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Fractional Calculus and Applied Analysis

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Volume 17, Issue 2


On fractional lyapunov exponent for solutions of linear fractional differential equations

Nguyen Cong / Doan Son
  • Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307, Ha Noi, Vietnam
  • Department of Mathematics, Imperial College London, 180 Queen’s Gate, SW7 2AZ, London, UK
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/ Hoang Tuan
Published Online: 2014-03-21 | DOI: https://doi.org/10.2478/s13540-014-0169-1


Our aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.

MSC: Primary 34A08; Secondary 34D08, 34D20

Keywords: fractional calculus; linear fractional differential equations; Mittag-Leffler type functions; Lyapunov exponent; stability

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About the article

Published Online: 2014-03-21

Published in Print: 2014-06-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 17, Issue 2, Pages 285–306, ISSN (Online) 1314-2224, DOI: https://doi.org/10.2478/s13540-014-0169-1.

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