# Analysis of solution trajectories of linear fractional order systems

## TL;DR

This paper investigates how solution trajectories of linear fractional order systems differ from classical systems, analyzing their relationships and geometric properties using Frenet apparatus in various cases.

## Contribution

It provides new insights into the relationship between trajectories in fractional systems and develops a Frenet framework for their geometric analysis.

## Key findings

- Trajectories in fractional systems do not follow the same paths as classical systems.
- The paper establishes a relation between two trajectories with specific initial conditions.
- Frenet apparatus is used to analyze the geometric properties of these trajectories.

## Abstract

The behavior of solution trajectories usually changes if we replace the classical derivative in a system by a fractional one. In this article, we throw a 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 $X(t_1)$ of trajectory $X(t)$. In contrast with classical systems, trajectories $X$ and $Y$ do not follow the same path. Further, we provide a Frenet apparatus of both trajectories in various cases and discuss their effect.