ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Anikin V. M. Representation of exact trajectory solutions for chaotic one-dimensional maps in Schroder form. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 2, pp. 128-142. DOI: 10.18500/0869-6632-003034, EDN: JVVGQR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
Full text PDF(En):
(downloads: 81)
Article type: 

Representation of exact trajectory solutions for chaotic one-dimensional maps in Schroder form

Anikin Valerij Mihajlovich, Saratov State University

Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps.

We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure.

Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented.

Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings. 

  1. Schroder E. Ueber unendlich viele Algorithmen zur Auflosung der Gleichungen. Mathematische Annalen. 1870;2(2):317–365 (in German). DOI: 10.1007/BF01444024.
  2. Schroder E. Ueber iterirte Functionen. Mathematische Annalen. 1870;3(2):296–322 (in German). DOI: 10.1007/BF01443992.
  3. Milnor J. Dynamics in One Complex Variable: Introductory Lectures. 3rd edition. Princeton: Princeton University Press; 2006. 320 p.
  4. Peitgen H-O, Richter PH. The Beauty of Fractals: Images of Complex Dynamical Systems. Berlin, Heidelberg, New York: Springer-Verlag; 1986. 202 p. DOI: 10.1007/978-3-642-61717-1.
  5. Crownover RM. Introduction to Fractals and Chaos. Boston, London: Jones and Bartlett Publishers; 1995. 350 p.
  6. Schroeder M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: Dover Publications; 2009. 448 p.
  7. Alexander DS. A History of Complex Dynamics: From Schroder to Fatou and Julia. Vol. E24 of Aspects of Mathematics. Braunschweig: Friedr. Vieweg & Sohn; 1994. 165 p.
  8. Alexander DS, Iavernaro F, Rosa A. Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906–1942. Vol. 38 of History of Mathematics. Providence, RI, London: London Mathematical Society; 2012. 454 p.
  9. Kuczma M, Choczewski B, Ger R. Iterative Functional Equations. Cambridge: Cambridge University Press; 1990. 576 p. DOI: 10.1017/CBO9781139086639.
  10. Kuczma M. Functional Equations in a Single Variable. Warszawa: PWN-Polish Scientific Publishers; 1968. 383 p.
  11. de Bruijn NG. Asymptotic Methods in Analysis. New York: Dover Publications; 1981. 296 p.
  12. Yanpol’skii AR. Hyperbolic Functions. Moscow: Fizmatgiz; 1960. 195 p. (in Russian).
  13. Billingsley P. Ergodic Theory and Information. New York: John Wiley & Sons; 1965. 193 p.
  14. Anikin VM, Golubentsev AF. Analytical Models of Deterministic Chaos. Moscow: FIZMATLIT; 2007. 328 p. (in Russian).
  15. Ulam SM. A Collection of Mathematical Problems. New York, London: Interscience Publishers; 1960. 150 p.
  16. Ulam SM, von Neumann J. On combination of stochastic and deterministic processes. Bulletin of the American Mathematical Society. 1947;53(11):1120.
  17. von Neumann J. Collected Works. Vol. 5. New York: Macmillan; 1963. P. 768–770.
  18. Schuster HG, Just W. Deterministic Chaos: An Introduction. 4th, Revised and Enlarged Edition. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA; 2005. 299 p. DOI: 10.1002/3527604804.
  19. Ermakov SM. Monte Carlo Method in Computational Mathematics. Saint Petersburg: Nevskii Dialect; Moscow: BINOM, Laboratorija Znanij; 2009. 192 p. (in Russian).
  20. Kuipers L, Niederreiter H. Uniform Distribution of Sequences. Mineola, New York: Dover Publications; 2006. 416 p.
  21. Golubentsev AF, Anikin VM. The explicit solutions of Frobenius-Perron equation for the chaotic infinite maps. International Journal of Bifurcation and Chaos. 1998;8(5):1049–1051. DOI: 10.1142/S0218127498000863.
  22. Golubentsev AF, Anikin VM. Special functions in the theory of deterministic chaos. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(3):50–58 (in Russian).
  23. Whittaker ET, Watson GN. A Course of Modern Analysis. 4th Edition. Cambridge: Cambridge University Press; 1996. 616 p. DOI: 10.1017/CBO9780511608759.
  24. Anikin VM, Arkadakskii SS, Remizov AS. Non-Self-Adjoint Linear Operators in Chaotic Dynamics. Saratov: Saratov University Publishing; 2015. 96 p. (in Russian).
Available online: