Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems

TL;DR

This paper investigates the long-term behavior of non-commensurate fractional-order planar systems, providing new stability conditions and demonstrating their effectiveness through numerical examples.

Contribution

It introduces novel sufficient conditions for stability and attractivity in fractional-order planar systems using complex analysis and fixed point methods.

Findings

Derived explicit stability conditions for linear systems.

Established Mittag-Leffler stability criteria for nonlinear systems.

Validated theoretical results with numerical examples.

Abstract

This paper is devoted to studying non-commensurate fractional order planar systems. Our contributions are to derive sufficient conditions for the global attractivity of non-trivial solutions to fractional-order inhomogeneous linear planar systems and for the Mittag-Leffler stability of an equilibrium point to fractional order nonlinear planar systems. To achieve these goals, our approach is as follows. Firstly, based on Cauchy's argument principle in complex analysis, we obtain various explicit sufficient conditions for the asymptotic stability of linear systems whose coefficient matrices are constant. Secondly, by using Hankel type contours, we derive some important estimates of special functions arising from a variation of constants formula of solutions to inhomogeneous linear systems. Then, by proposing new weighted norms combined with the Banach fixed point theorem for appropriate Banach spaces, we get the desired conclusions. Finally, numerical examples are provided to illustrate the effect of the main theoretical results.