LettersNo Access

RENORMALIZATION GROUP APPROACH TO THE EFFECT OF NOISE AT THE BIRTH OF A STRANGE NONCHAOTIC ATTRACTOR

Kotel'nikov Institute of Radio-Engineering and Electronics of RAS, Saratov Division, Zelenaya 38, Saratov, 410019, Russian Federation

Search for more papers by this author

and

IGOR R. SATAEV

Kotel'nikov Institute of Radio-Engineering and Electronics of RAS, Saratov Division, Zelenaya 38, Saratov, 410019, Russian Federation

Search for more papers by this author

https://doi.org/10.1142/S0219477507003921Cited by:0 (Source: Crossref)

Abstract

Scaling regularities associated with additive noise are examined in a model of the pitch-fork bifurcation map with multiplicative quasiperiodic driving (Grebogi et al., Physica D13, 261) with the golden-mean frequency ratio at the birth of a strange nonchaotic attractor (SNA). This case of the onset of SNA termed as the blowout bifurcation route was discussed in the context of realistic systems governed by non-autonomous differential equations (Yalçynkaya and Lai, Phys. Rev. Lett., 77, 5039). Our method taking into the account of noise is based on renormalization group (RG) analysis of the birth of SNA (Kuznetsov et al., Phys. Rev. E51, 1629) with application of an appropriate generalization of the approach of Crutchfield et al. (Phys. Rev. Lett., 46, 933) and Shraiman et al. (Phys. Rev. Lett., 46, 935) originally developed for the period doubling transition to chaos. A constant γ=7.4246 is evaluated that determines the scaling law regarding the intensity of noise: A decrease of the noise amplitude by this factor allows resolving one more level of the fractal-like structure associated with the characteristic time scale which is increased by a factor of . Numeric results demonstrating evidence of the expected regularities are presented, e.g. portraits of the noisy attractors in different scales.

Keywords:

References

M. J. Feigenbaum, J. Stat. Phys. 21, 669 (1979), DOI: 10.1007/BF01107909. Web of Science, Google Scholar
P. Collet, J.-P. Eckmann and H. Koch, Physica D 3, 457 (1981), DOI: 10.1016/0167-2789(81)90033-6. Web of Science, Google Scholar
A. P. Kuznetsov, S. P. Kuznetsov and I. R. Sataev, Physica D 109, 91 (1997), DOI: 10.1016/S0167-2789(97)00162-0. Web of Science, Google Scholar
B. Hu and J. Rudnik, Phys. Rev. Lett. 48, 1645 (1982), DOI: 10.1103/PhysRevLett.48.1645. Web of Science, Google Scholar
D. Randet al., Phys. Rev. Lett. 49, 132 (1982), DOI: 10.1103/PhysRevLett.49.132. Web of Science, Google Scholar
C. Grebogiet al., Physica D 13, 261 (1984), DOI: 10.1016/0167-2789(84)90282-3. Web of Science, Google Scholar
U. Feudel , S. Kuznetsov and A. Pikovsky , Strange Nonchaotic Attractors. Dynamics between Order and Chaos in Quasiperiodically Forced Systems ( World Scientific , Singapore , 2006 ) . Google Scholar
S. P. Kuznetsov, A. S. Pikovsky and U. Feudel, Phys. Rev. E 51, 1629 (1995), DOI: 10.1103/PhysRevE.51.R1629. Web of Science, Google Scholar
S. P. Kuznetsov, U. Feudel and A. S. Pikovsky, Phys. Rev. E 57, 1585 (1998), DOI: 10.1103/PhysRevE.57.1585. Web of Science, Google Scholar
S. P. Kuznetsovet al., Phys. Rev. E 62, 1995 (2000), DOI: 10.1103/PhysRevE.62.1995. Web of Science, Google Scholar
S. P. Kuznetsov, Phys. Rev. E 65, 066209 (2002), DOI: 10.1103/PhysRevE.65.066209. Web of Science, Google Scholar
J. P. Crutchfield, M. Nauenberg and J. Rudnik, Phys. Rev. Lett. 46, 933 (1981), DOI: 10.1103/PhysRevLett.46.933. Web of Science, Google Scholar
B. Shraiman, C. E. Wayne and P. C. Martin, Phys. Rev. Lett. 46, 935 (1981), DOI: 10.1103/PhysRevLett.46.935. Web of Science, Google Scholar
G. Györgyi and N. Tishby, Phys. Rev. Lett. 58, 527 (1987). Web of Science, Google Scholar
J. V. Kapustinaet al., Phys. Rev. E 64, 066207 (2001), DOI: 10.1103/PhysRevE.64.066207. Web of Science, Google Scholar
A. Hamm and R. Graham, Phys. Rev. A 46, 6323 (1992), DOI: 10.1103/PhysRevA.46.6323. Web of Science, Google Scholar
O. B. Isaeva, S. P. Kuznetsov and A. H. Osbaldestin, Phys. Rev. E 69, 036216 (2004), DOI: 10.1103/PhysRevE.69.036216. Web of Science, Google Scholar
T. Yalçynkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1996). Web of Science, Google Scholar
D. Fiel, J. Phys. A: Math. Gen. 20, 3209 (1987), DOI: 10.1088/0305-4470/20/11/024. Google Scholar