An instability theorem for nonlinear fractional differential systems

TL;DR

This paper establishes a criterion for instability in nonlinear fractional differential systems, showing that certain spectral conditions on the linearization imply the instability of equilibrium points.

Contribution

It provides a new instability theorem specifically for nonlinear Caputo fractional differential systems based on spectral analysis.

Findings

Eigenvalues in a specific sector imply instability of equilibrium

The criterion applies to systems with fractional order <

Advances understanding of stability in fractional differential systems

Abstract

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.