maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2025-1-6-15mais-1912Research ArticleComputing Methodologies and ApplicationsАлгоритм построения асимптотики периодических решений в моделях лазеров с быстро осциллирующей задержкойAlgorithm for constructing asymptotics of periodic solutions in laser models with a rapidly oscillating delayhttps://orcid.org/0000-0003-4543-2598ГригорьеваЕлена ВикторовнаGrigorievaElena V.grigorieva@tutby.newshttps://orcid.org/0000-0003-0511-5088ГлазковДмитрий ВладимировичGlazkovDmitry V.d.glazkov@uniyar.ac.ruhttps://orcid.org/0000-0001-5668-3929ТолбейАнна ОлеговнаTolbeyAnna O.a.tolbey@uniyar.ac.ruБелорусский государственный экономический университетРоссияBelarus State Economic UniversityRussian FederationЯрославский государственный университет им. П.Г. ДемидоваРоссияP.G. Demidov Yaroslavl State UniversityRussian Federation202522032025321615Copyright © Григорьева Е.В., Глазков Д.В., Толбей А.О., 20252025Григорьева Е.В., Глазков Д.В., Толбей А.О.Grigorieva E.V., Glazkov D.V., Tolbey A.O.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/1912

Рассматривается задача об устойчивости состояния равновесия в лазерной системе с быстро осциллирующими коэффициентами. Построена усредненная по быстрым осцилляциям система с распределенным запаздыванием. Выделены критические случаи в задаче об устойчивости состояния равновесия. Показано, что пороговое значение коэффициента обратной связи, при котором состояние равновесия становится неустойчивым, увеличивается вследствие быстрых осцилляций по сравнению с соответствующим значением при отсутствии модуляции. В критических случаях построены нормальные формы — уравнения для медленной амплитуды периодических решений. Выявлены условия существования, устойчивости и неустойчивости циклов.

The problem of stability of the equilibrium state in a laser system with fast oscillating coefficients is considered. A system averaged over fast oscillations and with a distributed delay is constructed. Critical cases in the problem of the stability of the equilibrium state are singled out. It is shown that the threshold value of the feedback coefficient at which the equilibrium state becomes unstable increases due to rapid oscillations compared to the corresponding value in the absence of modulation. In critical cases, normal forms are constructed – equations for the slowly varying amplitude of the periodic solutions. The conditions for the existence, stability and instability of cycles are revealed.

бифуркационный анализволновые структурызапаздываниелазерная динамикаbifurcation analysiswave structuresdelaylaser dynamicsРабота выполнена в рамках реализации программы развития регионального научно-образовательного математического центра (ЯрГУ) при финансовой поддержке Министерства науки и высшего образования РФ (Соглашение о предоставлении из федерального бюджета субсидии № 075-02-2024-1442).This work was carried out within the framework of a development programme for the Regional Scientific and Educational Mathematical Center of the Yaroslavl State University with financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement on provision of subsidy from the federal budget No. 075-02-2024-1442).ReferencesA. Kittel, K. Pyragas, and R. Richter, “Prerecorded history of a system as an experimental tool to control chaos,” Physical Review E, vol. 50, no. 1, pp. 262–268, 1994, doi: 10.1103/PhysRevE.50.262.A. Kittel, K. Pyragas, and R. Richter, “Prerecorded history of a system as an experimental tool to control chaos,” Physical Review E, vol. 50, no. 1, pp. 262–268, 1994, doi: 10.1103/PhysRevE.50.262.K. Pyragas, “Control of chaos via an unstable delayed feedback controller,” Physical Review Letters, vol. 86, no. 11, pp. 2265–2268, 2001, doi: 10.1103/PhysRevLett.86.2265.K. Pyragas, “Control of chaos via an unstable delayed feedback controller,” Physical Review Letters, vol. 86, no. 11, pp. 2265–2268, 2001, doi: 10.1103/PhysRevLett.86.2265.K. Pyragas, V. Pyragas, I. Z. Kiss, and J. L. Hudson, “Stabilizing and tracking unknown steady states of dynamical systems,” Physical Review Letters, vol. 89, no. 24, p. 244103, 2002, doi: 10.1103/PhysRevLett.89.244103.K. Pyragas, V. Pyragas, I. Z. Kiss, and J. L. Hudson, “Stabilizing and tracking unknown steady states of dynamical systems,” Physical Review Letters, vol. 89, no. 24, p. 244103, 2002, doi: 10.1103/PhysRevLett.89.244103.A. Ahlborn and U. Parlitz, “Controlling dynamical systems using multiple delay feedback control,” Physical Review E, vol. 71, no. 1, p. 016206, 2005, doi: 10.1103/PhysRevE.72.016206.A. Ahlborn and U. Parlitz, “Controlling dynamical systems using multiple delay feedback control,” Physical Review E, vol. 71, no. 1, p. 016206, 2005, doi: 10.1103/PhysRevE.72.016206.H. G. Schuster and M. P. Stemmler, “Control of chaos by oscillating feedback,” Physical Review E, vol. 56, no. 6, pp. 6410–6417, 1997, doi: 10.1103/PhysRevE.56.6410.H. G. Schuster and M. P. Stemmler, “Control of chaos by oscillating feedback,” Physical Review E, vol. 56, no. 6, pp. 6410–6417, 1997, doi: 10.1103/PhysRevE.56.6410.A. Gjurchinovski and V. Urumov, “Variable-delay feedback control of unstable steady states in retarded time-delayed systems,” Physical Review E, vol. 81, no. 1, p. 016209, 2010, doi: 10.1103/PhysRevE.81.016209.A. Gjurchinovski and V. Urumov, “Variable-delay feedback control of unstable steady states in retarded time-delayed systems,” Physical Review E, vol. 81, no. 1, p. 016209, 2010, doi: 10.1103/PhysRevE.81.016209.T. Jungling, A. Gjurchinovski, and V. Urumov, “Experimental time-delayed feedback control with variable and distributed delays,” Physical Review E, vol. 86, no. 4, p. 046213, 2012, doi: 10.1103/PhysRevE.86.046213.T. Jungling, A. Gjurchinovski, and V. Urumov, “Experimental time-delayed feedback control with variable and distributed delays,” Physical Review E, vol. 86, no. 4, p. 046213, 2012, doi: 10.1103/PhysRevE.86.046213.A. V. Skripal, D. A. Usanov, V. A. Vagarin, and M. Y. Kalinkin, “Autodyne detection in a semiconductor laser as the external reflector is moved,” Technical Physics, vol. 44, pp. 66–68, 1999, doi: 10.1134/1.1259253.A. V. Skripal, D. A. Usanov, V. A. Vagarin, and M. Y. Kalinkin, “Autodyne detection in a semiconductor laser as the external reflector is moved,” Technical Physics, vol. 44, pp. 66–68, 1999, doi: 10.1134/1.1259253.J. Martin-Regalado, G. H. M. Tartwijk, S. Balle, and M. S. Miguel, “Mode control and pattern stabilization in broad-area lasers by optical feedback,” Physical Review A, vol. 54, no. 6, pp. 5386–5393, 1996, doi: 10.1103/PhysRevA.54.5386.J. Martin-Regalado, G. H. M. Tartwijk, S. Balle, and M. S. Miguel, “Mode control and pattern stabilization in broad-area lasers by optical feedback,” Physical Review A, vol. 54, no. 6, pp. 5386–5393, 1996, doi: 10.1103/PhysRevA.54.5386.T. Yang, C. W. Wu, and L. O. Chua, “Cryptography based on chaotic systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 5, pp. 469–472, 1997, doi: 10.1109/81.572346.T. Yang, C. W. Wu, and L. O. Chua, “Cryptography based on chaotic systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 5, pp. 469–472, 1997, doi: 10.1109/81.572346.J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by delayed feedback tunable laser diode,” Physical Review Letters, vol. 80, no. 10, pp. 2249–2252, 1998, doi: 10.1103/PhysRevLett.80.2249.J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by delayed feedback tunable laser diode,” Physical Review Letters, vol. 80, no. 10, pp. 2249–2252, 1998, doi: 10.1103/PhysRevLett.80.2249.N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations. Delhi : Hindustan Publishing Corporation (India), 1961.N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations. Delhi : Hindustan Publishing Corporation (India), 1961.A. Stephenson, “On a new type of dynamical stability,” Memoirs and Proceedings of the Manchester Literary and Philosophical Society, vol. 52, no. 8, pp. 1–10, 1908.A. Stephenson, “On a new type of dynamical stability,” Memoirs and Proceedings of the Manchester Literary and Philosophical Society, vol. 52, no. 8, pp. 1–10, 1908.J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Optics Communications, vol. 96, no. 4--6, pp. 259–266, 1993, doi: 10.1016/0030-4018(93)90272-7.J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Optics Communications, vol. 96, no. 4--6, pp. 259–266, 1993, doi: 10.1016/0030-4018(93)90272-7.R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE Journal of Quantum Electronics, vol. 16, no. 3, pp. 347–357, 1980, doi: 10.1109/JQE.1980.1070479.R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE Journal of Quantum Electronics, vol. 16, no. 3, pp. 347–357, 1980, doi: 10.1109/JQE.1980.1070479.A. Levine, G. H. M. Tartwijk, D. Lenstra, and T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Physical Review A, vol. 52, no. 5, pp. R3436–R3439, 1995, doi: 10.1103/PhysRevA.52.R3436.A. Levine, G. H. M. Tartwijk, D. Lenstra, and T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Physical Review A, vol. 52, no. 5, pp. R3436–R3439, 1995, doi: 10.1103/PhysRevA.52.R3436.E. V. Grigorieva, A. A. Kashchenko, and S. A. Kashchenko, Local analysis of the dynamics of distributed laser models. Moscow: LENAND, 2024.E. V. Grigorieva, A. A. Kashchenko, and S. A. Kashchenko, Local analysis of the dynamics of distributed laser models. Moscow: LENAND, 2024.E. Grigorieva, “Instabilities of periodic orbits in lasers with oscillating delayed feedback,” Nonlinear Phenomena in Complex Systems, vol. 4, no. 1, pp. 6–12, 2001.E. Grigorieva, “Instabilities of periodic orbits in lasers with oscillating delayed feedback,” Nonlinear Phenomena in Complex Systems, vol. 4, no. 1, pp. 6–12, 2001.Y. A. Mitropol'skii, The method of averaging in nonlinear mechanics. Kiev : Naukova Dumka, 1971.Y. A. Mitropol'skii, The method of averaging in nonlinear mechanics. Kiev : Naukova Dumka, 1971.Y. S. Kolesov, V. S. Kolesov, and I. I. Fedik, Avtokolebaniya v sistemah s raspredelennymi parametrami. Kiev : Naukova Dumka, 1979.Y. S. Kolesov, V. S. Kolesov, and I. I. Fedik, Avtokolebaniya v sistemah s raspredelennymi parametrami. Kiev : Naukova Dumka, 1979.

The authors declare that there are no conflicts of interest present.