On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system

Anastacia Dlamini; Emile F. Doungmo Goufo; Melusi Khumalo; Anastacia Dlamini; Emile F. Doungmo Goufo; Melusi Khumalo

doi:10.3934/math.2021717

AIMS Mathematics

2021, Volume 6, Issue 11: 12395-12421. doi: 10.3934/math.2021717

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On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system

Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

Received: 13 March 2021 Accepted: 26 July 2021 Published: 30 August 2021
MSC : 26A33, 34A08, 34B15

The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.
- Caputo-Fabrizio fractal-fractional derivatives and integral operators,
- Lorenz chaotic system,
- well-posedness,
- Adams-Bashforth method
Citation: Anastacia Dlamini, Emile F. Doungmo Goufo, Melusi Khumalo. On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system[J]. AIMS Mathematics, 2021, 6(11): 12395-12421. doi: 10.3934/math.2021717

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Abstract

The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.

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