The Fractional Form of the Tinkerbell Map Is Chaotic

The Fractional Form of the Tinkerbell Map Is Chaotic

This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilizat...

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Main Authors: Adel Ouannas, Amina-Aicha Khennaoui, Samir Bendoukha, Thoai Phu Vo, Viet-Thanh Pham, Van Van Huynh
Format: Article
Language:English
Published: MDPI AG 2018-12-01
Series:Applied Sciences
Subjects:
fractional discrete calculus
discrete chaos
Tinkerbell map
bifurcation
stabilization
Online Access:https://www.mdpi.com/2076-3417/8/12/2640
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spelling doaj-ba8aea0d01784f998ff592118c4f525c2020-11-24T21:45:39ZengMDPI AGApplied Sciences2076-34172018-12-01812264010.3390/app8122640app8122640The Fractional Form of the Tinkerbell Map Is ChaoticAdel Ouannas0Amina-Aicha Khennaoui1Samir Bendoukha2Thoai Phu Vo3Viet-Thanh Pham4Van Van Huynh5Department of Mathematics and Computer Science, University of Larbi Tebessi, Tebessa 12002, AlgeriaDepartment of Mathematics and Computer Sciences, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, AlgeriaElectrical Engineering Department, College of Engineering at Yanbu, Taibah University, Medina 42353, Saudi ArabiaFaculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, VietnamNonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, VietnamModeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, VietnamThis paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings.https://www.mdpi.com/2076-3417/8/12/2640fractional discrete calculusdiscrete chaosTinkerbell mapbifurcationstabilization
collection DOAJ
language English
format Article
sources DOAJ
author Adel Ouannas
Amina-Aicha Khennaoui
Samir Bendoukha
Thoai Phu Vo
Viet-Thanh Pham
Van Van Huynh
spellingShingle Adel Ouannas
Amina-Aicha Khennaoui
Samir Bendoukha
Thoai Phu Vo
Viet-Thanh Pham
Van Van Huynh
The Fractional Form of the Tinkerbell Map Is Chaotic
Applied Sciences
fractional discrete calculus
discrete chaos
Tinkerbell map
bifurcation
stabilization
author_facet Adel Ouannas
Amina-Aicha Khennaoui
Samir Bendoukha
Thoai Phu Vo
Viet-Thanh Pham
Van Van Huynh
author_sort Adel Ouannas
title The Fractional Form of the Tinkerbell Map Is Chaotic
title_short The Fractional Form of the Tinkerbell Map Is Chaotic
title_full The Fractional Form of the Tinkerbell Map Is Chaotic
title_fullStr The Fractional Form of the Tinkerbell Map Is Chaotic
title_full_unstemmed The Fractional Form of the Tinkerbell Map Is Chaotic
title_sort fractional form of the tinkerbell map is chaotic
publisher MDPI AG
series Applied Sciences
issn 2076-3417
publishDate 2018-12-01
description This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings.
topic fractional discrete calculus
discrete chaos
Tinkerbell map
bifurcation
stabilization
url https://www.mdpi.com/2076-3417/8/12/2640
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