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

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Anikin V. M., Trubetskov D. I. Problems of deterministic chaos theory in А. F. Goloubentsev’s works. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 5, pp. 120-123. DOI: 10.18500/0869-6632-2013-21-5-120-123

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Language: 
Russian
Heading: 
History of Nonlinear Dynamics. Personalia
Article type: 
Personalia
UDC: 
519.24
DOI: 
10.18500/0869-6632-2013-21-5-120-123

Problems of deterministic chaos theory in А. F. Goloubentsev’s works

Autors: 
Anikin Valerij Mihajlovich, Saratov State University
Trubetskov Dmitriy Ivanovich, Saratov State University
Abstract: 

A short review of contribution to the deterministic chaos theory, that had been made by professor Alexander F. Goloubentsev (Saratov University), is given.

Key words: 
Chaotic maps
non-self-adjoint Perron–Frobenius operator
spectral problem
Reference: 
  1. Anikin VM, Gulyaev UV, Trubetskov DI. and others. In memory of Alexander Fedorovich Golubentsev. Radiotehnika and electronika. 2004;49(3):383–384.
  2. Babenko KI. Fundamentals of Numerical Analysis. Moscow: Nauka; 1986. 743 p. (In Russian).
  3. Golubentsev AF, Anikin VM. Euclid, Gauss and deterministic chaos. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2003;3(2):166–176.
  4. Anikin VM. Gaussian Mapping: Evolutionary and Probabilistic Properties. Saratov: Saratov University Publishing; 2007. 80 p. (In Russian).
  5. Goloubentsev AF, Anikin VM, Arkadaksky SS. On some properties of the Frobenius - Perron operator for the Bernoulli shifts. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(2):67–73.
  6. 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.
  7. Anikin VM, Golubentsev AF. Analytical models of deterministic chaos. Ed. Trubetskov DI. Moscow: Fizmatlit; 2007. 328 p. (In Russian).
  8. Anikin VM, Arkadakskij SS, Remizov AS. Analytical solution of spectral problem for the Perron – Frobenius operator of piece-wise linear chaotic maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(2):16–34. DOI: 10.18500/0869-6632-2006-14-2-16-34.
  9. Anikin VM, Arkadaksky SS, Remizov AS. FEATURES OF SOLVING SPECTRAL PROBLEM FOR THE PERRON-FROBENIUS OPERATOR, CAUSED BY CRITICAL COMBINATIONS OF CHAOTIC MAP PARAMETERS. Teoriticheskaya fizika. 2007;8:176–183.
  10. Anikin VM, Arkadakskij SS, Remizov AS, Kupcov SN, Vasilenko LP. Investigation of structure of invariant density for Renyi map by Gauss method. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(6):46–56. DOI: 10.18500/0869-6632-2008-16-6-46-56.
  11. Anikin VM, Arkadakskii SS, Remizov AS. et all. Lyapunov exponent for chaotic 1D maps with uniform invariant distribution. Bulletin of the Russian Academy of Sciences: Physics. 2008;72(12):1684–1688. DOI: 10.3103/S106287380812023X.
  12. Anikin VM, Arkadakskii SS, Remizov AS. et all. Classification of one-dimensional chaotic models. Bulletin of the Russian Academy of Sciences: Physics. 2009;73(12):1681–1683. DOI: 10.3103/S1062873809120302.
  13. Anikin VM. Spectral problems for the Perron–Frobenius operator. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(4):35–48. DOI: 10.18500/0869-6632-2009-17-4-35-48.
  14. Anikin VM, Arkadakskii SS, Remizov AS. et all. Relaxation properties of chaotic dynamical systems: Operator approach. Bulletin of the Russian Academy of Sciences: Physics. 2009;73(12):1632–1637. DOI: 10.3103/S106287380912020X.
  15. Goloubentsev AF, Anikin VM. The explicit solutions of Frobenius–Perron equation for the chaotic infinite maps. Int. J. Bifurcation and Chaos. 1998;8(5):1049–1051. DOI: 10.1142/S0218127498000863.
  16. Golubentsev AF, Anikin VM, Bogomolov AV. Chaotic generators of biological rhythms. Biomedicine radioengineering. 2000;2:38–41.
  17. Goloubentsev AF, Anikin VM. SPECIAL FUNCTIONS IN THE THEORY OF DETERMINISTIC CHAOS. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(3):50–58.
  18. Goloubentsev AF, Anikin VM, Arkadaksky SS. Ergodic maps with Lyapunov exponent equal to zero. 2nd International Conference «Control of Oscillation and Chaos», July 5-7, 2000, St. Petersburg, Russia: Proceedings. Ed. Chernousko FL, Fradkov AL. 2000. Vol. 1. 44 p.
  19. Goloubentsev AF, Anikin VM, Arkadaksky SS. On the convergence of nonstationary solutions of the Frobenius–Perron equations to the invariant density. Ibid. 2000;1:142–143. DOI: 10.1109/COC.2000.873537.
  20. Goloubentsev AF, Anikin VM, Barulina YA. Difference scheme with instant transition from order to chaos. Int. Conf. «Physics and Control–2003». St. Petersburg, Russia, August 20-22, 2003. St. Petersburg: Proceedings. 2003;2:446–447. DOI: 10.1109/PHYCON.2003.1236864.
  21. Goloubentsev AF, Anikin VM, Barulina YA. Chaotic maps generating white noise. Ibid. P. 2003;2:452–455. DOI: 10.1109/PHYCON.2003.1236865.
  22. Goloubentsev AF, Anikin VM, Noyanova SA, Barulina YA. Baker transformation as autoregression system. Ibid. P. 2003;2:654–656. DOI: 10.1109/PHYCON.2003.1236911.
  23. Goloubentsev AF, Anikin VM, Noyanova SA. MODIFICATIONS OF THE BAKER TRANSFORMATION AND THEIR ASIMPTOTIC PROPERTIES. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004;12(3):45–57.
  24. Golubentsev AF, Anikin VM. On the chaotic model of the early evolution of the Universe. Journal Radioengineering. 2005;4:50–55.
Received: 
15.09.2013
Accepted: 
15.09.2013
Published: 
31.12.2013
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