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Chaos in a nonlinear Bloch system with Atangana–Baleanu fractional derivatives
Author(s) -
GómezAguilar J.F.
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22219
Subject(s) - mathematics , uniqueness , nonlinear system , fractional calculus , mathematical analysis , constant (computer programming) , variable (mathematics) , stability (learning theory) , kernel (algebra) , order (exchange) , pure mathematics , physics , quantum mechanics , computer science , finance , machine learning , economics , programming language
In this article, a nonlinear model of the Bloch equation to include both fractional derivatives with variable‐order, constant‐order, and time delays was considered. The fractional derivative with the generalized Mittag‐Leffler function as kernel is introduced due to the nonlocality of the dynamical system. To find a numerical solution of the delay variable‐order model, a predictor corrector method had been developed to solve this system. The existence and uniqueness of the numerical scheme was discussed in detail. For the constant‐order, we presented the existence and uniqueness of a positive set of the solutions for the new model and the Adams–Moulton rule was considered to solved numerically the fractional equations. The behavior of the fractional commensurate order nonlinear delay‐dependent Bloch system with total order less than 3, which exhibits chaos and transient chaos, was presented. In addition, it is found that the presence of fractional variable‐order in the nonlinear Bloch system exhibit more complicated dynamics can improve the stability of the solutions.