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Fractional Calculus and Applied Analysis

, Volume 17, Issue 2, pp 285–306 | Cite as

On fractional lyapunov exponent for solutions of linear fractional differential equations

  • Nguyen Dinh CongEmail author
  • Doan Thai Son
  • Hoang The Tuan
Research Paper

Abstract

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.

Key Words and Phrases

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

MSC 2010

Primary 34A08 Secondary 34D08, 34D20 

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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  • Nguyen Dinh Cong
    • 1
    Email author
  • Doan Thai Son
    • 1
    • 2
  • Hoang The Tuan
    • 1
  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHa NoiVietnam
  2. 2.Department of MathematicsImperial College LondonLondonUK

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