Advertisement

Stability Analysis of Two-Dimensional Incommensurate Systems of Fractional-Order Differential Equations

  • Oana Brandibur
  • Eva KaslikEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Recently obtained necessary and sufficient conditions for the asymptotic stability and instability of the null solution of a two-dimensional autonomous linear incommensurate fractional-order dynamical system with Caputo derivatives are reviewed and extended. These theoretical results are then applied to investigate the stability properties of a two-dimensional fractional-order conductance-based neuronal model. Moreover, the occurrence of Hopf bifurcations is also discussed, choosing the fractional orders as bifurcation parameters. Numerical simulations are also presented to illustrate the theoretical results.

References

  1. 1.
    Anastasio, T.J.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72(1), 69–79 (1994)CrossRefGoogle Scholar
  2. 2.
    Armanyos, M., Radwan, A.G.: Fractional-order fitzhugh-nagumo and izhikevich neuron models. In: 2016 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), pp. 1–5. IEEE (2016)Google Scholar
  3. 3.
    Bonnet, C., Partington, J.R.: Coprime factorizations and stability of fractional differential systems. Syst. Control. Lett. 41(3), 167–174 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brandibur, O., Kaslik, E.: Stability properties of a two-dimensional system involving one caputo derivative and applications to the investigation of a fractional-order Morris-Lecar neuronal model. Nonlinear Dyn. 90(4), 2371–2386 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brandibur, O., Kaslik, E.: Stability of Two-Component Incommensurate Fractional-Order Systems and Applications to the Investigation of a FitzHugh-Nagumo Neuronal Model. Math. Methods Appl. Sci. 41(17), 7182–7194 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Čermák, J., Kisela, T.: Stability properties of two-term fractional differential equations. Nonlinear Dyn. 80(4), 1673–1684 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cottone, G., Di Paola, M., Santoro, R.: A novel exact representation of stationary colored gaussian processes (fractional differential approach). J. Phys. A: Math. Theor. 43(8), 085002 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Datsko, B., Luchko, Y.: Complex oscillations and limit cycles in autonomous two-component incommensurate fractional dynamical systems. Math. Balk. 26, 65–78 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2004)Google Scholar
  10. 10.
    Doetsch, G.: Introduction to the Theory and Application of the Laplace Transformation. Springer, Berlin (1974)CrossRefGoogle Scholar
  11. 11.
    Maolin, D., Wang, Z., Haiyan, H.: Measuring memory with the order of fractional derivative. Sci. Rep. 3, 3431 (2013)CrossRefGoogle Scholar
  12. 12.
    Engheia, N.: On the role of fractional calculus in electromagnetic theory. IEEE Antennas Propag. Mag. 39(4), 35–46 (1997)CrossRefGoogle Scholar
  13. 13.
    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)CrossRefGoogle Scholar
  14. 14.
    Gorenflo, R., Mainardi, F.: Fractional calculus, integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. CISM Courses and Lecture Notes, vol. 378, pp. 223–276. Springer, Wien (1997)CrossRefGoogle Scholar
  15. 15.
    Henry, B.I., Wearne, S.L.: Existence of turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math. 62, 870–887 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Heymans, N., Bauwens, J.-C.: Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta 33, 210–219 (1994)CrossRefGoogle Scholar
  17. 17.
    Huang, S., Xiang, Z.: Stability of a class of fractional-order two-dimensional non-linear continuous-time systems. IET Control Theory Appl. 10(18), 2559–2564 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  19. 19.
    Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)Google Scholar
  20. 20.
    Li, C., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71(4), 621–633 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J.-Spec. Top. 193, 27–47 (2011)CrossRefGoogle Scholar
  22. 22.
    Li, Y., Chen, Y.Q., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11(11), 1335–1342 (2008)CrossRefGoogle Scholar
  24. 24.
    Mainardi, F.: Fractional relaxation-oscillation and fractional phenomena. Chaos Solitons Fractals 7(9), 1461–1477 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In Computational Engineering in Systems Applications, pp. 963–968 (1996)Google Scholar
  26. 26.
    Mozyrska, D., Wyrwas, M.: Explicit criteria for stability of fractional h-difference two-dimensional systems. Int. J. Dyn. Control. 5(1), 4–9 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mozyrska, D., Wyrwas, M.: Stability by linear approximation and the relation between the stability of difference and differential fractional systems. Math. Methods Appl. Sci. 40(11), 4080–4091 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Petras, I.: Stability of fractional-order systems with rational orders (2008). arXiv preprint arXiv:0811.4102
  29. 29.
    Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)Google Scholar
  30. 30.
    Radwan, A.G., Elwakil, A.S., Soliman, A.M.: Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits Syst. I: Regul. Pap. 55(7), 2051–2063 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rivero, M., Rogosin, S.V., Tenreiro Machado, J.A., Trujillo, J.J.: Stability of fractional order systems. Math. Probl. Eng. 2013 (2013)Google Scholar
  32. 32.
    Sabatier, J., Farges, C.: On stability of commensurate fractional order systems. Int. J. Bifurc. Chaos 22(04), 1250084 (2012)CrossRefGoogle Scholar
  33. 33.
    Trächtler, A.: On BIBO stability of systems with irrational transfer function (2016). arXiv preprint arXiv:1603.01059
  34. 34.
    Wang, Z., Yang, D., Zhang, H.: Stability analysis on a class of nonlinear fractional-order systems. Nonlinear Dyn. 86(2), 1023–1033 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Weinberg, S.H.: Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model. PloS one 10(5), e0126629 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.West University of TimişoaraTimişoaraRomania
  2. 2.Academy of Romanian ScientistsBucharestRomania
  3. 3.Institute e-Austria TimisoaraTimişoaraRomania

Personalised recommendations