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

For citation:

Kuznetsov A. P., Savin A. V. About the problem of chaos boundary and typical structures at the parameter plane of non-autonomic discrete maps with period doubling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 4, pp. 25-36. DOI: 10.18500/0869-6632-2000-8-4-25-36

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 2000no4p025.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2000-8-4-25-36

About the problem of chaos boundary and typical structures at the parameter plane of non-autonomic discrete maps with period doubling

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Savin Aleksej Vladimirovich, Saratov State University
Abstract: 

The series of Lyapunov graphs for non—autonomic systems with period doubling under non—periodical influence is shown. With its help the structure of chaos boundary 15 discussed, the new type of branching structures, which are typical for self—similar signals, and the phenomenon of criticality crisis are found. Scaling at Lyapunov graph for map under noise influence are demonstrated.

Key words: 
-
Acknowledgments: 
The work was supported by the RFBR (grant №99-02-17735).
Reference: 
  1. Marcus M. Chaos in maps with continuous and discontinuous maxima. Computers in physics. 1990;4(5):481–493. DOI: 10.1063/1.4822940.
  2. Bastos de Figueireido JC, Malta CP. Lyapunov graph for two-parameter map: Application to the circle map. Int. J. of Bifurcation and Chaos. 1998;8(2):281–293. DOI: 10.1142/S0218127498000176.
  3. Ivankov NYu. The scaling properties of the parameter space of the periodically forced logistic map. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(2—3):118-127. (in Russian).
  4. Kuznetsov AP, Turukina LV. Dynamical systems of different classes as models of the kicked non-linear oscillator. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(2);31-42. (in Russian).
  5. Ikeda K, Daido H, Akimoto О. Optical turbulence: Chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 1980;45(9):709-712. DOI: 10.1103/PhysRevLett.45.709.
  6. Valley R, Delisle C, Chrostowski J. Noise versus chaos in acousto—optic bistability. Phys. Rev. A. 1984;30(1):336-342. DOI: 10.1103/PhysRevA.30.336.
  7. Kuznetsov АР, Kuznetsov SP, Sataev IR. Period doubling system under fractal signal. Bifurcation in the renormalization group equation. Chaos, Solitons & Fractals. 1991;1(4):355-367. DOI: 10.1016/0960-0779(91)90026-6.
  8. Kuznetsov АР, Kuznetsov SP, Sataev IR. Effect of the fractal signal on the Feigenbaum system and bifurcation in the renormalisation equation. Radiophysics and Quantum Electronics. 1991. Т. 34, No 6. С. 661-670. (in Russian).
  9. Kuznetsov АР, Kuznetsov SP, Sataev IR. Fractal signal and dynamics periodic-doubling systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 1995;3(5):64-86. (in Russian).
  10. Grebogy C, Ott E, Pelikan S, Yorke JA. Strange attractors that are not chaotic. Physica D. 1984;13(1-2):261-268.
  11. Anishchenko VS, Vadivasova TE, Astakhov VV. Nonlinear dynamics of chaotic and stochastic systems. Saratov: Saratov University Publishing; 1999. 368 p.
  12. Kuznetsov АР, Kuznetsov SP, Sataev IR. Dynamics of unidirectionally connected systems at the threshold of hyperchaos. Bicritical attractor. Radiophysics and Quantum Electronics. 1992;35(5):398-406. (in Russian).
  13. Kuznetsov АР, Kuznetsov SP, Sataev IR. Bicritical dynamics of period— doubling systems with unidirectional coupling. Int. J. of Bif. and Chaos 1991;1(4):839-848. DOI: 10.1142/S0218n127491000610.
  14. Kuznetsov АР, Kuznetsov SP. Fractal signal generator. Tech. Phys. Lett. 1992;18(24):19-22. (in Russian).
  15. 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.
  16. Mira C, Carcasses J. On the “crossroad аrеа — saddle area” and “crossroad area — spring area” transitions. Int. J. of Bif. and Chaos. 1991;1(3):641-655. DOI: 10.1142/S0218127491000464.
  17. Chang SJ, Pendley PR. Scaling and universal behavior on the bifurcation attractor. Phys. Rev. A. 1986;33(6):4092-4103. DOI: 10.1103/PhysRevA.33.4092.
  18. Mackay RS, Tresser С. Some flesh оn the skeleton - the bifurcation structure of bimodal maps. Physica D. 1987;27(3):412-422. DOI: 10.1016/0167-2789(87)90040-6.
  19. Komuro M, Tokunaga R, Matsumoto T, Chua L, Hotta А. Global bifurcation analysis of the double scroll circuit. Int. J. of Bif. and Chaos. 1991;1:139-182. DOI: 10.1142/S0218127491000105.
  20. Parlitz U. Common dynamical features оf periodically driven strictly dissipative oscillators. Int. J. of Bif. and Chaos. 1993;3(3):703-715. DOI: 10.1142/S0218127493000611.
  21. Scheffczyk C, Parlitz U, Kurz T, Knop W, Lauterborn W. Comparison оf bifurcation structures оf driven dissipative nonlinear oscillators. Phys. Rev. А. 1991;43(12):6495-6502. DOI: 10.1103/PhysRevA.43.6495.
  22. Ringland J, Schell M. Universal geometry in the parameter space of dissipative dynamical systems. Europhys. Lett. 1990;12(7):595-601. DOI: 10.1209/0295-5075/12/7/004.
  23. Gallas JAC. Structure оf thе parameter space of the Henon map. Phys. Rev. Lett. 1993;70(18):2714-2717. DOI: 10.1103/PhysRevLett.70.2714.
  24. Gallas JAC. Structure оf the parameter space оf а ring cavity. Appl. Phys. B. 1995;60:203-213.
  25. Shraiman B, Wayne C, Martin P. Scaling theory for noisy period—doubling systems. Phys. Rev. Lett. 1981;46(14):935-939. DOI: 10.1103/PhysRevLett.46.935.
  26. Schroeder M. Fractals, Chaos, Power Laws. N.Y.: W.H. Freeman and Company, 1990. 429 p.
Received: 
02.02.2000
Accepted: 
20.04.2000
Published: 
23.10.2000
  • 338 reads