Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel. (November 2021)

• Record Type: Journal Article
• Authors: Saifullah, Sayed; Ali, Amir; Franc Doungmo Goufo, Emile
• Is Part Of: Chaos, solitons and fractals. Volume 152(2021)
• Publication Date: 2021-11
• Subjects: Chaotic attractor -- fractal-fractional Mittag-Leffler law -- Schauder fixed-point theorem -- Banach fixed point theorem -- Ulam-Hyers stability -- Adam-Bashforth scheme -- Lagrange piecewise interpolation; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2021.111332
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract : In this article, the behaviour of a chaotic attractor in the fractal-fractional Mittag-Leffler perspective is studied. The diverse features of the chaotic attractor are observed with different fractal and fractional orders. The existence and uniqueness of the system are presented by using Schauder and Banach fixed point theorems. The stability of the system is verified by using Ulam-Hyers stability. The numerical scheme based on Adam-Bashforth the scheme is established with Lagrangian piecewise interpolation. The fractal dimension shows a significant impact on the dynamical system, such that a small change in fractal order causes a rapid change in the system's behaviour. The complex behaviour of the considered system is numerically illustrated using various fractal and fractional orders. Abstract: This article aims to study the behaviour of a chaotic attractor in the fractal-fractional Mittag-Leffler perspective. The different aspects of the chaotic attractor are observed with different fractal and fractional orders. The existence and uniqueness of the system are presented by using Schauder and Banach fixed point theorems. The stability analysis of the equilibrium points of the system is presented together with Ulam-Hyers stability for the system under consideration. The Lyapunov spectra and the bifurcation in the system with respect to control parameter ζ are studied. The numerical scheme based on the Adam-Bashforth method is established with Lagrangian piecewise interpolation. The complex behaviour of the considered system is numerically illustrated using various fractal and fractional orders. It is observed that, the chaotic attractor self-replicates its pattern in the fractal process when fractal dimension varies.