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 carefully chosen 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.

本文致力于研究非通约分数阶平面系统。我们的贡献是为分数阶非齐次线性平面系统的非平凡解的全局吸引力和分数阶非线性平面系统的平衡点的 Mittag-Leffler 稳定性推导出充分条件。为了实现这些目标，我们的方法如下。首先，根据复分析中的柯西论证原理，我们得到了系数矩阵为常数的线性系统渐近稳定的各种显式充分条件。其次，通过使用 Hankel 型等值线，我们推导出了一些由非齐次线性系统解的常数公式变化引起的特殊函数的重要估计。然后，通过为适当的 Banach 空间提出精心选择的加权范数并结合 Banach 不动点定理，我们得到了预期的结论。最后，通过数值例子来说明主要理论结果的影响。