Analysis of solution trajectories of fractional-order systems

The behavior of solution trajectories usually changes if we replace the classical derivative in a system with a fractional one. In this article, we throw light on the relation between two trajectories X(t) and Y(t) of such a system, where the initial point Y(0) is at some point X(t1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X(t_1)$$\end{document} of the trajectory X(t). In contrast with classical systems, these trajectories X and Y do not follow the same path. Further, we provide a Frenet apparatus for both trajectories in various cases and discuss their effect.