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

For citation:

Doroshenko V. M. Strange nonchaotic attractor of Hunt and Ott type in a system with ring geometry. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, iss. 1, pp. 16-30. DOI: 10.18500/0869-6632-2016-24-1-16-30

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 2016no1p016.pdf
(downloads: 144)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2016-24-1-16-30

Strange nonchaotic attractor of Hunt and Ott type in a system with ring geometry

Autors: 
Doroshenko Valentina Mihajlovna, Saratov State University
Abstract: 

The physical realizable system of ring structure, with a fixed irrational ratio of basic frequencies of external driving (the golden mean) manifests a strange nonchaotic attractor (SNA), similar to the attractor in the abstract map on a torus proposed and analyzed earlier by Hunt and Ott as an example of robust SNA. Simulation of the dynamics is provided basing on the numerical integration of the corresponding non-autonomous system of differential equations with quasi-periodic coefficients. It has been demonstrated that in terms of appropriately chosen phase variables the dynamics on the characteristic period is consistent with the topology of the mapping of Hunt and Ott. It has been shown that the birth of SNA corresponds to the criterion of Pikovsky and Feudel. Numerical calculations show that the Fourier spectra in sustained mode is of intermediate class between the continuous and discrete spectra (the singular continuous spectrum).

Key words: 
Strange nonchaotic attractor
Hunt-Ott map
robustness
fractal structure
singular continuous spectrum
Reference: 
  1. Grebogi C., Ott E., Pelikan S., Yorke J.A. Strange attractors that are not chaotic // Physica D. 1984. Vol. 13, No 1–2. P. 261.
  2. Kuznetsov S.P., Pikovsky A.S., Feudel U. Strange nonchaotic attractor // Nonlinear waves – 2004 / Eds A.V. Gaponov-Grekhov and V.I. Nekorkin. Nizhny Novgorod: IAP RAS, 2005. Vol. 25. P. 484 (in Russian).
  3. Bondeson A., Ott E., Antonsen T.M. Quasiperiodically forced damped pendula and Schrodinger equations with quasiperiodic potentials: implications of their equivalence  ? // Phys. Rev. Lett. 1985. Vol. 55. No 20. P. 2103.
  4. Ding M., Grebogi C., Ott E. Dimensions of strange nonchaotic attractors // Phys. Lett. A. 1989. Vol. 137, No 4–5. P. 167.
  5. Ding M., Grebogi C., Ott E. Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic // Phys. Rev. A. 1989. Vol. 39, No 5. P. 2593.
  6. Pikovsky A.S., Feudel U. Correlations and spectra of strange nonchaotic attractors // Phys. A: Math. Gen. 1994. Vol. 27. P. 5209.
  7. Pikovsky A.S., Feudel U. Characterizing strange nonchaotic attractors // Chaos. 1995. Vol. 5, No 1. P. 253.
  8. Pikovsky A.S., Zaks M.A., Feudel U., Kurths J. Singular continuous spectra in dissipative dynamics // Phys. Rev. E. 1995. Vol. 52, No 1. P. 285.
  9. Feudel U., Pikovsky A.S., Kurths J. Strange non-chaotic attractor in a quasiperiodically forced circle map // Physica D. 1995. Vol. 88. P. 176.
  10. Pokorny P., Schreiber I., Marek M. On the route to strangeness without chaos in the quasiperiodically forced van der Pol oscillator // Chaos, Solitons and Fractals. 1996. Vol. 7, No 3. P. 409.
  11. Kaneko K., Nishikawa T. Fractalization of a torus as a strange nonchaotic attractor // Phys. Rev. E. 1996. Vol. 54, No 6. P. 6114.
  12. Glendinning P. Intermittency and strange nonchaotic attractors in quasi-periodically forced circle maps // Phys. Lett. A. 1998. Vol. 244. P.545.
  13. Osinga H., Wiersig J., Glendinning P., Feudel U. Multistability and nonsmooth bifurcations in the quasiperiodically forced circle map // Int. J. of Bifurcation and Chaos. 2001. Vol. 11, No 12. P. 3085.
  14. Prasad A., Negi S.S., Ramaswamy R. Strange nonchaotic attractors // Int. J. of Bifurcation and Chaos. 2001. Vol. 11. P. 291.
  15. Hunt B.R., Ott E. Fractal properties of robust strange nonchaotic attractors // Phys. Rev. Lett. 2001. Vol. 87, No 25. P. 254101.
  16. Kim J-W., Kim S.-Y., Hunt B., Ott E. Fractal properties of robust strange nonchaotic attractors in maps of two or more dimensions // Phys. Rev. E. 2003. Vol. 67. P. 036211.
  17. Kim S.-Y., Lim W., Ott E. Mechanism for the intermittent route to strange nonchaotic attractors // Phys. Rev. E. 2003. Vol. 67. P. 056203.
  18. Ditto W.L., Spano M.L., Savage H.T. et al. Experimental observation of a strange nonchaotic attractor // Phys. Rev. Lett. 1990. Vol. 65, No 5. P. 533.
  19. Vohra S.T., Bucholtz F., Koo K.P., Dagenais D.M. Experimental observation of period-doubling suppression in the strain dynamics of a magnetostrictive ribbon // Phys. Rev. Lett. 1991. Vol. 66, No 2. P. 212.
  20. Zhou T., Moss F., Bulsara A. Observation of a strange nonchaotic attractor in a multistable potential // Phys. Rev. A. 1992. Vol. 45, No 8. P. 5394.
  21. Zeyer K.-P., Miinster A.F., Schneider F.W. Quasiperiodic forcing of a chemical reaction: experiments and calculations // J. Phys. Chem. 1995. Vol. 99. P. 13173.
  22. Ding W.X., Deutsch H., Dinklage A., Wilke C. Observation of a strange nonchaotic attractor in a neon glow discharge // Phys. Rev. E. 1997. Vol. 55, No 3. P. 3769.
  23. Yang T., Bilimgut K. Experimental results of strange nonchaotic phenomenon in a second-order quasi-periodically forced electronic circuit // Phys. Lett. A. 1997. Vol. 236. P. 494.
  24. Yu Y.H., Kim D.C., Ryu J.Y., Hong S.R. Experimental study on the blowout bifurcation route to strange nonchaotic attractor // J. of the Korean Phys. Society. 1999. Vol. 34, No 2. P. 130.
  25. Bezruchko B.P., Kuznetsov S.P., Seleznev Y.P. Experimental observation of dynamics near the torus-doubling terminal critical point // Phys. Rev. E. 2000. Vol. 62, No 6. P. 7828.
  26. Sanchez D., Platero G., Bonilla L.L. Quasiperiodic current and strange attractors in ac-driven superlattices // Phys. Rev. B. 2001. Vol. 63. P. 201 306.
  27. Vaszlenko A., Feely O. Dynamics of phase-locked loop with fm input and low modulating frequency // Int. J. of Bifurcation and Chaos. 2002. Vol. 12, No 7. P. 1633.
  28. Jalnine A.Yu., Kuznetsov S.P. On the realization of the Hunt–Ott strange nonchaotic attractor in a physical system // Zhurnal Tekhnicheskooe Fiziki, 2007, Vol. 77, No 4. P. 10 (in Russian).
  29. Kuznetsov S.P. Dynamic chaos. Moscow: Fizmatlit., 2001. 296 s. (in Russian).
Received: 
26.02.2016
Accepted: 
28.02.2016
Published: 
28.02.2016
Short text (in English):
PDF icon a2016no1p016.pdf
(downloads: 120)
  • 1727 reads