On fractional–order discrete–time systems: Chaos, stabilization and synchronization

Khennaoui, Amina–Aicha; Ouannas, Adel; Bendoukha, Samir; Grassi, Giuseppe; Lozi, René; Pham, Viet–Thanh

doi:10.1016/j.chaos.2018.12.019

Cited by 102 publications

(63 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The earlier release of the FoLDS was established in [55] using the ν-Caputo delta DDO. It turned out that this map, which possesses two nonlinear terms, is actually chaotic for some proper values of its parameters (α, β) and fractional-orders ν, where ν ∈ (0, 1].…”

Section: D-foldsmentioning

confidence: 99%

See 1 more Smart Citation

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

et al. 2020

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: D-foldsmentioning

confidence: 99%

“…where C h ν a denotes the Caputo h-DDO, t ∈ (hN) a+ (1-ν)h , a ∈ R is the starting point, and α and β are the system's parameters. Map (5), however, can be regarded as a generalized form of the FoLDS constructed in [55]. Its solution, moreover, can be obtained via employing the fractional h-sum operator.…”

Section: D-foldsmentioning

confidence: 99%

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

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…The fractional-order Hénon map is defined as:

where

and a and b are the bifurcation parameters. Here, the Caputo difference

[ 55 ] is used. By employing the numerical solution scheme, its solution is denoted as:

where

.…”

Section: Applications To Chaotic Systemsmentioning

confidence: 99%

Design of a Network Permutation Entropy and Its Applications for Chaotic Time Series and EEG Signals

Sun

2019

Entropy

View full text Add to dashboard Cite

Measuring the complexity of time series provides an important indicator for characteristic analysis of nonlinear systems. The permutation entropy (PE) is widely used, but it still needs to be modified. In this paper, the PE algorithm is improved by introducing the concept of the network, and the network PE (NPE) is proposed. The connections are established based on both the patterns and weights of the reconstructed vectors. The complexity of different chaotic systems is analyzed. As with the PE algorithm, the NPE algorithm-based analysis results are also reliable for chaotic systems. Finally, the NPE is applied to estimate the complexity of EEG signals of normal healthy persons and epileptic patients. It is shown that the normal healthy persons have the largest NPE values, while the EEG signals of epileptic patients are lower during both seizure-free intervals and seizure activity. Hence, NPE could be used as an alternative to PE for the nonlinear characteristics of chaotic systems and EEG signal-based physiological and biomedical analysis.Entropy 2019, 21, 849 2 of 18 algorithm; thus, complexity can be analyzed in a multiple scale sense. After this, many different multiscale complexity algorithms such as the multiscale PE (MPE) algorithm [27], the multiscale SampEn algorithm [28], the multiscale permutation Rényi entropy [29], and the multivariate multiscale entropy [30] were designed. Among those methods, the PE algorithm and MPE algorithm have a fast estimation speed and are gaining more and more academic attention. Therefore, this article will concentrate on the PE algorithm.In 2002, Bandt and Pompe [23] proposed the PE algorithm based on the patterns deduced from constructed vectors. However, it was indicated in many literature works that the PE algorithm has some drawbacks that cannot always measure complexity effectively. For instance, Zunino et al. [31] figured out that the Bandt-Pompe method for processing equal values could lead to erroneous conclusions. Until now, how to improve the PE algorithm is still an open project. At present, several improved PE algorithms have been proposed. Bian et al. [32] modified the PE algorithm by mapping the equal value onto the same symbol (rank), and the modified PE (mPE) algorithm was designed. Fadlallah et al. [33] proposed the weighted PE (WPE) algorithm, and EEG signals were analyzed to verify its effectiveness. Azami et al. [34] indicated that PE does not consider the average of the amplitude values and equal amplitude values and designed the amplitude-aware permutation entropy (AAPE). Chen et al. [35]proposed the improved PE (IPE) algorithm by introducing a symbolic process and combining some advantages of previous modifications of PE. As a result, IPE has more patterns and better robustness for noise-polluted time series. Those algorithms have better measuring results when comparing with the original PE algorithm. However, they do not consider the relationship between different patterns since they actually treat the vectors as independent units. Currently, appro...

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

On fractional–order discrete–time systems: Chaos, stabilization and synchronization

Cited by 102 publications

References 34 publications

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

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

Design of a Network Permutation Entropy and Its Applications for Chaotic Time Series and EEG Signals

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

Contact Info

Product

Resources

About