Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Othman Almatroud ◽  
Amina-Aicha Khennaoui ◽  
Adel Ouannas ◽  
Viet-Thanh Pham
Keyword(s):  
Fractional Order ◽  
Chaotic Dynamics ◽  
Approximate Entropy ◽  
Interesting Property ◽  
Chaotic Attractors ◽  
The Past ◽  
Offset Boosting ◽  
Discrete Time Systems ◽  
Infinite Line ◽  
Time Systems

Abstract The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention in the past years. In this paper, we propose a new 2D fractional map with the simplest algebraic structure reported to date and with an infinite line of equilibrium. The conceived map possesses an interesting property not explored in literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of periodic, chaotic and hyper-chaotic attractors. Bifurcation diagrams, computation of the maximum Lyapunov exponents, phase plots and 0–1 test are reported, with the aim to analyse the dynamics of the 2D fractional map as well as to highlight the coexistence of initial-boosting chaotic and hyperchaotic attractors in commensurate and incommensurate order. Results show that the 2D fractional map has an infinite number of coexistence symmetrical chaotic and hyper-chaotic attractors. Finally, the complexity of the fractional-order map is investigated in detail via approximate entropy.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Amina Aicha Khennaoui ◽  
A. Othman Almatroud ◽  
Adel Ouannas ◽  
M. Mossa Al-sawalha ◽  
Giuseppe Grassi ◽  
...  
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Test Method ◽  
Order System ◽  
Chaotic Attractors ◽  
Discrete Time System ◽  
Discrete Time Systems ◽  
Time Systems

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.

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

2019 ◽  
Vol 119 ◽  
pp. 150-162 ◽  
Author(s):  
Amina-Aicha Khennaoui ◽  
Adel Ouannas ◽  
Samir Bendoukha ◽  
Giuseppe Grassi ◽  
René Pierre Lozi ◽  
...  
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Discrete Time Systems ◽  
Time Systems ◽  
Chaos Stabilization

scholarly journals Chaos Synchronization between Fractional-Order Unified Chaotic System and Rossler Chaotic System

2012 ◽  
Vol 562-564 ◽  
pp. 2088-2091
Author(s):  
Xian Yong Wu ◽  
Yi Long Cheng ◽  
Kai Liu ◽  
Xin Liang Yu ◽  
Xian Qian Wu
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Chaotic Attractors ◽  
System Parameters ◽  
Unified Chaotic System ◽  
Simulation Results ◽  
Drive Systems ◽  
Asymptotic Synchronization

The chaotic dynamics of the unified chaotic system and the Rossler system with different fractional-order are studied in this paper. The research shows that the chaotic attractors can be found in the two systems while the orders of the systems are less than three. Asymptotic synchronization of response and drive systems is realized by active control through designing proper controller when system parameters are known. Theoretical analysis and simulation results demonstrate the effective of this method.

Approximating Chaotic Attractors by Period-Three Cycles in Discrete Stochastic Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550138 ◽  
Author(s):  
Irina Bashkirtseva ◽  
Lev Ryashko
Keyword(s):  
Stochastic Systems ◽  
Original System ◽  
Sensitivity Function ◽  
Function Technique ◽  
Chaotic Attractors ◽  
Suggested Approach ◽  
One Dimensional ◽  
Discrete Time Systems ◽  
Random States ◽  
Time Systems

We study a distribution of random states forced outwards of the general deterministic attractor (regular or chaotic) for one-dimensional discrete-time systems with unimodal map. To approximate this distribution, we replace the original system by the appropriate modeling system with stable 3-cycle and use the stochastic sensitivity function technique. Constructive abilities of the suggested approach are demonstrated in the analysis of noise-induced transitions from chaos to order.

scholarly journals Generalized convergence analysis of the fractional order systems

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 404-411
Author(s):  
Ahmad Ruzitalab ◽  
Mohammad Hadi Farahi ◽  
Gholamhossien Erjaee
Keyword(s):  
Fractional Order ◽  
Fractional Order Systems ◽  
Necessary And Sufficient Condition ◽  
Lyapunov Matrix Equation ◽  
Discrete Time Systems ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Contraction Theory ◽  
Time Systems

Abstract The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

scholarly journals Puu System of Fractional Order and Its Chaos Suppression

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 340 ◽  
Author(s):  
Marius-F. Danca
Keyword(s):  
Periodic Solutions ◽  
Fractional Order ◽  
Discrete Time ◽  
Continuous Time ◽  
Control Algorithm ◽  
Chaos Control ◽  
Chaos Suppression ◽  
Discrete Time Systems ◽  
Time Systems ◽  
Order Continuous

In this paper, the fractional-order variant of Puu’s system is introduced, and, comparatively with its integer-order counterpart, some of its characteristics are presented. Next, an impulsive chaos control algorithm is applied to suppress the chaos. Because fractional-order continuous-time or discrete-time systems have not had non-constant periodic solutions, chaos suppression is considered under some numerical assumptions.

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