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 ${}_0^C{\mathrm{D}}_t^{\alpha }X(t)=AX(t)$, 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.

12 pages, 22 figures

References in corpus (4)

• A New Approach to Generalized Fractional Derivatives
• Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions
• Cusp-shaped Elastic Creases and Furrows
• On Fractional Eulerian Numbers and Equivalence of Maps with Long Term Power-Law Memory (Integral Volterra Equations of the Second Kind) to Gr$\ddot{u}$nvald-Letnikov Fractional Difference (Differential) Equations

Cited by in corpus (3)

• Can we split fractional derivative while analyzing fractional differential equations?
• Nonexistence of invariant manifolds in fractional order dynamical systems
• Analysis of solution trajectories of linear fractional order systems