Multiple attractors and periodicity on the Vallis model for El Niño/La Niña-Southern oscillation model

Abstract

In this paper, we obtain multiple attractors and periodicity using differential and integral operators with power-law and Mittag-Leffler law for the coupled dynamical El Niño/La Niña-Southern oscillation model and the continuous-time Vallis model for El Niño. Also, we consider the extension of these models considering a stochastic approach where the given parameters are converted to normal distributions. Additionally, we consider for both models novel differential and integral operators with fractional order and fractal dimension. These novel operators predict chaotic behaviors involving the fractal derivative in convolution with power-law and the Mittag-Leffler function, also, these operators can capture self-similarities for both chaotic attractors. We have presented the conditions of existence of uniquely exact solutions of the system using the fixed-point theorem approach. Each model is solved numerical via the Adams-Bashforth-Moulton, Adams-Moulton and the Atangana-Toufik schemes. We presented numerical simulations for different values of fractional order to show the applicability and computational efficiency of these methods. The results obtained presents more information that were not revealed in the models with local derivative.