An instability theorem of nonlinear fractional differential systems

In this paper, we give a criterion on instability of an equilibrium of nonlinear Caputo fractional differential systems. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector $$\left\{\lambda\in\C\setminus\{0\}:|\arg{(\lambda)}|<\frac{\alpha \pi}{2}\right\},$$ where $\alpha\in (0,1)$ is the order of the fractional differential systems, then the equilibrium of the nonlinear systems is unstable.