×

On stability, persistence, and Hopf bifurcation in fractional order dynamical systems. (English) Zbl 1175.37084

Motivation for the investigations of fractional order dynamical systems is explained by rich applications in mathematical biology [A. M. Nakhushev, Equations of mathematical biology. Moskva: Vysshaya Shkola (1995; Zbl 0991.35500)]. Therefore it is essential to study bifurcation, stability and chaos in fractional order dynamical systems. Since persistence and seasonability are important concepts in biology, it is relevant to study the persistence in biologically motivated nonautonomous fractional equations.
In this article after some relations between complex adaptive systems and fractional mathematics sufficient conditions for the persistence of some biologically inspired fractional nonautonomous equations are derived. Then Poincaré-Andronov-Hopf bifurcation in fractional order systems is studied. Also the local stability questions in functional equations are considered.

MSC:

37N25 Dynamical systems in biology
92B99 Mathematical biology in general
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

Citations:

Zbl 0991.35500

Software:

FracPECE
PDFBibTeX XMLCite
Full Text: DOI arXiv

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) · Zbl 1105.82026
[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) · Zbl 1142.30303
[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) · Zbl 1142.30303
[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) · Zbl 1105.65122
[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] · Zbl 1028.65081
[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)
[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)
[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) · Zbl 1009.65049
[10] Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004) · Zbl 1055.65098
[11] Edelstein-Keshet, L.: Introduction to Mathematical Biology. Siam Classics in Appl. Math. SIAM, Philadelphia (2004)
[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) · Zbl 1062.65073
[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) · Zbl 1213.34025
[14] El-Sayed, A., El-Mesiry, A., EL-Saka, H.: On the fractional-order logistic equation. Appl. Math. Lett. 20, 817–823 (2007) · Zbl 1140.34302
[15] Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge (1998) · Zbl 0914.90287
[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)
[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)
[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) · Zbl 1066.34050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.