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

For citation:

Kuznetsov A. P., Turukina L. V. Dynamical systems of different classes as models of the kicked non-linear oscillator. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 2, pp. 31-42. DOI: 10.18500/0869-6632-2000-8-2-31-42

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2000no2p031.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2000-8-2-31-42

Dynamical systems of different classes as models of the kicked non-linear oscillator

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Turukina L. V., Saratov State University
Abstract: 

Correspondence between the models as dynamical systems of different classes is discussed by the example of nonlinear dissipative kicked oscillator. 1D map is investigated in details: Feigenbaum’s period doubling is studied and the possibility of non—Feigenbaum’s period doubling is shown, corresponding illustration in the form of bifurcation diagrams and set of iteration diagrams are given, tricritical points (terminal points of the Feigenbaum’s critical curves) are found in parameter space. The correlation with the properties of 2D map, the phenomenon of tricritically dynamics was demonstrated to take place only on definite areas of the parameter space.

Key words: 
-
Acknowledgments: 
The authors would like to thank S.P. Kuznetsov for fruitful discussion of the article. The work was supported by the RFBR (grant № 97-02-16414), Federal Target Program "Integration" (grant № 696.3) and Ministry of Education of the Russian Federation (grant № 97-0-8.3-88).
Reference: 
  1. Kuznetsov AP, Kuznetsov SP, Sataev IR. А variety of period—doubling universality classes in multi—parameter analysis оf transition to chaos. Physica D. 1997;109(1):91-112. DOI:10.1016/S0167-2789(97)00162-0.
  2. Kuznetsov AP, Kuznetsov SP, Sataev IR. Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems. Regul. Chaotic Dyn. 1997;2(3-4):90-105. DOI: 10.1070/RD1997v002n04ABEH000050
  3. Kuznetsov S.P. Tricriticality in two—dimensional maps. Phys. Lett. A. 1992;169(6):438-444. DOI: 10.1016/0375-9601(92)90824-6.
  4. Berge P, Pomeau Y, Vidal C. Order within Chaos. Weinheim: Wiley;1987. 329 p.
  5. Schuster G. Deterministic Chaos: An Introduction. Weinheim: Wiley;1998. 270 p.
  6. Moon FC. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Weinheim: Wiley;1987. 309 p.
  7. Kuznetsov SP, Erastova EN. Feigenbaum's theory. In: Vol. 2 of Lectures on Microwave Electronics and Radiophysics. Saratov: Saratov University Publishing; 1983. P. 3-22. (in Russian).
  8. Heagy JF. A physical interpretation of the Henon mар. Physica. 1992;57(3-4):436-446.
  9. Bezruchko BP, Prokhorov MD, Seleznev EP. Model of a dissipative nonlinear oscillator in the form of a one-dimensional mapping with three parameters. Tech. Phys. Lett. 1994;20(12):78-82. (in Russian).
  10. Ikeda K, Daido H, Akimoto О. Optical turbulence: chaotic behavior of transmitted light from а ring cavity. Phys. Pev. Lett. 1980;45(9):709-712. DOI:10.1103/PhysRevLett.45.709.
  11. Carcasses J, Mira C, Bosch M, Simo C, Tatjer JC. “Crossroad area — spring area” transition. I: Parameter plane representation. Int. J. Bifurc. & Chaos. 1991;1(1):183-196. DOI:10.1142/S0218127491000117.
  12. Carr Y, Eilbech YC. One-dimensional approximations for а quadratic Ikeda map. Phys. Lett. A. 1984;104(2):59-62. DOI: 10.1016/0375-9601(84)90962-9.
  13. Chang SJ, Wortis M, Wright JA. Iterative properties оf а one-dimensional quartic map. Critical lines and tricritical behaviour. Phys. Rev. A. 1981;24(5):2669-2684.
  14. Mosekilde E. Topics in Nonlinear dynamic. Singapore: World Scientific Publishing; 1996. 380 p. DOI: 10.1142/3194.
  15. Parlitz U. Common dynamical features оf periodically driven strictly dissipative oscillators. Int. J.Bifurc. & Chaos. 1993;3(3):703-715. DOI:10.1142/S021812749300060X.
  16. Parlitz U, Scheffczyk C, Kurz T, Lauterborn W. Two-dimensional maps modelling periodically driven strictly dissipative oscillator. In: Seydel R, Schneider FW, Küpper T, Troger H, editors. Bifurcation and Chaos: Analysis, Algorithms, Applications. Vol. 97 of  International Series of Numerical Mathematics. Birkhauser;1991. P. 283-287.
  17. Vallee R, Delisle C, Chrostowski J. Noise versus chaos in аn acousto—optic bistability. Phys. Rev. A. 1984;30(1):336-342. DOI: 10.1103/PhysRevA.30.336.
  18. Kuznetsov AP, Kuznetsov SP, Sataev IR, Chua LO. Multi—parameter criticality in Chua’s circuit at period—doubling transition to chaos. Int. J.Bifurc. & Chaos. 1996;6(1):119-148. DOI: 10.1142/S0218127496001880.
Received: 
25.09.1999
Accepted: 
24.01.2000
Published: 
25.05.2000
  • 404 reads