Chaos on the Vallis Model for El Niño with Fractional Operators

Badr Alkahtani,Abdon Atangana

doi:10.3390/e18040100

Badr Alkahtani, Abdon Atangana

Open Access

https://doi.org/10.3390/e18040100

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Journal: Entropy	Publication Date: Mar 23, 2016
Citations: 25	License type: CC BY 4.0

Affiliation: King Saud University, University of the Free State

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.

Highlights

Claude Shannon in 1948 established the concept of information theory
The Vallis model for El Niño was considered with three-different definitions of derivative in
The Vallis model for El Niño was considered with three-different definitions of derivative in this work

Summary

Introduction

Claude Shannon in 1948 established the concept of information theory. The concept has been used in many scientific fields, for instance in signal and image processing [1,2,3,4,5,6]. The model has been used with great success, it is perhaps important to mention that, anomalous or irregular phenomenon cannot accurately be described with local derivative. This has been revealed in many research papers, monograms and books in the last decade [11,12,13,14,15,16,17,18,19,20]. According to Caputo, the fractional derivative of a continuous and n-time differentiable function f is given as: Dtα p f ptqq “.

B 2C p

Existence of Exact Solution

Numerical Simulations

Figures and

Analysis of Vallis Model with Caputo Derivative

Analysis of the Vallis Model with Caputo–Fabrizio Derivative

Conclusions