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On stability, persistence, and Hopf bifurcation in fractional order dynamical systems

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Abstract

This is a preliminary study about the bifurcation phenomenon in fractional order dynamical systems. Persistence of some continuous time fractional order differential equations is studied. A numerical example for Hopf-type bifurcation in a fractional order system is given, hence we propose a modification of the conditions of Hopf bifurcation. Local stability of some biologically motivated functional equations is investigated.

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References

  1. Ahmed, E., El-Sayed, A.M.A., El-Mesiry, E.M., El-Saka, H.A.A.: Numerical solution for the fractional replicator equation. Int. J. Mod. Phys. 16(7), 1–9 (2005)

    Google Scholar 

  2. Ahmed, E., El-Sayed, A., El-Saka, H.: On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems. Phys. Lett. A 358, 1 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems. Phys. Lett. A 358, 1 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Diethelm, K.: Predictor–corrector strategies for single- and multi-term fractional differential equations. In: Lipitakis, E.A. (ed.) Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and its Applications, pp. 117–122. LEA Press, Athens (2002) [Zbl. Math. 1028.65081]

    Google Scholar 

  6. Diethelm, K., Ford, N.J.: The numerical solution of linear and non-linear Fractional differential equations involving Fractional derivatives several of several orders. Numerical Analysis Report 379, Manchester Center for Numerical Computational Mathematics

  7. Diethelm, K., Freed, A.: On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity. In: Keil, F., Mackens, W., Voß, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II—Computational Fluid Dynamics, Reaction Engineering, and Molecular properties, pp. 217–224. Springer, Heidelberg (1999)

    Google Scholar 

  8. Diethelm, K., Freed, A.: The FracPECE subroutine for the numerical solution of differential equations of fractional order. In: Heinzel, S., Plesser, T. (eds.) Forschung und wissenschaftliches Rechnen 1998. Gesellschaft für Wisseschaftliche Datenverarbeitung, pp. 57–71. Vandenhoeck & Ruprecht, Göttingen (1999)

    Google Scholar 

  9. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Edelstein-Keshet, L.: Introduction to Mathematical Biology. Siam Classics in Appl. Math. SIAM, Philadelphia (2004)

    Google Scholar 

  12. El-Mesiry, E.M., El-Sayed, A.M.A., El-Saka, H.A.A.: Numerical methods for multi-term fractional (arbitrary) orders differential equations. Appl. Math. Comput. 160(3), 683–699 (2005)

    MATH  MathSciNet  Google Scholar 

  13. El-Sayed, A.M.A., El-Mesiry, E.M., El-Saka, H.A.A.: Numerical solution for multi-term fractional (arbitrary) orders differential equations. Comput. Appl. Math. 23(1), 33–54 (2004)

    Article  MathSciNet  Google Scholar 

  14. El-Sayed, A., El-Mesiry, A., EL-Saka, H.: On the fractional-order logistic equation. Appl. Math. Lett. 20, 817–823 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge (1998)

    MATH  Google Scholar 

  16. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Eng. in Sys. Appl., vol. 2, p. 963. Lille, France (1996)

  17. Rocco, A., West, B.J.: Fractional calculus and the evolution of fractal phenomena. Physica A 265, 535 (1999)

    Article  Google Scholar 

  18. Smith, J.B.: A technical report of complex system. ArXiv:CS0303020 (2003)

  19. Stanislavsky, A.A.: Memory effects and macroscopic manifestation of randomness. Phys. Rev. E 61, 4752 (2000)

    Google Scholar 

  20. http://socserv2.socsci.mcmaster.ca/cesg2003/shimopaper.pdf

  21. Zhao, J., Jiang, J.: Average conditions for permanence and extinction in non-autonomous Lotka-Volterra system. J. Math. Anal. Appl. 299, 663 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Mansoura University, New Damietta, Egypt

    H. A. El-Saka

  2. Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt

    E. Ahmed & M. I. Shehata

  3. Mathematics Department, Faculty of Science, Alexandria University, Alexandria, Egypt

    A. M. A. El-Sayed

Authors
  1. H. A. El-Saka
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  2. E. Ahmed
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  3. M. I. Shehata
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  4. A. M. A. El-Sayed
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Correspondence to E. Ahmed.

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El-Saka, H.A., Ahmed, E., Shehata, M.I. et al. On stability, persistence, and Hopf bifurcation in fractional order dynamical systems. Nonlinear Dyn 56, 121–126 (2009). https://doi.org/10.1007/s11071-008-9383-x

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  • DOI: https://doi.org/10.1007/s11071-008-9383-x

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