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 α X ( t ) = A X ( 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 λ of 2 × 2 matrix A are at the boundary of stable region, i.e., | a r g ( λ ) | = α π 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 λ ∈ Region II.