Differences between fractional- and integer-order dynamics

Abstract

The qualitative theory of fractional-order dynamical systems and it applications to the sciences and engineering is a recent focus of interest of many researchers. In addition to natural similarities that can be drawn between fractional- and integer-order derivatives and fractional- and integer-order dynamical systems, very important differences arise as well. Since in many cases, qualitative properties of integer-order dynamical systems cannot be extended by generalization to fractional- order dynamical systems, the analysis of fractional-order dynamical systems is a very important field of research. This paper is devoted to presenting qualitative contrasts between fractional- and integer-order dynamical systems. For example, even though the integer-order derivative of a periodic function is obviously a periodic function of the same period, the fractional- order derivative of a non-constant periodic function cannot be a periodic function of the same period. As a consequence, periodic solutions do not exist in a wide class of fractional-order dynamical systems. Moreover, important differences will be highlighted concerning the asymptotic stability analysis of fractional- and integer-order linear delayed differential equations.