Singular points in the solution trajectories of fractional order dynamical systems

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 ${}_0^C{D}_t^{\alpha }X\left(t\right)=AX\left(t\right)$ , 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 $2\times 2$ matrix $A$ are at the boundary of stable region, i.e., $|arg\left(\lambda \right)|=\frac{\alpha \pi }{2}$ . Furthermore, we discuss the existence of singular points in the trajectories of such planar systems in a region of $C$ , viz. Region II. It is conjectured that there exists a singular point in the solution trajectories if and only if $\lambda \in$ Region II.