An instability theorem for nonlinear fractional differential systems

Nguyen Dinh Cong; Doan Thai Son; Stefan Siegmund; Hoang The Tuan

doi:10.3934/dcdsb.2017164

Article Contents

2017, Volume 22, Issue 8: 3079-3090. Doi: 10.3934/dcdsb.2017164

This issue Previous Article Integral conditions for nonuniform $μ$-dichotomy on the half-line Next Article Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity

An instability theorem for nonlinear fractional differential systems

1.
Institute of Mathematics, Vietnam Academy of Science and Technology, Viet Nam
2.
Department of Mathematics, Hokkaido University, Japan
3.
Department of Mathematics, Technische Universität Dresden, Dresden, Germany

Received: July 31, 2016

Revised: September 30, 2016

Published: October 2017

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Abstract

In this paper, we give a criterion on instability of an equilibrium of a nonlinear Caputo fractional differential system. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector

$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$

where $α∈ (0,1)$ is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.

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Mathematics Subject Classification: 34A08, 34K20.

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