Comparison Study of Dynamical System Using Different Kinds of Fractional Operators

Abstract

The dynamical system is one of the major research subjects, and many researchers and experts are attempting to evolve new models and approaches for its solution due to its vast applicability. Applied mathematics has been used to anticipate the chaotic behavior of some attractors using a novel operator termed fractal-fractional derivatives. They were made operating three distinct kernels: power low, exponential decay, and the generalized Mittag Leffler function. There are two parameters in the new operator. Fractional order is the first, while fractal dimension is the second. These derivatives will manage to detect self-similarities in chaotic attractors. We provided numerical approaches for solving such a nonlinear differential equation system. The solution’s existence and uniqueness are determined. Bifurcation analysis is also presented briefly. These new operators were tested in the chaotic attractor with numerical simulations for varied fractional order and fractal dimension, and the findings were quite interesting. We believe that this new notion is the way to go for modeling complexes with self-similarities in the future.