Chaos in a nonlinear Bloch system with Atangana-Baleanu fractional derivatives

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.