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

For citation:

Kuznetsov A. P., Kuznetsov S. P., Sataev I. R. Fractal signal and dynamics of periodic-doubling systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 1995, vol. 3, iss. 5, pp. 64-87. DOI: 10.18500/0869-6632-1995-3-5-64-87

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 1995no5p064.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1995-3-5-64-87

Fractal signal and dynamics of periodic-doubling systems

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Sataev Igor Rustamovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

The model of fractal signal having a phase portrait in a form of two-scale Cantor set provides a possibility to describe many real signals generating by dynamical systems at the onset of chaos and to treat them in a unified way. The points in the parameter plane of the fractal signal are outlined, which correspond to these real types of dynamical behavior. Simple electronic circuit admitting experimental realization is suggested, that generates the fractal signal with tunable parameters. Renormalization group analysis is developed for the case of period-doubling system forced by the fractal signal. It is shown that a bifurcation takes place in the RG equation, and the behavior at the onset of chaos may be described by either Feigenbaum or non-Feigenbaum fixed point solutions. The results of numerical simulations are presented to illustrate the scaling properties of the dynamics forced by the fractal signal.

Key words: 
-
Acknowledgments: 
The work was carried out with financial support from the Russian Foundation for Basic Research (project № 93-02-16169).
Reference: 
  1. Peinke J, Castaing B, Chabaud B, Chilla F, Hebral B, Naert A. On a fractal and an experimental арproach to turbulence. In:  Fractals in the Natural and Applied Sciences A41. 7-10 September 1993, London, UK. P. 295-304.
  2. Schroeder M. Fractals, Chaos, Power Laws. NY.: WH Freeman; 1991. 429 p.
  3. Ebeling W, Nicolis G. Entropy of symbolic sequences: The role of correlations. Europhys. Lett. 1991;14(3):191-196. DOI: 10.1209/0295-5075/14/3/001.
  4. Kuznetsov AP, Kuznetsov SP, Sataev IR. Influence of a fractal signal on a Feigenbaum system and bifurcation in renormalization group equations. Radiophys. Quantum Electron. 1991;34(6):556-563. DOI: 10.1007/BF01039580.
  5. Kuznetsov AP, Kuznetsov SP, Sataev IR. Period doubling system under fractal sygnal: bifurcation in the renormalization group equation. Chaos, Solitons and Fractals. 1991;1(4):355-367. DOI: 10.1016/0960-0779(91)90026-6.
  6. Kuznetsov AP, Kuznetsov SP. Fractal signal generator. Tech. Phys. Lett. 1992;18(24):19-22.
  7. Feigenbaum MJ. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 1978;19(1):25-52. DOI: 10.1007/BF01020332.
  8. Feigenbaum MJ. The universal metric properties of nonlinear transformations. J. Stat. Phys. 1979;21(6):669-706. DOI: 10.1007/BF01107909.
  9. Feigenbaum MJ. Universal Behavior in Nonlinear Systems. Los Alamos Science. 1980;1(1):4-27.
  10. Halsey ТS, Jensen МH, Kadanoff LP, Procaccia I, Shraiman BI. Fractal measures and their singularities. Phys. Rev. A. 1986;33(2):1141-1151. DOI: 10.1103/physreva.33.1141.
  11. Kuznetsov AP, Kuznetsov SP, Sataev IR. Bicritical dynamics of period-doubling system under the fractal sygnal. Int. J. Bif. Chaos. 1991;1(4):839-848. DOI: 10.1142/S0218127491000610.
  12. Kuznetsov SP. Cascade of period doublings in complex cubic mapping: renormal group analysis and quantitative universality. Izvestiya VUZ. Applied Nonlinear Dynamics. 1996;4(4):3-12.
  13. Huberman B, Zisook A. Power spectra of strange attractors. Phys. Rev. Lett. 1981;46(10):626-628. DOI: 10.1103/PhysRevLett.46.626.
  14. Nauenberg M, Rudnik J. Universality and the power spectrum at the onset of chaos. Phys. Rev. B. 1981;24(1):493-495. DOI: 10.1103/PhysRevB.24.493.
  15. Hu B, Mao JM. Period Doubling: universality and critical point order. Phys. Rev. A. 1982;25(6):3259 –3261. DOI: 10.1103/PhysRevA.25.3259.
  16. Van der Weele JP, Capel HW, Kluiving К. On the scaling factors alfa(z) and delta(z). Phys. Lett. A. 1986;119(1):15-20. DOI: 10.1016/0375-9601(86)90636-5.
  17. Neimark YuI, Landa PS. Stochastic and Chaotic Osciliations. Dordrecht: Springer; 1992. 500 p. DOI: 10.1007/978-94-011-2596-3.
  18. Chang SJ, Wortis M, Wright J. Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior. Phys. Rev. A. 1981;24(5):2669-2684. DOI: 10.1103/PhysRevA.24.2669.
  19. Kuznetsov AP, Kuznetsov SP. Tree of super-stable orbits and scaling in three-parameter mappings. Tech. Phys. Lett. 1992;18(21):34-36.
  20. Kuznetsov AP, Kuznetsov SP, Sataev IR. Three- parameter scaling for one-dimensional maps. Phys. Lett. A. 1994;189(5):367-373. DOI: 10.1016/0375-9601(94)90018-3.
  21. Eckmann JP, Koch H, Wittwer P. Existence of a fixed point of the doubling transformation for area-preserving map of the plane. Phys. Rev. A. 1982;26(1):720-722. DOI: 10.1103/PhysRevA.26.720.
  22. Kuznetsov SP, Sataev IR. New types of critical dynamics for two-dimensional maps. Phys. Lett. A. 1992;162(3):236-242. DOI: 10.1016/0375-9601(92)90440-W.
  23. Bezruchko BP, Gulyaev YuV, Kuznetsov SP, Seleznev ЕP. A new type of critical behaviour of related systems during the transition to chaos. Sov. Phys. Doklady. 1986;287(3):619-622.
  24. Kuznetrsov SP. A renormalization-group analysis for two unidirectionally coupled Feigenbaum systems at the hyperchaos threshold. Radiophys. Quantum Electron. 1990;33(7):578-581. DOI: 10.1007/BF01048488.
  25. Kuznetsov AP, Kuznetsov SP, Sataev IR. Variety of types of critical behavior and multistability in period doubling systems with uni-directional coupling near the onset of chaos. Int. J. Bif. Chaos. 1993;3(1):139-152. DOI: 10.1142/S0218127493000106.
  26. Kuznetsov AP, Kuznetsov SP, Sataev IR. Multi-parameter transition to chaos and fractal nature of critical attractors. In: Novak M, editor. Fractals in the Natural and Applied Sciences. Amsterdam: Elsevier; 1994. Р. 229-239.
  27. Erastova EN, Kuznetsov AP, Kuznetsov SP, Sataev IR. Two-parameter criticality in nonlinear systems near the onset of chaos. In: Proc. Int. Seminar «Nonlinear Circuits and Systems». Vol.2. М.; 1992. P. 131.
  28. MacKay RS, Van Zeijts JB. Period doubling for bimodal maps: A Horseshoe for a Renormalization Operator. Nonlinearity. 1988;1:253-277.
  29. Kuznetsov AP, Kuznetsov SP, Sataev IR, Chua L.O. Two-parameter study of transition to chaos in Chua’s circuit: Renormalization Group, Universality and Scaling. Int. J. Bif. Chaos. 1993;3(4):943-962. DOI: 10.1142/S0218127493000799.
  30. Kuznetsov SP. Critical quasi-attractor: an infinite self-similar set of stable cycles arising from a two-parameter analysis of the transition to chaos. Tech. Phys. Lett. 1994;20(10):11-15.
  31. Goldberg АI, Sinai YaG, Khanin KM. Universal properties for sequences of bifurcations of period three. Rus. Math. Surveys. 1983;38(1):187-188. DOI: 10.1070/RM1983v038n01ABEH003398.
  32. Cvitanovic P, Myrheim J. Universality for period n-tupling in complex mappings. Phys. Lett. A. 1983;94(8):329-333. DOI: 10.1016/0375-9601(83)90121-4.
  33. Cvitanovic P, Myrheim J. Complex universality. Commun. Math. Phys. 1989;121:225-254. DOI: 10.1007/BF01217804.
  34. Iooss G, Joseph DD. Elementary Stability and Bifurcation Theory. N.Y.: Springer; 1990. 324 p. DOI: 10.1007/978-1-4612-0997-3.
  35. Balescu R. Equilibrium and Non-Equilibrium Statistical Mechanics. N.Y.: Wiley; 1975. 742 p.
Received: 
13.01.1995
Accepted: 
12.08.1995
Published: 
21.10.1996
  • 257 reads