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

For citation:

Anikin V. M., Arkadakskij S. S., Remizov A. S. Analytical solution of spectral problem for the Perron – Frobenius operator of piece-wise linear chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, iss. 2, pp. 16-34. DOI: 10.18500/0869-6632-2006-14-2-16-34

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 78)
Article type: 

Analytical solution of spectral problem for the Perron – Frobenius operator of piece-wise linear chaotic maps

Anikin Valerij Mihajlovich, Saratov State University
Arkadakskij Sergej Sergeevich, Saratov State University
Remizov Aleksandr Sergeevich, Saratov State University

Spectral properties of the linear non-self-adjoint Perron – Frobenius operator of piece-wise linear chaotic maps having regular structure are investigated. Eigenfunctions of the operator are found in the form of Bernoulli and Euler polynomials. Corresponding eigenvalues are presented by negative powers of number of map brunches. The solution is obtained in general form by means of generating functions for eigenfunctions of the operator. Expressions for eigenfunctions and eigenvalues are different for original and inverse maps having even and odd number of branches. Results allow us to find analogous solution of the spectral problem for conjugate maps and to calculate analytically decay of correlations for such chaotic dynamical systems.

Key words: 
  1. Lichtenberg AJ, Lieberman MA. Regular and stochastic motion. New York: Springer; 1984. 528 p.
  2. Schuster G. Deterministic chaos: Introduction. Moscow: Mir; 1988. 240 p. (In Russian).
  3. Neimark YI, Landa PS. Stochastic and Chaotic Oscillations. Dordrecht: Springer; 1992. 500 p. DOI: 10.1007/978-94-011-2596-3.
  4. Malinetsky GG. Chaos. Structures. Computational experiment: Introduction to nonlinear dynamics. Moscow: Editorial URSS; 2000. 356 p. (In Russian).
  5. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
  6. Prigozhin IR, Stengers I. Time, chaos, quantum. To solve the paradox of time. Moscow: Progress; 1994. 272 p. (In Russian).
  7. Koronovsky AA, Trubetskov DI. Nonlinear dynamics in action. Saratov: Kolledg; 2002. 324 p. (In Russian).
  8. Lifshitz EM, Khalatnikov IM, Sinai YaG, Khanin KM, Shchur LN. On the stochastic properties of relativistic cosmological models near the singularity. JETP Letters. 1983;38(2):79–82.
  9. Golubentsev AF, Anikin VM. On the chaotic model of the early evolution of the Universe. Radioengineering. 2005;4:50–55. (In Russian).
  10. Godunov SK, Antonov AG, Kirilyuk OP, Kostin VI. Guaranteed accuracy of solving systems of linear equations in Euclidean spaces. Novosibirsk: Nauka, Siberian Branch; 1988. 456 p. (In Russian).
  11. Babenko KI. Fundamentals of numerical analysis. Moscow: Nauka; 1986. 743 p. (In Russian).
  12. Golubencev AF, Anikin VM. Invariant subspaces for linear evolution operators of chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(1-2):3-17. DOI: 10.18500/0869-6632-2005-13-1-3-37.
  13. Antoniou I, Tasaki S. Generalized spectral decomposition of mixing dynamical systems. Int. J. Quantum Chemistry. 1993;46(3):425–474. DOI: 10.1002/QUA.560460311.
  14. Golubentsev AF, Anikin VM, Arkadaksky SS. On some properties of the Frobenius-Perron operator for Bernoulli shifts. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(2):67—73 (in Russian).
  15. Goloubentsev AF, Anikin VM, Arkadaksky SS. On the convergence of nonstationary solutions of the Frobenius – Perron equations to the invariant density. Proceedings of 2nd International Conference «Control of Oscillation and Chaos». Edited by FL. Chernousko and AL. Fradkov. S.-Petersburg. 2000. Vol. 1. P. 142.
  16. Dorfle M. Spectrum and eigenfunctions for the Frobenius – Perron operator of thetent map. J. Stat. Phys. 1985;40(1/2):93–132. DOI: 10.1007/BF01010528.
  17. Golubentsev AF, Anikin VM, Arkadaksky SS. Conjugated chaotic displays: construction, trajectory, probabilistic and spectral characteristics. Problems of modern physics. Ed. AN. Sisakyan and DI. Trubetskova. Dubna: JINR; 2000. P. 172. (In Russian).
  18. Golubentsev AF, Anikin VM, Barulina YuA. To solve the spectral problem for the evolutionary operator by the method of producing functions. Modeling: Sat. scientific articles. Ed. BE. Zhelezovsky. Saratov: Istok-S; 2002. P. 24. (In Russian).
  19. Malinetsky GG, Potapov AB. Modern problems of nonlinear dynamics. Moscow: Editorial URSS; 2000. 336 p. (In Russian).
  20. Blank LM. Stability and Localization in Chaotic Dynamics. Moscow: MCCME; 2001. 352 p. (in Russian).
  21. Lasota A, Mackey MC. Probabilistic properties of deterministic systems. Cambridge: Cambridge University Press; 1985. Ch.4. DOI: 10.1017/CBO9780511897474.
  22. Iosifescu M, Kraaikamp C. Metrical Theory of Continued Fractions. Kluwer Boston, Inc. 2002. Chps. 1, 2.
  23. Handbook of mathematical functions with formular, graphs and mathematical tables. Ed. M. Abramowitz and I. Stigan. U.S. Department of Commerce, National Bureau of Standards. 1964.
Short text (in English):
(downloads: 51)