  Mathematical analysis with numerical application of the newly formulated
fractional version of the Adams-Bashforth method using the Atangana-Baleanu
derivative which has nonlocal and nonsingular properties is considered in this
paper. We adopt the fixed point theory and approximation method to prove the
existence and uniqueness of the solution via general two-component
time-fractional differential equations. The method is tested with three
nonlinear chaotic dynamical systems in which the integer-order derivative is
modeled with the proposed fractional-order case. Simulation result for
different $\alpha$ values in $(0,1]$ is presented.
