CHAOTIC BEHAVIOR OF BHALEKAR–GEJJI DYNAMICAL SYSTEM UNDER ATANGANA–BALEANU FRACTAL FRACTIONAL OPERATOR

Shabir Ahmad (), Aman Ullah (), AKGÃœL Ali () and Thabet Abdeljawad
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Shabir Ahmad: Department of Mathematics, University of Malakand, Chakdara Dir (L), 18000 Khyber Pakhtunkhwa, Pakistan
Aman Ullah: Department of Mathematics, University of Malakand, Chakdara Dir (L), 18000 Khyber Pakhtunkhwa, Pakistan
AKGÃœL Ali: Art and Science Faculty, Department of Mathematics, Siirt University, TR-56100 Siirt, Turkey
Thabet Abdeljawad: Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia4Department of Medical Research, China Medical University, 40402 Taichung, Taiwan5Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

FRACTALS (fractals), 2022, vol. 30, issue 01, 1-13

Abstract: In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal–fractional derivatives in which fractional derivative is taken in Atangana–Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam–Hyers stable under the new fractal–fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.

Keywords: Fractal–Fractional Operator; Bhalekar–Gejji Dynamical System; Ulam–Hyers Stability (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22400059

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