Abstract
We study the features of the stochastic dynamics of a Hamiltonian system with the potential x 2n subjected to an external monochromatic perturbation. Three regimes of stochastic diffusion, which differ in the value of the amplitude of the external perturbation, are detected. We demonstrate the possibility of chaotic regimes manifesting themselves in pendulum vibrations of the well of a water-moderated and-cooled nuclear power reactor as an application of the model being investigated. Finally, we propose a method of simple proportional control, which makes it possible to control the chaotic vibrations of the anharmonic oscillator.
Similar content being viewed by others
References
G. M. Zaslavsky, Chaos in Dynamical Systems, Harwood Academic, New York (1985).
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, Springer, Berlin (1992).
M. Tabor, Chaos and Integrability in Nonlinear Dynamics, Wiley, New York (1989).
L. E. Reichl, The Transition to Chaos, Springer, Berlin (1992).
N. B. Delone, V. P. Krainov, and D. L. Shepelyanskii, Usp. Fiz. Nauk 140, 355 (1983) [Sov. Phys. Usp. 26, 551 (1983)].
G. Casati et al., Phys. Rep. 154, 77 (1987).
R. V. Jensen, Phys. Rev. A 30, 386 (1984).
M. Guttzviller, Chaos in Classical and Quantum Mechanics, Springer, Berlin (1991).
P. V. Elyutin, Usp. Fiz. Nauk 155, 397 (1988) [Sov. Phys. Usp. 31, 597 (1988)].
Yu. L. Bolotin, V. Yu. Gonchar, and M. Ya. Granovskii, JETP Lett. 59, 651 (1994).
Yu. L. Bolotin, V. Yu. Gonchar, and M. Ya. Granovsky, Physica D 86, 500 (1995).
H. Breuer and M. Holthaus, Ann. Phys. (N.Y.) 211, 249 (1991).
L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Pergamon Press, Oxford (1976).
B. V. Chirikov, At. Energ. 6, 630 (1959).
T. H. Dupree, Phys. Fluids 9, 1773 (1966).
I. F. Akerhielm, R. Espefalt, and J. Lorenzen, Prog. Nucl. Energy 9, 453 (1982).
Proc. Symposium on Nuclear Reactor Surveillance and Diagnostics (SMORN VII), 19–23 June 1995, Avignon, France (1995).
V. V. Bulavin, V. I. Pavelko, and D. F. Gutsev, At. Energy 79, 343 (1995).
E. Alstadt, M. Scheffler, and F.-P. Weiss, Prog. Nucl. Energy 29, 129 (1995).
J. A. Thie, Nucl. Technol. 45, 5 (1979).
J. Vavrin, in Proc. Symposium on Nuclear Reactor Surveillance and Diagnostics (SMORN VI), Gatlinburg, Tennessee, USA (1991).
O. Thomson, N. S. Garis, and I. Pazsit, submitted to Nucl. Technol.
J. Runkel, D. Stegemann, J. Fiedler et al., in Proc. Symposium on Nuclear Reactor Surveillance and Diagnostics (SMORN VII), Avignon, France (1995).
G. Por, K. A. Kantor, and L. A. Sokolov, in Proc. Symposium on Nuclear Reactor Surveillance and Diagnostics (SMORN VII), Avignon, France (1995).
E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990).
K. Pyragas, Phys. Lett. A 170, 421 (1992).
J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
T. Shinbrot, Adv. Phys. 44, 71 (1995).
Y.-C. Lai, M. Ding, and C. Grebogi, Phys. Rev. E 47, 86 (1993).
C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 57, 1294 (1986).
C. F. F. Karney, Physica D 8, 360 (1983).
Author information
Authors and Affiliations
Additional information
Zh. Éksp. Teor. Fiz. 115, 361–377 (January 1999)
Rights and permissions
About this article
Cite this article
Bolotin, Y.L., Gonchar, V.Y., Granovskii, M.Y. et al. Dynamics of an anharmonic oscillator with a periodic perturbation. J. Exp. Theor. Phys. 88, 196–205 (1999). https://doi.org/10.1134/1.558782
Received:
Issue Date:
DOI: https://doi.org/10.1134/1.558782