Analysis of a chaotic system using fractal-fractional derivatives with exponential decay type kernels

Ihtisham Ul Haq; Nigar Ali; Hijaz Ahmad; Ihtisham Ul Haq; Nigar Ali; Hijaz Ahmad

doi:10.3934/mmc.2022019

Mathematical Modelling and Control

2022, Volume 2, Issue 4: 185-199. doi: 10.3934/mmc.2022019

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Research article

Analysis of a chaotic system using fractal-fractional derivatives with exponential decay type kernels

1.
Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan
2.
Near East University, Operational Research Center in Healthcare, Near East Boulevard, PC: 99138 Nicosia/Mersin 10, Turkey
3.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 39,00186 Roma, Italy

Received: 19 May 2022 Revised: 09 November 2022 Accepted: 14 November 2022 Published: 19 December 2022

In this article, we introduce and analyze a novel fractal-fractional chaotic system. We extended the memristor-based chaotic system to the fractal-fractional mathematical model using Atangana-Baleanu–Caputo and Caputo-Fabrizio types of derivatives with exponential decay type kernels. We established the uniqueness and existence of the solution through Banach's fixed theory and Schauder's fixed point. We used some new numerical methods to derive the solution of the considered model and study the dynamical behavior using these operators. The numerical simulation results presented in both cases include the two and three-dimensional phase portraits and the time-domain responses of the state variables to evaluate the efficacy of both kernels.
- fractal-fractional derivative,
- memristor-based chaotic system,
- Schauder's fixed point,
- Banach fixed theory
Citation: Ihtisham Ul Haq, Nigar Ali, Hijaz Ahmad. Analysis of a chaotic system using fractal-fractional derivatives with exponential decay type kernels[J]. Mathematical Modelling and Control, 2022, 2(4): 185-199. doi: 10.3934/mmc.2022019

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Abstract

In this article, we introduce and analyze a novel fractal-fractional chaotic system. We extended the memristor-based chaotic system to the fractal-fractional mathematical model using Atangana-Baleanu–Caputo and Caputo-Fabrizio types of derivatives with exponential decay type kernels. We established the uniqueness and existence of the solution through Banach's fixed theory and Schauder's fixed point. We used some new numerical methods to derive the solution of the considered model and study the dynamical behavior using these operators. The numerical simulation results presented in both cases include the two and three-dimensional phase portraits and the time-domain responses of the state variables to evaluate the efficacy of both kernels.

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