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Validity of fractal derivative to capturing chaotic attractors

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  • Atangana, Abdon
  • Khan, Muhammad Altaf

Abstract

Suggested independently with different definitions, fractal derivative and conformable derivative are α proportional. They have been applied in quit a few problems in many field of sciences in the last few years with great success. However, some researchers have pointed out some criticisms and even concluded that they were flawed. In this paper, while confirm the validity of the conformable and fractal derivatives and we present their applications to chaotic attractors. We considered a general non-linear Cauchy problem where the differential operator is that of fractal and conformable and present the derivation of conditions for which the existence and the uniqueness of the exact solution are reached. Several examples are considered, solved and numerical simulations depicting real world observations.

Suggested Citation

  • Atangana, Abdon & Khan, Muhammad Altaf, 2019. "Validity of fractal derivative to capturing chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 50-59.
    • Handle: RePEc:eee:chsofr:v:126:y:2019:i:c:p:50-59
    DOI: 10.1016/j.chaos.2019.06.002
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    References listed on IDEAS

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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. Kanno, Ryutaro, 1998. "Representation of random walk in fractal space-time," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 248(1), pages 165-175.
    3. Doungmo Goufo, Emile F. & Mbehou, Mohamed & Kamga Pene, Morgan M., 2018. "A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 170-176.
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    14. El-Nabulsi, Rami Ahmad, 2024. "Transition from circular to spiral waves and from Mexican hat to upside-down Mexican hat-solutions: The cases of local and nonlocal λ−ω reaction-diffusion-convection fractal systems with variable coef," Chaos, Solitons & Fractals, Elsevier, vol. 189(P2).
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