10.1007/s40096-022-00478-w

On the modeling and numerical discretizations of a chaotic system via fractional operators with and without singular kernels

  • Ndolane Sene*1,
  1. Laboratoire de Finances pour le Developpement, Department of Mathematics, Cheikh Anta Diop University, Dakar Fann, SN

Published in Issue 2022-06-24

How to Cite

Sene, N. (2022). On the modeling and numerical discretizations of a chaotic system via fractional operators with and without singular kernels. Mathematical Sciences, 17(4 (December 2023). https://doi.org/10.1007/s40096-022-00478-w
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Abstract

Abstract A new chaotic system with Caputo, Atangana–Baleanu, and Caputo–Fabrizio derivatives has been presented. The conditions for the existence and uniqueness of the new fractional chaotic system solutions have been provided for the Caputo, Atangana–Baleanu, and Caputo–Fabrizio operators. The bifurcation maps to detect chaotic regions according to the variations in the model’s parameters have been proposed. The Lyapunov exponents for the fractional-order chaotic systems have been calculated to characterize the behaviors of the dynamics of the considered fractional-order system. The stability analysis of the equilibrium points of the considered model has been investigated with two methods. The phase portraits of the fractional chaotic model studied in this paper have been obtained via the fractional linear multistep method and Adams–Basford method. The fractional operators in our modeling are the Caputo derivative, the fractional derivative with the Mittag–Leffler kernel, and the Caputo–Fabrizio fractional derivative. The circuit schematics for the fractional version of our presented model regarding resistors and capacitors have been proposed to confirm the theoretical results.

Keywords

  • Chaotic attractors,
  • Bifurcation maps,
  • Lyapunov exponents

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