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Entropy 2016, 18(4), 100; doi:10.3390/e18040100

Chaos on the Vallis Model for El Niño with Fractional Operators

1
Department of Mathematics, Colleges of Sciences, King Saud University, P.O. Box 1142, Riyadh 11989, Saudi Arabia
2
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9301, South Africa
*
Author to whom correspondence should be addressed.
Academic Editors: J. A. Tenreiro Machado and António M. Lopes
Received: 13 January 2016 / Revised: 9 March 2016 / Accepted: 10 March 2016 / Published: 23 March 2016
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Abstract

The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first time the exact solution for equilibrium points, and then we presented the exact solutions with the numerical simulations. We further examined the model within the scope of fractional order derivatives. The fractional derivatives used here are the Caputo derivative and Caputo–Fabrizio type. Within the scope of fractional derivatives, we presented the existence and unique solutions of the model. We derive special solutions of both models with Caputo and Caputo–Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo–Fabrizio derivative than other one with local and Caputo derivative. When compare the three models, we realized that, the Caputo derivative plays a role of low band filter when the Caputo–Fabrizio presents more information that were not revealed in the model with local derivative. View Full-Text
Keywords: Vallis model; chaotic behavior; Caputo–Fabrizio fractional derivative; analysis; numerical simulations Vallis model; chaotic behavior; Caputo–Fabrizio fractional derivative; analysis; numerical simulations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Alkahtani, B.S.T.; Atangana, A. Chaos on the Vallis Model for El Niño with Fractional Operators. Entropy 2016, 18, 100.

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