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Fractional dynamical system and its linearization theorem

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Abstract

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 system 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 Audounet–Matignon–Montseny conjecture. Several illustrative examples are given as well to support the theoretical analysis.

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Acknowledgements

The present job was financially supported by the Key Program of Shanghai Municipal Education Commission (No. 12ZZ084) and the Shanghai Leading Academic Discipline Project (No. S30104).

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. China

    Changpin Li & Yutian Ma

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  1. Changpin Li
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  2. Yutian Ma
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Correspondence to Changpin Li.

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Dedicated to Professor Ravi P. Agarwal on the Occasion of his 65th Birthday

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Li, C., Ma, Y. Fractional dynamical system and its linearization theorem. Nonlinear Dyn 71, 621–633 (2013). https://doi.org/10.1007/s11071-012-0601-1

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