decades, targets to compel the states of a slave system to tend towards the exact trajectories that are determined by a master system, assuming that both systems start from different initial points in phase space [19].…”

Section: Introductionmentioning

confidence: 99%

Different dimensional fractional-order discrete chaotic systems based on the Caputo h-difference discrete operator: dynamics, control, and synchronization

et al. 2020

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The paper investigates control and synchronization of fractional-order maps described by the Caputo h-difference operator. At first, two new fractional maps are introduced, i.e., the Two-Dimensional Fractional-order Lorenz Discrete System (2D-FoLDS) and Three-Dimensional Fractional-order Wang Discrete System (3D-FoWDS). Then, some novel theorems based on the Lyapunov approach are proved, with the aim of controlling and synchronizing the map dynamics. In particular, a new hybrid scheme is proposed, which enables synchronization to be achieved between a master system based on a 2D-FoLDS and a slave system based on a 3D-FoWDS. Simulation results are reported to highlight the effectiveness of the conceived approach.

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On chaos in the fractional-order Grassi–Miller map and its control

Cited by 39 publications

References 20 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

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

Different dimensional fractional-order discrete chaotic systems based on the Caputo h-difference discrete operator: dynamics, control, and synchronization

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