Clocking convergence of the fractional difference logistic map


The convergence of the fractional difference logistic map is studied in this paper. A computational technique based on the visualisation of the algebraic complexity of transient processes is employed for that purpose. It is demonstrated that the dynamics of the fractional difference logistic map is similar to the behaviour of the extended invertible logistic map in the neighbourhood of unstable orbits. This counter-intuitive result provides a new insight into the transient processes of the fractional difference logistic map.

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  1. 1.

    Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611 (2011)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Anastassiou, G.A.: Discrete fractional calculus and inequalities. arXiv e-prints arXiv:0911.3370 (2009)

  3. 3.

    Area, I., Losada, J., Nieto, J.J.: On fractional derivatives and primitives of periodic functions. Abstr. Appl. Anal. 2014, 392598 (2014)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Ausloos, M., Dirickx, M.: The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications. Springer, Berlin (2006)

    Google Scholar 

  5. 5.

    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int 13(5), 529–539 (1967)

    Google Scholar 

  6. 6.

    Chen, F., Luo, X., Zhou, Y.: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011(1), 1–12 (2010)

    Google Scholar 

  7. 7.

    Edelman, M.: Universal fractional maps and cascade of bifurcations type attractors. Chaos 23, 033127 (2013)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Edelman, M.: Caputo standard \(\alpha \)-family of maps: fractional difference vs. fractional. Chaos 24, 023137 (2014)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Edelman, M.: Fractional maps as maps with power-law memory. In: Afraimovich, V., et al. (eds.) Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity, vol. 8, pp. 79–120. Springer, New York (2014)

    Google Scholar 

  10. 10.

    Edelman, M.: Fractional maps and fractional attractors. Part II: fractional difference Caputo \(\alpha \)-families of maps. Interdiscip. J. Discontin. Nonlinearity Complex. 4, 391–402 (2015)

    MATH  Google Scholar 

  11. 11.

    Edelman, M.: Evolution of systems with power-law memory: do we have to die? arXiv e-prints arXiv:1904.13370 (2019)

  12. 12.

    Edelman, M., Tarasov, V.: Fractional standard map. Phys. Lett. A 374, 279 (2009)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)

    Google Scholar 

  14. 14.

    Jonnalagadda, J.M.: Periodic solutions of fractional nabla difference equations. Commun. Appl. Anal. 20, 585–609 (2016)

    MATH  Google Scholar 

  15. 15.

    Jonnalagadda, J.M.: Quasi-periodic solutions of fractional nabla difference systems. Fract. Differ. Calc. 7(2), 339–355 (2017)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kanso, A., Smaoui, N.: Logistic chaotic maps for binary numbers generations. Chaos Solitons Fractals 40(5), 2557–2568 (2009)

    MATH  Google Scholar 

  17. 17.

    Kocarev, L., Jakimoski, G.: Logistic map as a block encryption algorithm. Phys. Lett. A 289(4), 199–206 (2001)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Kurakin, A., Kuzmin, A., Nechavev, A.: Linear complexity of polinear sequences. J. Math. Sci. 76, 2793–2915 (1995)

    MathSciNet  Google Scholar 

  19. 19.

    Landauskas, M., Navickas, Z., Vainoras, A., Ragulskis, M.: Weighted moving averaging revisited: an algebraic approach. Comput. Appl. Math. 36, 1545–1558 (2017)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Landauskas, M., Ragulskis, M.: A pseudo-stable structure in a completely invertible bouncer system. Nonlinear Dyn. 78, 1629–1643 (2014)

    MathSciNet  Google Scholar 

  21. 21.

    López-Ruiz, R., Fournier-Prunaret, D., Nishio, Y., Grácio, C.: Nonlinear Maps and Their Applications: Selected Contributions from the NOMA 2013 International Workshop. Springer Proceedings in Mathematics & Statistics. Springer, Berlin (2015)

    Google Scholar 

  22. 22.

    Lu, G., Landauskas, M., Ragulskis, M.: Control of divergence in an extended invertible logistic map. Int. J. Bifurc. Chaos 28(10), 1850129 (2018)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    May, M.R.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)

    MATH  Google Scholar 

  24. 24.

    Miller, K.S., Ross, B.: Fractional difference calculus. In: Srivastava, H.M., Owa, S. (eds.) Univalent Functions, Fractional Calculus, and Their Applications, pp. 139–151. Ellis Horwood, Chichester (1989)

    Google Scholar 

  25. 25.

    Murillo-Escobar, M.A., Cruz-Hernández, C., Cardoza-Avendaño, L., Méndez-Ramírez, R.: A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dyn. 87(1), 407–425 (2017)

    MathSciNet  Google Scholar 

  26. 26.

    Pareek, N., Patidar, V., Sud, K.: Image encryption using chaotic logistic map. Image Vis. Comput. 24(9), 926–934 (2006)

    Google Scholar 

  27. 27.

    Peng, Y., Sun, K., He, S., Wang, L.: Comments on “Discrete fractional logistic map and its chaos” [Nonlinear Dyn. 75, 283–287 (2014)]. Nonlinear Dyn. 97(1), 897–901 (2019)

    Google Scholar 

  28. 28.

    Phatak, S.C., Rao, S.S.: Logistic map: a possible random-number generator. Phys. Rev. E 51, 3670–3678 (1995)

    Google Scholar 

  29. 29.

    Ragulskis, M., Navickas, Z.: The rank of a sequence as an indicator of chaos in discrete nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 16, 2894–2906 (2011)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Google Scholar 

  31. 31.

    Tavazoei, M.S., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45, 1886–1890 (2009)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Yazdani, M., Salarieh, H.: On the existence of periodic solutions in time-invariant fractional order systems. Automatica 47, 1834–1837 (2011)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Minvydas Ragulskis.

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Petkevičiūtė-Gerlach, D., Timofejeva, I. & Ragulskis, M. Clocking convergence of the fractional difference logistic map. Nonlinear Dyn 100, 3925–3935 (2020).

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  • Fractional difference logistic map
  • Convergence
  • H-rank
  • Temporal stabilisation