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Stability properties of two-term fractional differential equations

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Abstract

This paper formulates explicit necessary and sufficient conditions for the local asymptotic stability of equilibrium points of the fractional differential equation

$$\begin{aligned} {D}^{\alpha }y(t)+f(y(t),\, {D}^{\beta }y(t))=0,\quad t>0 \end{aligned}$$

involving two Caputo derivatives of real orders \(\alpha >\beta \) such that \(\alpha /\beta \) is a rational number. First, we consider this equation in the linearized form and derive optimal stability conditions in terms of its coefficients and orders \(\alpha ,\beta \). As a byproduct, a special fractional version of the Routh–Hurwitz criterion is established. Then, using the recent developments on linearization methods in fractional dynamical systems, we extend these results to the original nonlinear equation. Some illustrating examples, involving significant linear and nonlinear fractional differential equations, support these results.

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Acknowledgments

The authors are grateful to referees for reading this paper, suggestions and recommendations which considerably helped to improve its content. The research was supported by the grant P201/11/0768 of the Czech Science Foundation and by the project CZ.1.07/2.3.00/30.0039 of Brno University of Technology.

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  1. Institute of Mathematics, Brno University of Technology, Technická 2, 616 69 , Brno, Czech Republic

    Jan Čermák & Tomáš Kisela

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  1. Jan Čermák
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  2. Tomáš Kisela
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Correspondence to Tomáš Kisela.

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Čermák, J., Kisela, T. Stability properties of two-term fractional differential equations. Nonlinear Dyn 80, 1673–1684 (2015). https://doi.org/10.1007/s11071-014-1426-x

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