Singular points in the solution trajectories of fractional order dynamical systems

TL;DR

This paper reviews the solutions of linear fractional order dynamical systems with Caputo derivatives, analyzing their phase portraits, boundary behaviors, and the conjectured existence of singular points related to eigenvalues in specific regions.

Contribution

It provides a comprehensive review of solutions for linear fractional systems, describes phase portraits, and introduces a conjecture on the existence of singular points based on eigenvalue regions.

Findings

Exact solutions for canonical forms are described.

Behavior of trajectories at boundary eigenvalues is analyzed.

A conjecture on the existence of singular points based on eigenvalue regions is proposed.

Abstract

Dynamical systems involving non-local derivative operators are of great importance in Mathematical analysis and applications. This article deals with the dynamics of fractional order systems involving Caputo derivatives. We take a review of the solutions of linear dynamical systems ${}_0^C\mathrm{D}_t^\alpha X(t)=AX(t)$, where the coefficient matrix $A$ is in canonical form. We describe exact solutions for all the cases of canonical forms and sketch phase portraits of planar systems. We discuss the behavior of the trajectories when the eigenvalues $\lambda$ of $A$ are at the boundary of stable region i.e. $|arg(\lambda)|=\frac{\alpha\pi}{2}$. Further, we discuss the existence of singular points in the trajectories of such systems in a region of $\mathbb{C}$ viz. Region II. It is conjectured that there exists singular point in the solution trajectories if and only if $\lambda\in$ Region II.