CONTROLLING CHAOS IN IKEDA SYSTEM Spatio–temporal model
Cite this article as:
Ryskin N. M., Khavroshin O. S. CONTROLLING CHAOS IN IKEDA SYSTEM Spatio–temporal model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 2, pp. 87-98. DOI: https://doi.org/10.18500/0869-6632-2009-17-2-87-98
The method for controlling chaos in a ring resonator filled with a medium with cubic phase nonlinearity (Ikeda system), suggested in [1], is investigated within the framework of a distributed spatio-temporal model described by a Nonlinear Schr¨ odinger Equation with time-delayed boundary condition. Numerical results are presented which confirm the capability of the suggested method. For the case of weakly dispersive nonlinear medium, the results are in good agreement with the approximate theory based on the return map [1]. In the case of strong dispersion, when the non-stationary behavior is determined mainly by the modulation instability, the dynamics is more complicated due to competition of different resonator eigenmodes. It is demonstrated, that proper adjustment of the control feedback parameters allows suppressing self-modulation oscillations and provides stable single-frequency operation in a broad range of parameters.
1. Рыскин Н.М., Хаврошин О.С. Управление хаосом в системе Икеды. Упрощенная модель в виде точечного отображения // Изв. вузов. Прикладная нелинейная динамика. 2009. Т. 17, No 2. C. 66.
2. Ikeda K., Daido H., Akimoto O. Optical turbulence: chaotic behavior of transmitted light from a ring cavity // Phys. Rev. Lett. 1980. Vol. 45, No 9. P. 709.
3. Измайлов И.В., Лячин А.В., Пойзнер Б.Н. Детерминированный хаос в моделях кольцевого нелинейного интерферометра. Томск: Изд-во Том. ун-та, 2007.
4. Розанов Н.Н. Оптическая бистабильность и гистерезис в распределенных нелинейных системах. М.: Наука, Физматлит, 1997.
5. Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.S. Solitons and Nonlinear Wave Equations, Academic Press, London, 1984.
6. Рыскин Н.М., Трубецков Д.И. Нелинейные волны. М.: Наука, Физматлит, 2000.
7. Островский Л.А., Потапов А.И. Введение в теорию модулированных волн. М.: Физматлит, 2003.
8. Балякин А.А. Рыскин Н.М. Хаврошин О.С. Нелинейная динамика модуляционной неустойчивости в распределенных резонаторах под внешним гармоническим воздействием // Изв. вузов. Радиофизика. 2007. Т. 50, No 9. С. 800.
9. Балякин А.А., Рыскин Н.М. Смена характера модуляционной неустойчивости вблизи критической частоты // Письма в ЖТФ. 2004. Т. 30, No 5. С. 6.
10. Balyakin A.A., Ryskin N.M. Modulation instability in a nonlinear dispersive medium near cut-off frequency // Nonlinear Phenomena in Complex Systems. 2004. Vol. 7, No 1. P. 34.
11. Самарский А.А. Теория разностных схем. М.: Наука, Физматлит, 1989.
BibTeX
author = {N. M. Ryskin and O. S. Khavroshin},
title = {CONTROLLING CHAOS IN IKEDA SYSTEM Spatio–temporal model},
year = {2009},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {17},number = {2},
url = {https://old-andjournal.sgu.ru/en/articles/controlling-chaos-in-ikeda-system-spatio-temporal-model},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2009-17-2-87-98},pages = {87--98},issn = {0869-6632},
keywords = {Controlling chaos,nonlinear ring-loop resonator,delayed feedback,Nonlinear Schr¨ odinger equation,modulation instability.},
abstract = {The method for controlling chaos in a ring resonator filled with a medium with cubic phase nonlinearity (Ikeda system), suggested in [1], is investigated within the framework of a distributed spatio-temporal model described by a Nonlinear Schr¨ odinger Equation with time-delayed boundary condition. Numerical results are presented which confirm the capability of the suggested method. For the case of weakly dispersive nonlinear medium, the results are in good agreement with the approximate theory based on the return map [1]. In the case of strong dispersion, when the non-stationary behavior is determined mainly by the modulation instability, the dynamics is more complicated due to competition of different resonator eigenmodes. It is demonstrated, that proper adjustment of the control feedback parameters allows suppressing self-modulation oscillations and provides stable single-frequency operation in a broad range of parameters. }}