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 , where the coefficient matrix 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 of matrix are at the boundary of stable region, i.e., . Furthermore, we discuss the existence of singular points in the trajectories of such planar systems in a region of , viz. Region II. It is conjectured that there exists a singular point in the solution trajectories if and only if Region II.