Fractional dynamical system and its linearization theorem. (English) Zbl 1268.34019

Summary: Nowadays, it is known that the solution to a fractional differential equation can’t generally define a dynamical system in the sense of semigroup property due to the history memory induced by the weakly singular kernel. But we can still establish the similar relationship between a fractional differential equation and the corresponding fractional flow under a reasonable condition. In this paper, we firstly present some results on fractional dynamical systems defined by the fractional differential equation with Caputo derivative. Furthermore, the linearization and stability theorems of the nonlinear fractional system are also shown. As a byproduct, we prove the Audounet-Matignon-Montseny conjecture. Several illustrative examples are given as well to support the theoretical analysis.


34A08 Fractional ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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