Open AccessCHAOTIC BEHAVIOR OF BHALEKAR–GEJJI DYNAMICAL SYSTEM UNDER ATANGANA–BALEANU FRACTAL FRACTIONAL OPERATOR
Author(s) -
Shabir Ahmad,
Aman Ullah,
Ali Akgül,
Thabet Abdeljawad
Publication year - 2021
Publication title -
fractals
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.654
H-Index - 44
eISSN - 1793-6543
pISSN - 0218-348X
DOI - 10.1142/s0218348x22400059
Subject(s) - fractal , mathematics , fractional calculus , uniqueness , fixed point theorem , fractal dimension , operator (biology) , banach fixed point theorem , mathematical analysis , chaotic , pure mathematics , computer science , biochemistry , chemistry , repressor , artificial intelligence , transcription factor , gene
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.