Abstract
<jats:p>Many problems that describe Caputo fractional-order systems with state equations associate different orders, and an initial time is known to be important by going beyond the classical concepts of Newton-Leibniz calculus. In many applications of interest, suppose that a constant equilibrium is present in such systems. Can we predict the stability and instability of the simple equilibria of such systems? In this paper, first, by using a generalized Laplace transform, we formulate one stability and instability theorem for autonomous linear forms of such systems. Then, we generalize a well-known vector comparison approach by constructing stable and unstable Metzler matrices and propose both stability and instability theorems for general non-autonomous fractional-order systems. We put forward four examples that demonstrate typical applications to illustrate the effectiveness of our results.</jats:p>