On fractional lyapunov exponent for solutions of linear fractional differential equations

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• 作者：Nguyen Dinh Cong (1)
  Doan Thai Son (1) (2)
  Hoang The Tuan (1)
  
• 关键词：Primary 34A08 ; Secondary 34D08 ; 34D20 ; fractional calculus ; linear fractional differential equations ; Mittag ; Leffler type functions ; Lyapunov exponent ; stability
• 刊名：Fractional Calculus and Applied Analysis
• 出版年：2014
• 出版时间：June 2014
• 年：2014
• 卷：17
• 期：2
• 页码：285-306
• 全文大小：286 KB
• 参考文献：1. L.Ya. Adrianova, / Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs 46, Americal Mathematical Society, 1995.
  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. dx.doi.org/10.1017/CBO9781107326026" target="_blank" title="It opens in new window">CrossRef
  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鈥?8. dx.doi.org/10.1016/j.amc.2006.08.104" target="_blank" title="It opens in new window">CrossRef
  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鈥?68. dx.doi.org/10.1016/j.amc.2013.10.010" target="_blank" title="It opens in new window">CrossRef
  5. L. Cveticanin and M. Zukovic, Melnikov鈥檚 criteria and chaos in systems with fractional order deflection. / J. Sound Vibration 326 (2009), 768鈥?79. dx.doi.org/10.1016/j.jsv.2009.05.012" target="_blank" title="It opens in new window">CrossRef
  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鈥?033. dx.doi.org/10.1016/j.jmaa.2006.06.007" target="_blank" title="It opens in new window">CrossRef
  7. W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations. / Nonlinear Anal. 72, No 3鈥? (2010), 1768鈥?777. dx.doi.org/10.1016/j.na.2009.09.018" target="_blank" title="It opens in new window">CrossRef
  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.
  9. K. Diethelm and N.J. Ford, Analysis of fractional differential equations. / J. Math. Anal. Appl. 265, No 2 (2002), 229鈥?48. dx.doi.org/10.1006/jmaa.2000.7194" target="_blank" title="It opens in new window">CrossRef
  10. R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function / E _{/ 伪, / 尾} ( / z) and its derivative. / Fract. Calc. Appl. Anal. 5, No 4 (2002), 491鈥?18. Correction in: / Fract. Calc. Appl. Anal. 6, No 1 (2003), 111鈥?12.
  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. dx.doi.org/10.1016/S0304-0208(06)80001-0" target="_blank" title="It opens in new window">CrossRef
  12. V. Lakshmikantham, S. Leela and J. Vasundhara Devi, / Theory of Fractional Dynamic Systems. Cambridge Scientific Pub., Cambridge, 2009.
  13. C. Li and G. Chen, Chaos in the fractional order Chen system and its control. / Chaos Solitons Fractals 22 (2004), 549鈥?54. dx.doi.org/10.1016/j.chaos.2004.02.035" target="_blank" title="It opens in new window">CrossRef
  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.
  15. Ch. Li and Y. Ma, Fractional dynamical system and its linearization theorem. / Nonlinear Dynam. 71, No 4 (2013), 621鈥?33; DOI: 10.1007/s11071-012-0601-1. dx.doi.org/10.1007/s11071-012-0601-1" target="_blank" title="It opens in new window">CrossRef
  16. Z.M. Odibat, Analytic study on linear systems of fractional differential equations. / Comput. Math. Appl. 59 (2010), 1171鈥?183. dx.doi.org/10.1016/j.camwa.2009.06.035" target="_blank" title="It opens in new window">CrossRef
  17. V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. / Trans. Moscow Math. Soc. 19 (1968), 197鈥?31.
  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.
  19. H. Pollard, The completely monotonic character of the Mittag-Leffler function / E _伪(鈭?em class="a-plus-plus">x). / Bull. Amer. Math. Soc. 54 (1948), 1115鈥?116. dx.doi.org/10.1090/S0002-9904-1948-09132-7" target="_blank" title="It opens in new window">CrossRef
  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鈥?212. dx.doi.org/10.1016/j.chaos.2005.10.073" target="_blank" title="It opens in new window">CrossRef
  21. B.J. West, M. Bologna and P. Grigolini, / Physics of Fractal Operators. Springer, 2003. dx.doi.org/10.1007/978-0-387-21746-8" target="_blank" title="It opens in new window">CrossRef
  
• 作者单位：Nguyen Dinh Cong (1)
  Doan Thai Son (1) (2)
  Hoang The Tuan (1)
  
  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307, Ha Noi, Vietnam
  2. Department of Mathematics, Imperial College London, 180 Queen鈥檚 Gate, SW7 2AZ, London, UK
  

文摘

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.