The Fractional Form of the Tinkerbell Map Is Chaotic

Adel Ouannas,Thoai Vo,Samir Bendoukha,Viet-Thanh Pham,Amina-Aicha Khennaoui,Van Huynh

doi:10.3390/app8122640

Abstract

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.

Highlights

Throughout the last 50 years, chaotic dynamical systems have attracted increasing attention due to their applicability in a range of diverse and multidisciplinary fields
The main objective of this paper is to investigate the fractional Caputo-difference form of the Tinkerbell map in order to benefit from the added degrees of freedom due to the fractional nature
The bifurcation diagrams obtained using the same parameter and initial condition values from earlier are depicted in. Even though these bifurcation diagrams suggest the existence of chaos in the fractional Tinkerbell map, they are not definitive

Summary

Introduction

Throughout the last 50 years, chaotic dynamical systems have attracted increasing attention due to their applicability in a range of diverse and multidisciplinary fields. A dynamical system is said to be chaotic if its states are extremely sensitive to small variations in the initial conditions Another important property of chaotic systems is that they have attractors characterized by a complicated set of points with a fractal structure commonly referred to as a strange attractor. The Tinkerbell map has been studied by many as it exhibits very rich dynamics including a chaotic behavior and a range of periodic states. These diagrams confirm that the map exhibits a range of different behaviors.

Fractional Tinkerbell Map

Dynamics of the Fractional Tinkerbell Map

Control of the Fractional Tinkerbell Map

Conclusions