Stabilization of chaotic dynamics of one-dimensional maps by a cyclic parametric transformation

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 6, Issue 4, 1996, Pages 725-735

  Loskutov, Alexander Yu ^a   Tereshko, Valery M ^a   Vasiliev, Konstantin A ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041115404     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127496000382     Document Type: Article

References (41)

1. Abarbanel, H. D. I & Linsay, P. S. [1993] "Secure communications and unstable periodic orbits of strange attractors," IEEE Trans. Circuits Syst. 40, 643-645.
2. Alekseev, V. V. & Loskutov, A. Yu. [1985] "Destochastization of a system with a strange attractor by parametric interaction," Moscow Univ. Phys. Bull. 40, 46-49.
3. Alekseev, V. V. & Loskutov, A. Yu. [1987] "Control of a system with a strange attractor through periodic parametric action," Sov. Phys. Dokl. 32, 270-271.
4. Bayly, P. V & Virgin, L. N. [1994] "Practical considerations in the control of chaos," Phys. Rev. E50, 604-607.
5. Braiman, Y. & Goldhirsh, I. [1991] "Taming chaotic dynamics with weak periodic perturbations," Phys. Rev. Lett. 66, 2545-2548.
6. Collet, P. & Eckmann, J.-P. [1980] Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston).
7. Guemez, J. & Matias, M. A. [1993] "Control of chaos in unidimensional maps," Phys. Lett. A181, 29-32.
8. Hayes, S., Grebogi, C. & Ott, E. [1993] "Communicating with chaos," Phys. Rev. Lett. 70, 3031-3034.
9. Hayes, S., Grebogi, C., Ott, E. & Mark, A. [1994] "Experimental control of chaos for communication," Phys. Rev. Lett. 73, 1781-1784.
10. Huang, W.-H. [1994] "On the bifurcation in a 'modulated' logistic map," Phys. Lett. A194, 57-58.
11. Hübler, A. [1989] "Adaptive control of chaotic systems," Helv. Phys. Acta 62, 343-346.
12. Jackson, E. A. & Hübler, A. [1990] "Periodic entrainment of chaotic logistic map dynamics," Physica D44, 407-420.
13. Jackson, E. A. & Hübler, A. [1992] "Entrainment and migration controls of two-dimensional maps," Physica D54, 253-265.
14. Jakobson, M. [1989] "Ergodic theory of one-dimensional maps," in Dynamical Systems II (Springer Verlag, Berlin).
15. Kivshar, Yu. S., Rödelsperger, B. & Benner, H. [1994] "Suppression of chaos by nonresonant parametric perturbations," Phys. Rev. E49, 319-324.
16. Lasota, A. & Mackey, M. C. [1994] Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics (Springer, Berlin).
17. Lima, R. & Pettini, M. [1990] "Suppression of chaos by resonant parametric perturbation," Phys. Rev. A41, 726-733.
18. Liu, Y. & Rios Leite, J. R. [1994] "Control of Lorenz chaos," Phys. Lett. A185, 35-37.
19. Loskutov, A. Yu. [1993] "Dynamics control of chaotic systems by parametric destochastization," J. Phys. A: Math. Gen. 26, 4581-4594.
20. Loskutov, A. Yu. & Shishmarev, A. I. [1993] "On a property of the quadratic maps family with the stochastic dynamics at parametric perturbation," Uspekhi Mat. Nauk 48, 169-170 (in Russian).
21. Loskutov, A. Yu. & Shishmarev, A. I. [1994] "Control of dynamical systems behavior by parametric perturbations: An analytic approach," Chaos 4, 391-395.
22. Loskutov, A. Yu. & Rybalko, S. D. [1994] "Parametric suppression of chaos in n-dimensional maps," ICTP Preprint No. IC/94/347. Trieste, Italy, Nov. 1994.
23. Loskutov, A. Yu. [1996] to be published.
24. Loskutov, A. Yu. & Tereshko, V. M. [1993] "Processing information encoded in chaotic sets of dynamical systems," Proc. of the SPIE 1993 Annual Meeting, San Diego, California, July 1993, pp. 263-272.
25. Loskutov, A. Yu., Tereshko, V. M. & Vasiliev, K. A. [1995] "Predicted dynamics for cyclic cascades of chaotic deterministic automata," Int. J. of Neural Systems.
26. Loskutov, A. Yu. & Tereshko, V. M. [1996] in preparation.
27. Lüsher, E. & Hübler, A. [1989] "Resonant stimulations of complex systems," Helv. Phys. Acta 62, 544-551.
28. de Melo, W. & van Strien, M. [1993] One-Dimensional Dynamics (Springer Verlag, Berlin).
29. Misiurewicz, M. [1981] "Absolutely continuous measures for a certain maps of an interval," Publ. Math. I.H.E.S. 53, 17-51.
30. Ognev, A. I. [1981] "Metrical properties of some class mappings of an interval," Mat. Zametki 30, 723-736 (in Russian).
31. Ott, E., Grebogi, C. & Yorke, J. A. [1990] "Controlling chaos," Phys. Rev. Lett. 64, 1196-1199.
32. Pecora, L. M. [1993] ed. Proc. of the SPIE Annual Meeting Chaos in Communications, July 1993, San Diego, California, SPIE Proc., Vol. 238.
33. Rajasekar, S. & Lakshmanan, M. [1992] "Controlling chaos in Boenhoffer-van der Pol oscillator," Int. J. Bifurcation and Chaos 2, 201-204.
34. Romeiras, F. J., Grebogi, C., Ott, E. & Dayawansa, W. R. [1992] "Controlling chaotic dynamical systems," Physica D58, 165-192.
35. Rössler, O. E., Kiwi, M., Hess, B. & Markus, M. [1989] "Modulated nonlinear processes and a novel mechanism to induce chaos," Phys. Rev. A39, 5954-5959.
36. Sataev, E. A. [1992] "Invariant measures for hyperbolic maps with singularities," Uspekhi Mat. Nauk 47, 147-202 (in Russian).
37. Sanju & Varma, V. S. [1993] "Quadratic map modulated by additive periodic forcing," Phys. Rev. E48, 1670-1675.
38. Shinbrot, T. [1993] "Chaos: Unpredictable yet controllable?" Nonlin. Sci. Today 3, 1-8.
39. Shinbrot, T., Grebogi, C., Ott, E. & Yorke, J. A. [1993] "Using small perturbations to control chaos," Nature 363, 411-417.
40. Tereshko, V. M. [1996] in preparation.
41. Vul, E. B., Sinai, Ya. G. & Khanin, K. M. [1984] "Feigenbaum universality and thermodynamical formalism," Uspekhi Mat. Nauk 39, 3-37 (in Russian).