For citation:
Golubencev A. F., Anikin V. M., Noyanova . A. Modifications of the baker transformation and their asimptotic properties. Izvestiya VUZ. Applied Nonlinear Dynamics, 2004, vol. 12, iss. 3, pp. 45-57. DOI: 10.18500/0869-6632-2004-12-3-45-57
Modifications of the baker transformation and their asimptotic properties
Some modifications of the baker transformation are introduced. The equations for the brunches of the maps are solved exactly. It is shown that the y-component of baker transformations is represented by a linear autoregression equation of the first order where digits of the initial value x₀, play the role of ап excitation (input signal) if x₀, is considered as а random value. It is shown that the digital filter corresponding to the baker transformation is causal, stable and reversible one. The asymptotic regime of baker transform dynamics does not depend оп the distribution of the initial value y₀,. Investigations of asymptotic properties of the baker map are important for chaos-based key cryptography schemes for digital communication.
1. Hopf E. Ergodic Theory. Russian Mathematical Surveys. 1949;4(2–39):113–182.
2. Halmos PR. Lectures on Ergodic Theory. Izhevsk: Regular and chaotic dynamics; 2001. 132 p.
3. Arnold VI, Avets A. Ergodic Problems in Classical Mechanics, Regular and Chaotic Dynamics. Izhevsk: Regular and chaotic dynamics; 1999. 284 p. (in Russian).
4. Rid M, Saimon B. Methods of modern mathematical physics. Functional analysis. Moscow: Mir; 1977. 360 p. (in Russian).
5. Kornfeld IP, Sinai YaG, Fomin SV. Ergodic theory. Springer, New York, 1982. 482 p.
6. Lichtenberg A, Liberman M. Regular and Stochastic Dynamics. Moscow: Mir; 1984. 528 p. (in Russian).
7. Lasota А, Mackey MC. Probabilistic properties of deterministic systems. Cambridge: Cambridge University Press; 1985. 358 p.
8. Nikolis G, Prigozhin I. Exploring Complexity. Moscow: Mir; 1990. 334 p. (in Russian).
9. Prigozhin I. The End of Certainty: Time, Chaos, and the New Laws of Nature. Izhevsk: Regular and chaotic dynamics; 2000. 208 p. (in Russian).
10. Tabor M. Chaos and integratability in the nonlinear dynamics. Moscow: Editorial URSS Publ.; 2001. 320 p. (in Russian).
11. Manfred R Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Izhevsk: Regular and chaotic dynamics; 2001. 528 p. (in Russian).
12. Kuznetsov SP. Cynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
13. Gаspard P. Diffusion, effusion and chaotic scattering: An exactly solvable Liouvillian dynamics. J. Stat. Phys.1992;68(5/6):673–747.
14. Schuster G. Deterministic Chaos: An Introduction. Moscow: Mir; 1988. 240 p. (in Russian).
15. Graham RL, Knuth DE, Patashnik O. Concrete Mathematics: A Foundation for Computer Science. Moscow: Mir; 1998. 703 p. (in Russian).
16. Goloubentsev AF, Anikin VM, Noyanova SA. On the connection between the baker's transformation and the first-order autoregressive model. In: Proceedings of the Second International Conference «Fundamental problems of physics». 9–14 October 2000, Saratov, Russia. Saratov: College, 2000. P. 63–64. (in Russian).
17. Goloubentsev AF, Anikin VM, Noyanova SA, Barulina YA. Baker transformation аs autoregression system. In: Proceedings of the Second International Conference «Physics and Control». 20–22 August 2003, Saint Petersburg, Russia. Saint Petersburg: Institute of Electrical and Electronics Engineers Inc.; 2003. P. 654–656. DOI: 10.1109/PHYCON.2003.1236911. (in Russian).
18. Kac M. Statistical Independence in Probability, Analysis, and Number Theory. Moscow: Foreign literature; 1963. 165 p. (in Russian).
19. Goloubentsev AF, Anikin VM. Euclid, Gauss and deterministic chaos. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2003;3(2):166–176. (in Russian).
20. Rabiner LR, Gold B. Theory and Application of Digital Signal Processing. Moscow: Mir; 1978. 848 p. (in Russian).
21. Driebe DJ, Ordonez G.E. Using symmetries оf the Frobenius-Perron operator to determine spectral decompositions. Phys. Lett. А. 1996;211:204–210.
22. Goloubentsev AF, Anikin VM, Noyanova SA. «Инверсное» отображение пекаря. In: Proceedings of the Sixth International School оn chaotic oscillations and pattern formation. 2–7 October 2001, Saratov, Russia. Saratov: College, 2001. P. 59. (in Russian).
23. Antoniou I, Tasaki S. Generalized spectral decomposition оf mixing dynamical systems. Int. J. of Quantum Chemistry. 1993;46:425–474.
- 165 reads