- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Central limit behavior at the edge of chaos in the z-logistic map

Abbas Ali Saberi1,2,3,*, Ugur Tirnakli4,5,†, and Constantino Tsallis6,7,8,‡

  • *Contact author: asaberi@constructor.university
  • †Contact author: ugur.tirnakli@ieu.edu.tr
  • ‡Contact author: tsallis@cbpf.br
PDF

Phys. Rev. E 112, 064209 – Published 9 December, 2025

DOI: https://doi.org/10.1103/gtlz-67cf

Abstract

We focus on the Feigenbaum-Coullet-Tresser point of the dissipative one-dimensional z-logistic map xt+1=1−a|xt|z(z≥1). We show that sums of iterates converge to q-Gaussian distributions Pq(y)=Pq(0)expq(−βqy2)=Pq(0)[1+(q−1)βqy2]1/(1−q)(q≥1;βq>0), which optimize the nonadditive entropic functional Sq under simple constraints. We propose and justify heuristically a closed-form prediction for the entropic index, q(z)=1+2/(z+1), and validate it numerically via data collapse for typical z values. The formula captures how the limiting law depends on the nonlinearity order and implies finite variance for z>2 and divergent variance for 1≤z≤2. These results extend edge-of-chaos central limit behavior beyond the standard (z=2) case and provide a simple predictive law for unimodal maps with varying maximum order.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (32)

  1. S. H. Strogatz, Nonlinear Dynamics and Chaos (CRC Press, London, 2024).
  2. R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, New York, 2000).
  3. C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems: An Introduction (Cambridge University Press, Cambridge, UK, 1993).
  4. M. Y. Hamada, Investigating the dynamics of generalized discrete logistic map, Math. Models Methods Appl. Sci. 48, 5325 (2025).
  5. M. Lawnik, Generalized logistic map and its application in chaos based cryptography, J. Phys.: Conf. Ser. 936, 012017 (2017).
  6. K.-J. Moon and S. D. Choi, Reducible expansions and related sharp crossovers in Feigenbaum's renormalization field, Chaos 18, 023104 (2008).
  7. B. Hu and J. M. Mao, Period doubling: Universality and critical-point order, Phys. Rev. A 25, 3259 (1982).
  8. B. Hu, Introduction to real-space renormalization-group methods in critical and chaotic phenomena, Phys. Rep. 91, 233 (1982).
  9. P. R. Hauser, C. Tsallis, and E. M. F. Curado, Criticality of the routes to chaos of the 1−a|x|z map, Phys. Rev. A 30, 2074 (1984).
  10. J. P. Van Der Weele, H. W. Capel, and R. Kluiving, Period doubling in maps with a maximum of order z, Physica A 145, 425 (1987).
  11. K. Briggs, A precise calculation of the Feigenbaum constant, Math. Comp. 57, 435 (1991).
  12. V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics: Vol. 16 (World scientific, Singapore, 2000), pp. 324.
  13. U. M. S. Costa, M. L. Lyra, A. R. Plastino, and C. Tsallis, Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity, Phys. Rev. E 56, 245 (1997).
  14. U. Tirnakli, C. Beck, and C. Tsallis, Central limit behavior of deterministic dynamical systems, Phys. Rev. E 75, 040106(R) (2007).
  15. S. Umarov, C. Tsallis, and S. Steinberg, On a q-central limit theorem consistent with nonextensive statistical mechanics, Milan J. Math. 76, 307 (2008).
  16. S. Umarov, C. Tsallis, M. Gell-Mann, and S. Steinberg, Generalization of symmetric α-stable Lévy distributions for q>1, J. Math. Phys. 51, 033502 (2010).
  17. K. P. Nelson and S. Umarov, Nonlinear statistical coupling, Physica A 389, 2157 (2010).
  18. M. G. Hahn, X. X. Jiang, and S. Umarov, On q-Gaussians and exchangeability, J. Phys. A 43, 165208 (2010).
  19. S. Umarov and C. Tsallis, Mathematical Foundations of Nonextensive Statistical Mechanics (World Scientific, Singapore, 2022).
  20. D. Prato and C. Tsallis, Nonextensive foundation of Lévy distributions, Phys. Rev. E 60, 2398 (1999).
  21. C. Tsallis, Introduction to Nonextensive Statistical Mechanics–Approaching a Complex World, 2nd ed. (Springer, 2023).
  22. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988).
  23. O. Afsar and U. Tirnakli, Generalized Huberman-Rudnick scaling law and robustness of qq-Gaussian probability distributions, Europhys. Lett. 101, 20003 (2013).
  24. B. A. Huberman and J. Rudnick, Scaling behavior of chaotic flows, Phys. Rev. Lett. 45, 154 (1980).
  25. U. Tirnakli, C. Tsallis, and C. Beck, Closer look at time averages of the logistic map at the edge of Chaos, Phys. Rev. E 79, 056209 (2009).
  26. D. Lai and G. Chen, Generating different statistical distributions by the Chaotic skew tent map, Int. J. Bifurcat. Chaos 10, 1509 (2000).
  27. C. Beck, U. Tirnakli, and C. Tsallis, Generalization of the Gauss map: A jump into Chaos with universal features, Phys. Rev. E 110, 064213 (2024).
  28. M. A. Pires, C. Tsallis, and E. M. F. Curado, Composing α-Gauss and logistic maps: Gradual and sudden transitions to Chaos, Phys. Rev. E 112, 034209 (2025).
  29. A. Bountis, J. J. Veerman, and F. Vivaldi, Cauchy distributions for the integrable standard map, Phys. Lett. A 384, 126659 (2020).
  30. L. F. Burlaga and A. F. Vinas, Multi-scale probability distributions of solar wind speed fluctuations at 1 AU described by a generalized Tsallis distributions, Geophys. Res. Lett. 31, L16807 (2004).
  31. C. Tsallis, M. Gell-Mann, and Y. Sato, Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive, Proc. Natl. Acad. Sci. USA 102, 15377 (2005).
  32. J.-P. Gazeau and C. Tsallis, Moebius transforms, cycles and q-triplets in statistical mechanics, Entropy 21, 1155 (2019).

Outline

Information

Phys. Rev. E 112, 064209– Published 9 December, 2025
  • Received 6 August 2025
  • Accepted 12 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation