Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag-Leffler Memories Involving the Truncated M-Derivative

In this research, novel M-truncated fractional derivatives with three orders have been proposed. These operators involve truncated Mittag-Leffler function to generalize the Khalil conformable derivative as well as the M-derivative. The new operators proposed are the convolution of truncated M-derivative with a power law, exponential decay and the complete Mittag-Leffler function. Numerical schemes based on Lagrange interpolation to predict chaotic behaviors of Rucklidge, Shimizu-Morioka and a hybrid strange attractors were considered. Additionally, numerical analysis based on 0-1 test and sensitive dependence on initial conditions were carried out to verify and show the existence of chaos in the chaotic attractor. These results showed that these novel operators involving three orders, two for the truncated M-derivative and one for the fractional term, depict complex chaotic behaviors.