{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T19:48:33Z","timestamp":1780084113860,"version":"3.54.0"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,3,23]],"date-time":"2016-03-23T00:00:00Z","timestamp":1458691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Vallis model for El Ni\u00f1o 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\u2013Fabrizio 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\u2013Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo\u2013Fabrizio 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\u2013Fabrizio presents more information that were not revealed in the model with local derivative.<\/jats:p>","DOI":"10.3390\/e18040100","type":"journal-article","created":{"date-parts":[[2016,3,23]],"date-time":"2016-03-23T11:40:54Z","timestamp":1458733254000},"page":"100","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Chaos on the Vallis Model for El Ni\u00f1o with Fractional Operators"],"prefix":"10.3390","volume":"18","author":[{"given":"Badr","family":"Alkahtani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Colleges of Sciences, King Saud University, P.O. Box 1142, Riyadh 11989, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdon","family":"Atangana","sequence":"additional","affiliation":[{"name":"Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9301, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2016,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s11071-010-9724-4","article-title":"Entropy analysis of integer and fractional dynamical systems","volume":"62","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"109","DOI":"10.5890\/JAND.2012.03.001","article-title":"Entropy analysis of fractional derivatives and their approximation","volume":"1","year":"2012","journal-title":"J. Appl. 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