Chaos on the Vallis Model for El Niño with Fractional Operators

Abstract

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.

Chaotic dynamics of fractional Vallis system for El-Niño

“…Only a couple of papers have appeared recently. Alkahtani et al [10] have studied Vallis model with local derivatives, Caputo derivatives and Caputo-Fabrizio derivative. They have also drawn phase portraits for commensurate fractional Vallis system with Caputo derivative for p = 0.3 and α = 0.85, 0.55, 0.25.…”

A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

“…In recent years, fractional calculus has received much attention due to fractional derivatives providing more accurate models than their integer-order counterparts. Many examples have been found in different interdisciplinary fields [25], ranging from the description of viscoelastic anomalous diffusion in complex liquids, D-decomposition technique for control problems, chaotic systems; to macroeconomic models with dynamic memory, forecast of the trend of complex systems, and so on [26][27][28][29][30][31][32][33][34]. Those works have demonstrated that fractional derivatives provide an excellent approach to describing the memory and hereditary properties of real physical phenomena.…”

Generalized finite-time function projective synchronization of two fractional-order chaotic systems via a modified fractional nonsingular sliding mode surface

“…Since then, it has been shown that fractional-order dynamics can be observed for instance in viscoelastic systems, in dielectric polarization, in electromagnetic waves, just to name some [3]- [5]. Meanwhile another prolific field of science, namely chaos theory, has acquired a new development to address the challenging problem of chaos synchronization due to its potential applications in physics, chemistry, telecommunication, biology, medicine and so on [6]- [18]. The streaming of research within this area aims at achieving master-slave synchronization between two chaotic systems by choosing various kinds of methods to extend and enrich the pioneering work of Pecora and Carroll [19].…”