Abstract
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.
Similar content being viewed by others
References
Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611 (2011)
Anastassiou, G.A.: Discrete fractional calculus and inequalities. arXiv e-prints arXiv:0911.3370 (2009)
Area, I., Losada, J., Nieto, J.J.: On fractional derivatives and primitives of periodic functions. Abstr. Appl. Anal. 2014, 392598 (2014)
Ausloos, M., Dirickx, M.: The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications. Springer, Berlin (2006)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int 13(5), 529–539 (1967)
Chen, F., Luo, X., Zhou, Y.: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011(1), 1–12 (2010)
Edelman, M.: Universal fractional maps and cascade of bifurcations type attractors. Chaos 23, 033127 (2013)
Edelman, M.: Caputo standard \(\alpha \)-family of maps: fractional difference vs. fractional. Chaos 24, 023137 (2014)
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)
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)
Edelman, M.: Evolution of systems with power-law memory: do we have to die? arXiv e-prints arXiv:1904.13370 (2019)
Edelman, M., Tarasov, V.: Fractional standard map. Phys. Lett. A 374, 279 (2009)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)
Jonnalagadda, J.M.: Periodic solutions of fractional nabla difference equations. Commun. Appl. Anal. 20, 585–609 (2016)
Jonnalagadda, J.M.: Quasi-periodic solutions of fractional nabla difference systems. Fract. Differ. Calc. 7(2), 339–355 (2017)
Kanso, A., Smaoui, N.: Logistic chaotic maps for binary numbers generations. Chaos Solitons Fractals 40(5), 2557–2568 (2009)
Kocarev, L., Jakimoski, G.: Logistic map as a block encryption algorithm. Phys. Lett. A 289(4), 199–206 (2001)
Kurakin, A., Kuzmin, A., Nechavev, A.: Linear complexity of polinear sequences. J. Math. Sci. 76, 2793–2915 (1995)
Landauskas, M., Navickas, Z., Vainoras, A., Ragulskis, M.: Weighted moving averaging revisited: an algebraic approach. Comput. Appl. Math. 36, 1545–1558 (2017)
Landauskas, M., Ragulskis, M.: A pseudo-stable structure in a completely invertible bouncer system. Nonlinear Dyn. 78, 1629–1643 (2014)
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)
Lu, G., Landauskas, M., Ragulskis, M.: Control of divergence in an extended invertible logistic map. Int. J. Bifurc. Chaos 28(10), 1850129 (2018)
May, M.R.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)
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)
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)
Pareek, N., Patidar, V., Sud, K.: Image encryption using chaotic logistic map. Image Vis. Comput. 24(9), 926–934 (2006)
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)
Phatak, S.C., Rao, S.S.: Logistic map: a possible random-number generator. Phys. Rev. E 51, 3670–3678 (1995)
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)
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)
Tavazoei, M.S., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45, 1886–1890 (2009)
Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)
Yazdani, M., Salarieh, H.: On the existence of periodic solutions in time-invariant fractional order systems. Automatica 47, 1834–1837 (2011)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Petkevičiūtė-Gerlach, D., Timofejeva, I. & Ragulskis, M. Clocking convergence of the fractional difference logistic map. Nonlinear Dyn 100, 3925–3935 (2020). https://doi.org/10.1007/s11071-020-05703-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05703-6