Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2017: 2.865
5-year IMPACT FACTOR: 3.323

CiteScore 2017: 3.06

SCImago Journal Rank (SJR) 2017: 1.967
Source Normalized Impact per Paper (SNIP) 2017: 1.954

Mathematical Citation Quotient (MCQ) 2017: 0.98

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 17, Issue 2

Issues

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Hoang Tuan
Published Online: 2014-03-21 | DOI: https://doi.org/10.2478/s13540-014-0169-1

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.

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

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

  • [1] L.Ya. Adrianova, Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs 46, Americal Mathematical Society, 1995. Google Scholar

  • [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. http://dx.doi.org/10.1017/CBO9781107326026CrossrefGoogle Scholar

  • [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. http://dx.doi.org/10.1016/j.amc.2006.08.104CrossrefGoogle Scholar

  • [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. http://dx.doi.org/10.1016/j.amc.2013.10.010CrossrefGoogle Scholar

  • [5] L. Cveticanin and M. Zukovic, Melnikov’s criteria and chaos in systems with fractional order deflection. J. Sound Vibration 326 (2009), 768–779. http://dx.doi.org/10.1016/j.jsv.2009.05.012Web of ScienceCrossrefGoogle Scholar

  • [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. http://dx.doi.org/10.1016/j.jmaa.2006.06.007CrossrefGoogle Scholar

  • [7] W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 72, No 3–4 (2010), 1768–1777. http://dx.doi.org/10.1016/j.na.2009.09.018CrossrefGoogle Scholar

  • [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] K. Diethelm and N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, No 2 (2002), 229–248. http://dx.doi.org/10.1006/jmaa.2000.7194CrossrefGoogle Scholar

  • [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. Google Scholar

  • [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. http://dx.doi.org/10.1016/S0304-0208(06)80001-0CrossrefGoogle Scholar

  • [12] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems. Cambridge Scientific Pub., Cambridge, 2009. Google Scholar

  • [13] C. Li and G. Chen, Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals 22 (2004), 549–554. http://dx.doi.org/10.1016/j.chaos.2004.02.035CrossrefGoogle Scholar

  • [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. CrossrefPubMedGoogle Scholar

  • [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. http://dx.doi.org/10.1007/s11071-012-0601-1Web of ScienceCrossrefGoogle Scholar

  • [16] Z.M. Odibat, Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59 (2010), 1171–1183. http://dx.doi.org/10.1016/j.camwa.2009.06.035CrossrefGoogle Scholar

  • [17] V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231. Google Scholar

  • [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. Google Scholar

  • [19] H. Pollard, The completely monotonic character of the Mittag-Leffler function E α(−x). Bull. Amer. Math. Soc. 54 (1948), 1115–1116. http://dx.doi.org/10.1090/S0002-9904-1948-09132-7CrossrefGoogle Scholar

  • [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. http://dx.doi.org/10.1016/j.chaos.2005.10.073CrossrefGoogle Scholar

  • [21] B.J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators. Springer, 2003. http://dx.doi.org/10.1007/978-0-387-21746-8CrossrefGoogle Scholar

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.

Export Citation

© 2014 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Tene Alain Giresse, Kofane Timoleon Crepin, and Tchoffo Martin
Chaos, Solitons & Fractals, 2019, Volume 118, Page 311
[2]
Michał Niezabitowski
Applied Mechanics and Materials, 2015, Volume 789-790, Page 1052

Comments (0)

Please log in or register to comment.
Log in