Descriptive set theory is a classical branch of mathematics formed at the beginning of the last century. We offer the basics of descriptive chaos theory. It is shown that if the topological entropy is positive, then a dynamical system:
(1) has many different trajectory attractors, namely, a continuum of attractors;
(2) the basins of most attractors have an overly complex structure, namely, are sets of the third class according to the terminology of descriptive set theory;
(3) the basins of different attractors are too strongly intertwined and they cannot be separated from each other by any open or closed sets, but only by sets of the second complexity class, and
(4) the set of all attractors of the dynamical system forms an attractor network (grid) in the space of closed sets of the state space (with the Hausdorff metric), the cells of which are created by Cantor sets from the attractors themselves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Ruette, “Chaos on the interval,” Amer. Math. Soc., Ser. Univ. Lect., 67 (2017).
A. N. Sharkovsky, “On attracting and attracted sets,” Dokl. Akad. Nauk SSSR, 160, No. 5, 1036–1038 (1965); English translation: Soviet Math. Dokl., 6, 268–270 (1965).
A. N. Sharkovsky, “A classification of fixed points,” Ukr. Mat. Zh., 17, No. 5, 80–95 (1965); English translation: Amer. Math. Soc. Transl. Ser. 2, 97, 159–179 (1970).
A. N. Sharkovsky, “Behavior of a mapping in the neighborhood of an attracting set,” Ukr. Mat. Zh., 18, No. 2, 60–83 (1966); English translation: Amer. Math. Soc. Transl. Ser. 2, 97, 227–258 (1970).
T.-Y. Li and J. A. Yorke, “Period three implies chaos,” Amer. Math. Monthly, 82, No. 10, 985–992 (1975).
P. Kloeden, M. Deakin, and A. Tirkel, “A precise definition of chaos,” Nature, 264, 295 (1976).
A. N. Sharkovsky, “Coexistence of the cycles of a continuous map of the line into itself,” Ukr. Math. J., 16, No. 1, 61–71 (1964); Intern. J. Bifurcation Chaos, 5, No. 5, 1263–1273 (1995); Reprint of the paper in World Sci. Ser. Nonlin. Sci. Ser. B, 8, Thirty years after Sharkovsky’s theorem: new perspectives (Murcia, 1994) (1995), pp. 1–11.
A. N. Sharkovsky, “Structure of an endomorphisms on an ω-limit set,” in: Abstr. of the International Mathematical Congr. (Moscow, 1966), Sec. 6, p. 51.
A. N. Shakovsky, “How complicated can be one-dimensional dynamical systems: descriptive estimates of sets,” in: Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, Warsaw (1989), pp. 447–453.
A. G. Sivak, “Descriptive estimates for statistically limit sets of dynamical systems,” in: Dynamical Systems and Turbulence [in Russian], Institute of Mathematics, Akad. Nauk Ukr. RSR, Kiev (1989), pp. 100–102.
A. G. Sivak, “On the structure of the set of trajectories generating an invariant measure,” in: Dynamical Systems and Nonlinear Phenomena [in Russian], Institute of Mathematics, Akad. Nauk Ukr. RSR, Kiev (1990), pp. 39–43.
A. G. Sivak, σ-Attractors of Trajectories and Their Basins [in Russian], Addition, Chap. 7, in [14], pp. 281–310.
A. N. Sharkovsky and A. G. Sivak, “Basins of attractors of trajectories,” J. Difference Equat. Appl., 22, No. 2, 159–163 (2016).
A. N. Sharkovsky, Attractors of Trajectories and Their Basins [in Russian], Naukova Dumka, Kiev (2013).
R. Baire, “Sur la representation des fonctions discontinues,” Acta Math., 30 (1905).
L. V. Keldysh, “Structure of B-sets,” Tr. Steklov Mat. Inst., 17 (1945).
A. N. Sharkovsky, “Partially ordered system of attracting sets,” Dokl. Akad. Nauk SSSR, 170, No. 6, 1276–1278 (1965); English translation: Soviet Math. Dokl., 7, 1384–1386 (1966)).
A. M. Blokh, A. M. Bruckner, P. D. Humke, and J. Smital, “Space of ω -limit sets of a continuous map of the interval,” Trans. Amer. Math. Soc., 348, No. 4, 1357–1372 (1996).
V. S. Bondarchuk, Invariant Sets of Smooth Dynamical Systems [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Kiev (1974).
V. S. Bondarchuk and A. N. Sharkovsky, “Reproducibility of extending endomorphisms in the system of ω -limit sets,” in: Dynamical Systems and the Problems of Stability of Solutions of Differential Equations [in Russian], Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1973), pp. 28–34.
A. N. Sharkovsky and V. S. Bondarchuk, “Partially ordered system of ω-limit sets of expanding endomorphisms,” in: Dynamical Systems and the Problems of Stability of Solutions of Differential Equations [in Russian], Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1973), pp. 128–164.
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 12, pp. 1709–1718, December, 2022. Ukrainian DOI:https://doi.org/10.37863/umzh.v74i12.6515.
O. M. Sharkovsky is deceased.
Translated from Ukrainian by Olena Sharkovska, Iryna Chumakova, and Dr. Elena Romanenko.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sharkovsky, O.M. Descriptive Theory of Deterministic Chaos. Ukr Math J 74, 1950–1960 (2023). https://doi.org/10.1007/s11253-023-02180-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-023-02180-z