CONTROL PARAMETER SPACE OF A NONLINEAR OSCILLATOR UNDER QUASIPERIODIC DRIVING
Cite this article as:
Seleznev Е. P., Zakharevich А. М. CONTROL PARAMETER SPACE OF A NONLINEAR OSCILLATOR UNDER QUASIPERIODIC DRIVING. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 6, pp. 17-35. DOI: https://doi.org/10.18500/0869-6632-2009-17-6-17-35
Dynamics and space of сontrol parameters for a nonlinear oscillator under quasiperiodic driving are investigated experimentally by using a nonlinear circuit with pn junction diode and numerically by using maps and differential equations. The dynamics of the systems under quasiperiodic driving is invariant due to initial driving phases, as a result the plane of the driving amplitudes is symmetrical. The basic element of the control parameter space is the set of torus doubling terminal points, which are the starting and end points of the torus doubling lines, transition to strange nonchaotic and chaotic attractors.
1. Grebodgi C., Ott E., Pelican S., Yorke J. Strange attractor that are not chaotic // Physica. 1984. Vol. D13. P. 261.
2. Romeiras F.J., and Ott E. Strange nonchaotic attractor of the damped pendulum with quasiperiodic forcing // Phys. Rev. 1987. Vol. A35. P. 4404.
3. Ding M., Grebogi C., and Ott E. Evolution of attractors in quasiperiodically forced system // Phys. Rev. 1989. Vol. A39. P. 2593.
4. Ditto W.L. et al. Experimantal observation of strange nonchaotic attractors // Phys. Rev. Lett. 1990. Vol. 65. P. 533.
5. Ding M., Grebogi C., Ott E. Dimensions of strange nonchaotic attractors // Phys. Lett. A. 1989. Vol. 137. P. 167.
6. Zhou T., Moss F. and Bulsara A. Observation of strange nonchaotic attractors in a multistable potential // Phys. Rev. 1992. Vol. A45. P. 5394.
7. Feudel U., Kurths J. and Pikovsky A. Strange nonchaotic attractors in quasiperiodically forced circle map // Physica. 1995. Vol. D88. P. 176.
8. Pikovsky A. and Feudel U. Characterizing strange nonchaotic attractors // CHAOS. 1995. Vol. 5. P. 253.
9. Pikovsky A., Feudel U. Correlations and spectra of strange nonchaotic attractors// J. Phys. A: Math., Gen. 1994. Vol. 27. P. 5209.
10. Ding M., Scott-Kelso J. Phase-resetting map and the dynamics of quasiperiodically forced biological oscillators // Int. J. Bif. Chaos. 1994. Vol. 4. P. 553.
11. Feudel U., Pikovsky A.S., Zaks M.A. Correlation properties of quasiperiodically forced two-level system // Phys. Rev. E. 1995. Vol. 51. P. 1762.
12. Kuznetsov S., Pikovsky A., Feudel U. Birth of a strange nonchaotic attractor: Renor-malization group analysis // Phys. Rev. E. 1995. Vol. 51. P. R1629.
13. Анищенко В.С., Вадивасова Т.Е., Сосоновцева О.Н. Механизмы рождения странного нехаотического аттрактора в отображении кольца с квазипериодическим воздействием. // Изв. вузов. Прикладная нелинейная динамика. 1995. Т. 3, No 3. С. 34.
14. Y.-C. Lai. Transition from strange nonchaotic attractor to strange chaotic attractor // Phys. Rev. 1996. Vol. E53. P. 57.
15. Nishikawa T. and Kaneko K. Fractalization of torus revisited as a strange nonchaotic attractor // Phys. Rev. 1996. Vol. E54. P. 6114.
16. Anishchenko V.S., Vadivasova T.E., Sosnovtseva O.N. Mechanisms of ergodic torus destruction and apperence of strange nonchaotic attractor // Phys. Rev. 1996. Vol. E53. P. 4451.
17. Feudel U., Pikovsky A., Politi A. Renormalization of correlations and spectra of a strange nonchaotic attractor // J. Phys. A. 1996. Vol. 29. P. 5297.
18. Sosnovtseva O., Feudel U., Kurths J., Pikovsky A. Multiband strange nonchaotic attractors in quasiperiodically forced systems// Phys. Lett. A. 1996. Vol. 218. P. 255.
19. Keller G. A note on strange nonchaotic attractors // Fundamenta Mathematicae. 1996. Vol. 151. P. 139.
20. Prasad A., Mehra V., Ramaswamy R. Intermittency route to strange nonchaotic attractors // Phys. Rev. Lett. 1997. Vol. 79, No 21. P. 4127.
21. Witt A., Feudel U., Pikovsky A. Birth of strange nonchaotic attractors due to interior crisis // Physica. 1997. Vol. 109D. P. 180.
22. Kuznetsov S., Feudel U., Pikovsky A. Renormalization group for scaling at the torus-doubling terminal point // Phys. Rev. E. 1998. Vol. 57. P. 1585.
23. Prasad A., Mehra V., Ramaswamy R. Strange nonchaotic attractors in the quasi-periodically forced logistic map // Phys. Rev. E. 1998. Vol. 57. P. 1576.
24. Negi S.S., Prasad A., Ramaswamy R. Bifurcations and transitions in the quasiperiodi-cally driven logistic map // Physica. 2000. Vol. 145D. P. 1.
25. Osinga H.M., Feudel U. Boundary crisis in quasiperiodically forced systems // Physica. 2000. Vol. 141D. P. 54.
26. Hunt B.R., Ott E. Fractal properties of robust strange nonchaotic attractors // Phys. Rev. Lett. 2001. Vol. 87, No 25.
27. Kuznetsov S.P., Neumann E., Pikovsky A., Sataev I.R. Critical point of tori collision in quasiperiodically forced systems // Phys. Rev. E. 2000. Vol. 62. P. 1995.
28. Безручко Б.П., Кузнецов С.П., Пиковский А.С., Фойдель У., Селезнев Е.П. О динамике нелинейных систем под внешним квазипериодическим воздействием вблизи точки окончания линии бифуркации удвоения тора // Изв. вузов. Прикладная нелинейная динамика. 1997. Т. 5, No 6. С. 3.
29. Bezruchko B.P., Kuznetsov S.P., Seleznev E.P. Experimental observation of dynamics near the torus-doubling terminal critical point // Phys. Rev. E. 2000. Vol. 62, No 6. Р. 7828.
30. Kuznetsov S.P. Torus fractalization and intermittency // Phys. Rev. E. 2002. Vol. 65. 066209.
31. Кузнецов С.П., Пиковский А.С. Фойдель У. Странный нехаотический аттрактор // «Нелинейные волны – 2004». M.: Наука, 2004.
32. Kuznetsov S., Feudel U., Pikovsky A. Strange nonchaotic attractors // World scientific series on Nonlinear Science. Series A. Vol. 56. 2006.
33. Bezruchko B.P., Prokhorov M.D., Seleznev Ye.P. Multiparameter model of a dissipative nonlinear oscillator in the form of one-dimensional map // Chaos, Solitons, Fractals. 1995. Vol. 5, No 11. P. 2095.
34. Linsay P.S. Period doubling and chaotic behaviour in a driven anharmonic oscillator // Phys. Rev. Lett. 1981. Vol. 47, No19. P. 1349.
35. Testa J., Perez J., Jeffries C. Evidence for universal behavior of a driven nonlinear oscillator // Phys. Rev. Lett. 1982. Vol. 48, No 11. P. 714.
36. Buskirk R., Jeffries C. Observation of chaotic dynamics of coupled nonlinear oscillators // Phys. Rev. A. 1985. Vol. 31, No 5. P. 3332.
37. Bocko M.F., Douglass D.H., Frutchy H.H. Bounded regions of chaotic behavior in the control parameter space of a driven nonlinear resonator // Phys. Lett. A. 1984. Vol. 104, No 8. P. 388.
38. Klinker T., Meyer-Ilse W., Lauterborn W. Period doubling and chaotic behavior in a driven Toda oscillator // Phys. Lett. A. 1984. Vol. 101, No 8. P. 371.
39. Астахов В.В., Безручко Б.П., Селезнев Е.П. Исследование динамики нелинейного колебательного контура при гармоническом воздействии // Радиотехника и электроника. 1987. Т. 32, No 12. С. 2558.
40. Baxter J.H., Bocko M.F., Douglass D.H. Behavior of a nonlinear resonator driven at subharmonic frequencies // Phys. Rev. A. 1990. Vol. 41, No 2. P. 619.
41. Безручко Б.П. Особенности возбуждения субгармонических и хаотических колебаний в контуре с диодом // Радиотехника и электроника. 1991. Т. 36, No 1. С. 39.
42. Daido H. Resonance and intermittent transition from torus to chaos in periodically forced system near intermittency threshold // Progr. Theor. Phys. Japan. 1983. Vol. 70, No 3. P. 879.
43. Picovsky A.S., Zaks M.A., Feuidel U., Kurth J. Singular continuous spectra in dissipative system // Phys. Rev. E. 1995. Vol. 52, No 1. P. 286.
44. Zaks M.A. Fractal Fourier spectra of cherry flows// Physica. 2001. Vol. D149. P. 237.
45. Ketzmerick R., Petschel G., Geisel T. Slow decay of temporal correlations in quantum systems with Cantor spectra // Phys. Rev. Lett. 1992. Vol. 69. P. 695.
46. Holschneider M. Fractal wavelet dimensions and localization // Communications in Mathematical Physics. 1994. Vol. 160, No 3. P. 457.
47. Makarov K.A. Asymptotic expansions for Fourier transform of singular self-affine measures // J. Math. An. and App. 1994. Vol. 186. P. 259.
48. Астахов В.В., Безручко Б.П., Ерастова Е.Н. Селезнев Е.П. Виды колебаний и их эволюция в диссипативно связанных фейгенбаумовских системах// ЖТФ. 1990. Т. 60, вып. 10. С. 19.
49. Астахов В.В., Безручко Б.П., Пудовочкин О.Б., Селезнев Е.П. Фазовая мульти-стабильность и установление колебаний в нелинейных системах с удвоением периода // Радиотехника и электроника. 1993. Т. 38, No. 2. С. 291.
50. Zakharevich A.M., Seleznev Ye.P. Sets of resonant cycles and their evolution in the nonlinear oscillator’s model under two?frequency action // Abstracts of the Second Interdisciplinary School on Nonlinear Dynamics for System and Signal Analysis (EUROATTRACTOR 2001). Warsaw, Poland, 2001. P. 71.
51. Захаревич А.М., Селезнев Е.П. Структура пространства управляющих параметров в модели нелинейного осциллятора при двухчастотном воздействии // Изв. вузов. Прикладная нелинейная динамика. 2001. Т. 9, No 2. С. 39.
BibTeX
author = {Е. P. Seleznev and А. М. Zakharevich},
title = {CONTROL PARAMETER SPACE OF A NONLINEAR OSCILLATOR UNDER QUASIPERIODIC DRIVING},
year = {2009},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {17},number = {6},
url = {https://old-andjournal.sgu.ru/en/articles/control-parameter-space-of-nonlinear-oscillator-under-quasiperiodic-driving},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2009-17-6-17-35},pages = {17--35},issn = {0869-6632},
keywords = {Strange nonchaotic attractor,torus doubling,torus doubling terminal point,singular continuous spectrum,rational approximation method,phase sensitivity method,Lyapunov exponent.},
abstract = {Dynamics and space of сontrol parameters for a nonlinear oscillator under quasiperiodic driving are investigated experimentally by using a nonlinear circuit with pn junction diode and numerically by using maps and differential equations. The dynamics of the systems under quasiperiodic driving is invariant due to initial driving phases, as a result the plane of the driving amplitudes is symmetrical. The basic element of the control parameter space is the set of torus doubling terminal points, which are the starting and end points of the torus doubling lines, transition to strange nonchaotic and chaotic attractors. }}