Chaos on the Vallis Model for El Nino with Fractional Operators

The Vallis model for El Nino 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.