Skip to main content

On stable manifolds for fractional differential equations in high-dimensional spaces

An Erratum to this article was published on 26 August 2016

Abstract

Our aim in this paper is to establish stable manifolds near hyperbolic equilibria of fractional differential equations in arbitrary finite-dimensional spaces.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Baleanu, D., Mustafa, O.G., Agarwal, R.P.: Asymptotic integration of \((1 + \alpha )\)-order fractional differential equations. Comput. Math. Appl. 62, 1492–1500 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Baleanu, D., Agarwal, R.P., Mustafa, O.G., Cosulschi, M.: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A: Math. Theor 44, 055203 (2011). (9 pp.)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Cong, N.D., Doan, T.S., Tuan, H.T.: On fractional Lyapunov exponent for solutions of linear fractional differential equations. Fract. Calc. Appl. Anal. 17(2), 285–306 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for planar fractional differential equations. Appl. Math. Comput. 226, 157–168 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Structure of the fractional Lyapunov spectrum for linear fractional differential equations. Adv. Dyn. Syst. Appl. 9, 149–159 (2014)

    MathSciNet  Google Scholar 

  8. 8.

    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifold for fractional differential equations in high dimensional space. Preprint IMH2015/03/01, Institute of Mathematics, Vietnam Academy of Science and Technology, Mar 2015. http://math.ac.vn/en/component/staff/?task=showPrePrint&year=2015

  9. 9.

    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Linearized asymptotic stability for fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 39, 1–13 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Deshpande, A., Daftardar-Gejji, V.: Local stable manifold theorem for fractional systems. Nonlinear Dyn. 83, 2435–2452 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Deng, W.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 72(3–4), 1768–1777 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Diethelm, K.: The analysis of fractional differential equations. In: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)

  13. 13.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)

    Google Scholar 

  14. 14.

    Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Li, C.P., Gong, Z., Qian, D., Chen, Y.: On the bound of the Lyapunov exponents for the fractional differential systems. Chaos 20(1), 013127 (2010). (7 pages)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Li, C.P., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71(4), 621–633 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)

    Article  Google Scholar 

  18. 18.

    Li, K., Peng, J.: Laplace transform and fractional differential equations. Appl. Math. Lett. 24, 2019–2023 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Autom. J. IFAC 45(8), 1965–1969 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Ma, L., Li, C.: Center manifold of fractional dynamical system. J. Comput. Nonlinear Dynam. 11(2), 021010 (2015). (6 pp)

    Article  Google Scholar 

  21. 21.

    Matignon, D.: Stability results for fractional differential equations with applications to control processing. Comput Eng. Syst. Appl. 2, 963–968 (1996)

    Google Scholar 

  22. 22.

    Nieto, J.J.: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23, 1248–1251 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Podlubny, I.: 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., San Diego (1999)

  25. 25.

    Shilov, G.E.: Linear Algebra. Dover Publications Inc., New York (1977)

    Google Scholar 

Download references

Acknowledgments

This research of the N. D. Cong, T. S. Doan, and H. T. Tuan is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2014.42.

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Siegmund.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s11071-016-3039-z.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cong, N.D., Doan, T.S., Siegmund, S. et al. On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dyn 86, 1885–1894 (2016). https://doi.org/10.1007/s11071-016-3002-z

Download citation

Keywords

  • Stable manifold
  • Fractional differential equation
  • Lyapunov stability