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


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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  1. [1]
    L.Ya. Adrianova, Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs 46, Americal Mathematical Society, 1995.zbMATHGoogle Scholar
  2. [2]
    L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.CrossRefzbMATHGoogle Scholar
  3. [3]
    B. Bonilla, M. Rivero and J.J. Trujillo, On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187 (2007), 68–78.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    N.D. Cong, T.S. Doan, S. Siegmund and H.T. Tuan, On stable manifolds for planar fractional differential equations. Appl. Math. Comput. 226 (2014), 157–168.CrossRefMathSciNetGoogle Scholar
  5. [5]
    L. Cveticanin and M. Zukovic, Melnikov’s criteria and chaos in systems with fractional order deflection. J. Sound Vibration 326 (2009), 768–779.CrossRefGoogle Scholar
  6. [6]
    V. Daftardar-Gejji and H. Jafari, Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J. Math. Anal. Appl. 328, No 2 (2007), 1026–1033.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 72, No 3–4 (2010), 1768–1777.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics 2004, Springer-Verlag, Berlin, 2010.Google Scholar
  9. [9]
    K. Diethelm and N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, No 2 (2002), 229–248.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function E α,β(z) and its derivative. Fract. Calc. Appl. Anal. 5, No 4 (2002), 491–518. Correction in: Fract. Calc. Appl. Anal. 6, No 1 (2003), 111–112.zbMATHMathSciNetGoogle Scholar
  11. [11]
    A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.CrossRefzbMATHGoogle Scholar
  12. [12]
    V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems. Cambridge Scientific Pub., Cambridge, 2009.zbMATHGoogle Scholar
  13. [13]
    C. Li and G. Chen, Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals 22 (2004), 549–554.CrossRefzbMATHGoogle Scholar
  14. [14]
    C. Li, Z. Gong, D. Qian and Y. Chen, On the bound of the Lyapunov exponents for the fractional differential systems. Chaos 20, No 1 (2010), # 013127, 7 p.Google Scholar
  15. [15]
    Ch. Li and Y. Ma, Fractional dynamical system and its linearization theorem. Nonlinear Dynam. 71, No 4 (2013), 621–633; DOI: 10.1007/s11071-012-0601-1.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Z.M. Odibat, Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59 (2010), 1171–1183.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.zbMATHGoogle Scholar
  18. [18]
    I. Podlubny, Fractional Differential equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of Their Applications. Mathematics in Science and Engineering, 198, Academic Press, Inc., CA, 1999.zbMATHGoogle Scholar
  19. [19]
    H. Pollard, The completely monotonic character of the Mittag-Leffler function E α(−x). Bull. Amer. Math. Soc. 54 (1948), 1115–1116.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Long-Jye Sheu, Hsien-Keng Chen, Juhn-Horng Chen and Lap-Mou Tam, Chaos in a new system with fractional order. Chaos Solitons Fractals 31 (2007), 1203–1212.CrossRefGoogle Scholar
  21. [21]
    B.J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators. Springer, 2003.CrossRefGoogle Scholar

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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