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

For citation:

Kuptsov P. V., Kuznetsov S. P. Wavelet analysis of critical attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 1999, vol. 7, iss. 5, pp. 10-25. DOI: 10.18500/0869-6632-1999-7-5-10-25

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 1999no5p010.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1999-7-5-10-25

Wavelet analysis of critical attractors

Autors: 
Kuptsov Pavel Vladimirovich, 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
Abstract: 

Wavelet analysis is developed for critical attractors of one— and two—dimensional maps аt the threshold of the chaos onset via period—doubling cascades. Numerically calculated wavelet diagrams are presented and discussed relating to critical attractors of different universality classes.

Key words: 
-
Acknowledgments: 
The work was supported by the RFBF, project № 97-02-16414.
Reference: 
  1. Kuznetsov АР, Kuzneisov SР, Sataev IR. A variety оf period-doubling universality classes in multi—parameter analysis оf transition to chaos. Physica D. 1997;109(1-2):91-112. DOI: 10.1016/S0167-2789(97)00162-0.
  2. Kuznetsov AP, Kuznetsov SP, Sataev IR. Three-parameter scaling for оne-dimensional maps. Phys. Lett. А. 1994;189(5):367-373. DOI: 10.1016/0375-9601(94)90018-3.
  3. Kuznetsov AP, Kuznetsov SP, Sataev IR. Codimension and typicality in the context of the problem of describing the transition to chaos through period doubling in dissipative dynamical systems. Regul. Chaotic Dyn. 1997;2(3-4):90-105. (in Russian).
  4. Arneodo А, Argoul F, Васrу E, Elezgaray J, Freysz E, Grasseau G, Muzy J-F, Pouligny В. Wavelet transform of fractals. In: Meyer Y, editor. Wavelets and Applications. Paris: Masson; 1992. P. 286-352.
  5. Datsenko NM, Sonechkin DM. Wavelet analysis of time series and atmospheric dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics. 1993;1(2):9-14. (in Russian).
  6. Arneodo А, Argoul F, Elezgaray J, Grasseau G. Wavelet transform analysis of fractals: application to nonequilibrium phase transitions. In: Turchetti G, editor. Nonlinear Dynamics. Singapore: World Scientific; 1989. P. 130-180.
  7. Feigenbaum MJ. Quantitative universality for а class of nonlinear transformations. J. Stat. Phys. 1978;19:25–52. DOI: 10.1007/BF01020332.
  8. Feigenbaum MJ. The universal metric properties оf nonlinear transformations. J. Stat. Phys. 1979;21:669-706. DOI: 10.1007/BF01107909.
  9. Feigenbaum MJ. Universal behavior in nonlinear systems. Physica D. 1983;7(1-3):16-39. DOI: 10.1016/0167-2789(83)90112-4.
  10. Holschneider M. On the wavelet transform of fractal objects. J. Stat. Phys. 1988;50:963-993. DOI: 10.1007/BF01019149.
  11. Arneodo А, Grasseau G, Holschneider M. Wavelet transform of multifractals. Phys. Rev. Lett. 1988;61(20):2281-2284. DOI: 10.1103/PhysRevLett.61.2281.
  12. Schroeder М. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. N.Y.: Henry Holt & С; 1992. 429 p.
  13. Feder J. Fractals. N.Y.: Springer; 1988. 284 p. DOI: 10.1007/978-1-4899-2124-6.
  14. Halsey TS, Jensen MH, Kadanoff LP, Procaccia I, Shraiman ВI. Fractal measures and their singularities. Phys.Rev. A. 1986;33(2):1141-1151. DOI: 10.1103/PhysRevA.33.1141.
  15. Eckmann JP, Koch H, Wittwer P. Existence of a fixed point of the doubling transformation for area-preserving maps оf the plane. Phys. Rev. A. 1982;26(1):720-722. DOI: 10.1103/PhysRevA.26.720.
  16. Kuznetsov AP, Kuznetsov SP, Sataev LR. Bicritical dynamics of period—doubling systems with unidirectional coupling. Int J. Bifurc. Chaos. 1991;1(4):839-848. DOI: 10.1142/S0218127491000610.
  17. Kuznetsov AP, Kuznetsov SP, Sataev IR. Variety оf types оf critical behavior and multistability in period—doubling systems with unidirectional coupling near the onset оf chaos. Int. J. Bifurc. Chaos. 1993;3(1):139-152. DOI: 10.1142/S0218127493000106.
  18. Kuznetsoy SP, Sataev IR. New types оf critical dynamics for two-dimensional maps. Phys. Lett. A. 1992;162(3):236-242. DOI: 10.1016/0375-9601(92)90440-W.
  19. Kuznetsov SP, Sataev IR. Period doubling for two—dimensional non-invertible maps: Renormalization group analysis and quantitative universality. Physica D. 1997;101(3-4):249-269. DOI: 10.1016/S0167-2789(96)00237-0.
  20. Crutchfield JP, Farmer JD, Huberman BA. Fluctuations and simple chaotic dynamics. Phys. Rep. 1982;92(2):45-82. DOI: 10.1016/0370-1573(82)90089-8.
  21. Coullet Р, Tresser СJ. Iterations d’endomorphismes et groupe de renormalisation. J. Phys. Colloques. 1978;39:25-28. DOI: 10.1051/jphyscol:1978513.
  22. Chang SJ, Fendley PR. Scaling and universal behavior оn bifurcation attractor. Phys. Rev. A. 1986;33(6):4092-4103. DOI: 10.1103/physreva.33.4092.
  23. Grassberger P. On the Hausdorff dimension of fractal attractors. J. Stat. Phys. 1981;26:173-179. DOI: 10.1007/BF01106792.
  24. Grassberger P, Procaccia I. Measuring the strangeness of strange attractors. Physica D. 1983;9(1-2):189-208. DOI: 10.1016/0167-2789(83)90298-1.
  25. Chang SJ, McCown J. Universal exponent and fractal dimensions of Feigenbaum attractors. Phys. Rev. A. 1984;30(2):1149-1151. DOI: 10.1103/PhysRevA.30.1149.
Received: 
03.06.1999
Accepted: 
05.10.1999
Published: 
01.12.1999
  • 203 reads