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

For citation:

Bezruchko B. P., Kuznetsov S. P., Pikovsky A. S., Seleznev E. P., Feudel U. On dynamics of nonlinear systems under external quasi-periodic force near the terminal point of the torus-doubling bifurcation curve. Izvestiya VUZ. Applied Nonlinear Dynamics, 1997, vol. 5, iss. 6, pp. 3-20. DOI: 10.18500/0869-6632-1997-5-6-3-20

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 1997no6p003.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1997-5-6-3-20

On dynamics of nonlinear systems under external quasi-periodic force near the terminal point of the torus-doubling bifurcation curve

Autors: 
Bezruchko Boris Petrovich, Saratov State University
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Pikovsky Arkady Samuilovich, Potsdam University
Seleznev Evgeny Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Feudel Ulrica, Humboldt University of Berlin
Abstract: 

A logistic тар under quasi~periodic force is investigated for the case of frequency given by the golden mean. It is found that the end point of torus—doubling bifurcation curve on the plane of control parameter and force amplitude is a critical point where regions of torus, doubled torus, strange nonchaotic attractor and chaos meet together. Attractor at this point is a fractal object — «critical torus». Using the renormalization group approach we reveal scaling properties both for the critical attractor and for the parameter plane topography near the critical point. We present also experimental results for electronic oscillator under quasi—periodic excitation and demonstrate qualitative correspondence with the theory.

Key words: 
-
Acknowledgments: 
The work was carried out with financial support from the Russian Foundation for Basic Research (projects 97-02-16414 and 96-02-16755).
Reference: 
  1. Landau LD. To the problem of turbulence. Soviet Phys. Doklady. 1944;44:339.
  2. Ruelle D, Takens F. On the nature of turbulence. Commun.Math. Phys. 1971;20:167-192. DOI: 10.1007/BF01646553.
  3. Grebogi C, Оtt E, Pelikan S, Yorke JA. Strange attractors that are not chaotic. Physica D. 1984;13(1-2):261-268. DOI: 10.1016/0167-2789(84)90282-3.
  4. Bondeson А, Ott E, Antonsen TM. Quasiperiodically forced damped pendula and Schrödinger equations with quasiperiodic potentials: Implications of their equivalence. Phys. Rev. Lett. 1985;55(20):2103-2106. DOI: 10.1103/PhysRevLett.55.2103.
  5. Romeiras FJ, Bondeson А, Ott E, Antonsen TM, Grebogi С. Quasiperiodically forced dynamical systems with strange nonchaotic attractor. Physica D. 1987;26(1-3):277-294. DOI: 10.1016/0167-2789(87)90229-6.
  6. Romeiras FJ, Оtt Е. Strange nonchaotic attractors оf the damped pendulum with quasiperiodic forcing. Phys. Rev. A. 1987;З5(10):4404-4413. DOI: 10.1103/physreva.35.4404.
  7. Ding M, Grebogi C, Оtt Е. Evolution of attractors in quasiperiodically forced systems. Phys. Rev. A. 1989;39(5):2593-2598. DOI: 10.1103/physreva.39.2593.
  8. Ding M, Grebogi C, Ott Е. Dimensions оf strange nonchaotic attractors. Phys. Lett. A. 1989;137(4-5):167-172. DOI: 10.1016/0375-9601(89)90204-1.
  9. Kapitaniak T, Ponce E, Wojewoda J. Route to chaos via strange non—chaotic attractors. J. Phys. А: Math. Gen. 1990;23(8):L383-L387. DOI: 10.1088/0305-4470/23/8/006.
  10. Heagy JF, Hammel SM. The birth of strange nonchaotic attractor. Physica D. 1994;70(1-2):140-153. DOI: 10.1016/0167-2789(94)90061-2.
  11. Pikovsky А, Feudel U. Characterizing strange nonchaotic attractors. CHAOS. 1995;5(1):253-260. DOI: 10.1063/1.166074.
  12. Feudel U, Kurths J, Pikovsky А. Strange non-chaotic attractors in а quasiperiodically forced circle mар. Physica D. 1995;88(3-4):176-186. DOI: 10.1016/0167-2789(95)00205-I.
  13. Kuznetsov SP, Pikovsky АS, Feudel U. Birth оf а strange nonchaotic attractor: Renormalization group analysis. Phys. Rev. Е. 1995;51(3):R1629-R1632. DOI: 10.1103/physreve.51.r1629.
  14. Pikovsky А, Feudel U. Correlations and spectra of strange nonchaotic аttractors. J. Phys. A: Math. Gen. 1994;27(15):5209. DOI: 10.1088/0305-4470/27/15/020.
  15. Ding M, Scott—Kelso J. Phase-resetting map and the dynamics оf quasiperiodically forced biological oscillators. Int. J. Bif. Chaos. 1994;4(3):553-567. DOI: 10.1142/S0218127494000393.
  16. Lai Y—C. Transition from strange nonchaotic to strange chaotic attractors. Phys. Rev. Е. 1996;53(1):57–65. DOI: 10.1103/PhysRevE.53.57.
  17. Nishikawa T, Kaneko K. Fractalization of torus revisited as а strange non-chaotic attractor. Phys. Rev. Е. 1996;54(6):6114-6124. DOI: 10.1103/PhysRevE.54.6114.
  18. Anishchenko VS, Vadivasova TE, Sosnovtseva ОN. Mechanisms of birth of a strange non-chaotic attractor in the reflection of the ring by quasi-periodic influence. Izvestiya VUZ. Applied Nonlinear Dynamics. 1995;3(3):34-43. (In Russian).
  19. Anishchenko VS, Vadivasova TE, Sosnovtseva ОN. Mechanisms of ergodic torus destruction and арреarance of strange nonchaotic attractor. Phys.Rev. E. 1996;53(5):4451-4456. DOI: 10.1103/physreve.53.4451.
  20. Ditto WL. Experimental observation of a strange nonchaotic attractor. Phys. Rev. Lett. 1990;65(5):533-536. DOI: 10.1103/PhysRevLett.65.533.
  21. Zhou T, Moss F, Bulsara A. Observation of a strange nonchaotic attractor in a multistable potential. Phys. Rev. A. 1992;45(8):5394-5400. DOI: 10.1103/PhysRevA.45.5394.
  22. Feudel U, Pikovsky А, Politi А. Renormalization of correlations and spectra of a strange non-chaotic attractor. J. Phys. А: Math. Gen. 1996;29(17):5297-5311. DOI: 10.1088/0305-4470/29/17/008.
  23. Ketoja JA, Satija II. Private message.
  24. Feudel U, Pikovsky AS, Zaks MA. Correlation properties of quasiperiodically forced two—level system. Phys. Rev. Е. 1995;51(3):1762-1769. DOI: 10.1103/PhysRevE.51.1762.
  25. Keller G. A note on strange nonchaotic attractors. Fund. Math. 1996;151(2):139-148.
  26. Stark J. Private message.
  27. Kaneko K. Doubling of torus. Prog. Theor. Phys. 1983;69(6):1806-1810. DOI: 10.1143/PTP.69.1806.
  28. Kuznetsov SP. Effect of a periodic external perturbation on a system which exhibits an order-chaos transition through period-doubling-bifurcations metal-insulator-semiconductor. JETP Lett. 1984;39(3):133-136.
  29. Arneodo А. Scaling for а periodic forcing of a period—doubling system. Phys. Rev. Lett. 1985;54(1):86-86. DOI: 10.1103/PhysRevLett.54.86.2.
  30. Kuznetsov SP, Pikovsky AS. Renormalization group for the response function and spectrum of the period—doubling system. Phys. Lett. А. 1989;140(4):166-172. DOI: 10.1016/0375-9601(89)90887-6.
  31. Kaneko K. Collapse of Tori and Genesis of Chaos in Dissipative Systems. Singapore: World Scientific; 1986. 276 p. DOI: 10.1142/0175.
  32. Anishchenko VS, Letchford ТЕ, Safonova MA. Stochasticity and the disruption of quaiperiodic motion due to doubling in a system of coupled generators. Radiophys. Quantum Electron. 1984;27(5):381-390. DOI: 10.1007/BF01044783.
  33. Broer H, Huitema GB, Takens F. Unfoldings and bifurcations оf quasi—periodic tori. Mem. Amer. Math. Soc. 1990;421:1-82.
  34. Chastell PR, Glendinning PA, Stark J. Locating bifurcations in quasiperiodically forced systems. Phys. Lett. А. 1995;200(1):17-26. DOI: 10.1016/0375-9601(95)00107-E.
  35. Feigenbaum MJ, Kadanoff LP, Shenker SJ. Quasiperiodicity in dissipative systems: а renormalization group analysis. Physica D. 1982;5(2-3):370-386. DOI: 10.1016/0167-2789(82)90030-6.
  36. Rand Р, Ostlund S, Sethna J, Siggia ED. Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. Physica D. 1983;8(3):303-342. DOI: 10.1016/0167-2789(83)90229-4.
  37. Linsay PS. Period doubling and chaotic behavior in а driven anharmonic oscillator. Phys. Rev. Lett. 1981;47(19):1349-1352. DOI: 10.1103/PhysRevLett.47.1349.
  38. Testa J, Perez J, Jeffries С. Evidence for universal chaotic behavior оf а drivеn nonlinear oscillator. Phys. Rev. Lett. 1982;48(11):714-717. DOI: 10.1103/PhysRevLett.48.714.
  39. Astakhov VV, Bezruchko BP, Seleznev EP. Study of the dynamics of a nonlinear oscillatory circuit under harmonic influence. Soviet J. Comm. Tech. Electron. 1987;32(12):2558-2566. (In Russian).
  40. Bezruchko BP. Features of excitation of subharmonic and chaotic oscillations in the circuit with a diode. Soviet J. Comm. Tech. Electron. 1991;36(11):2171-2174. (In Russian).
Received: 
24.11.1997
Accepted: 
12.02.1998
Published: 
18.03.1998
  • 330 reads