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BY-NC-ND 3.0 license Open Access Published by De Gruyter March 21, 2014

On fractional lyapunov exponent for solutions of linear fractional differential equations

  • Nguyen Cong EMAIL logo , Doan Son and Hoang Tuan

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.

[1] L.Ya. Adrianova, Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs 46, Americal Mathematical Society, 1995. 10.1090/mmono/146Search in 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/CBO978110732602610.1017/CBO9781107326026Search in Google 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.2006.08.10410.1016/j.amc.2006.08.104Search in Google 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.2013.10.01010.1016/j.amc.2013.10.010Search in Google 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.2009.05.01210.1016/j.jsv.2009.05.012Search in Google 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.2006.06.00710.1016/j.jmaa.2006.06.007Search in Google 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.2009.09.01810.1016/j.na.2009.09.018Search in Google 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. 10.1007/978-3-642-14574-2_8Search in 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.2000.719410.1006/jmaa.2000.7194Search in Google 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. Search in 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)80001-010.1016/S0304-0208(06)80001-0Search in Google Scholar

[12] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems. Cambridge Scientific Pub., Cambridge, 2009. Search in 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.2004.02.03510.1016/j.chaos.2004.02.035Search in Google 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. 10.1063/1.3314277Search in Google Scholar PubMed

[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/s11071-012-0601-110.1007/s11071-012-0601-1Search in Google 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.2009.06.03510.1016/j.camwa.2009.06.035Search in Google Scholar

[17] V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231. Search in 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. Search in 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/S0002-9904-1948-09132-710.1090/S0002-9904-1948-09132-7Search in Google 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.2005.10.07310.1016/j.chaos.2005.10.073Search in Google Scholar

[21] B.J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators. Springer, 2003. http://dx.doi.org/10.1007/978-0-387-21746-810.1007/978-0-387-21746-8Search in Google Scholar

Published Online: 2014-3-21
Published in Print: 2014-6-1

© 2014 Diogenes Co., Sofia

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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