Abstract
We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the stability of the zero equilibrium state. We note that the number of roots of the characteristic equation of the linearized problem with the real part that is close to zero increases without bound for order parameter values tending to bifurcational ones. The asymptotics of such roots determines the asymptotic representation of solutions of the original problem that appear in a neighborhood of zero. An explicit change of variables allows finally obtaining equations of the special form for slow amplitudes that are independent of a small parameter and satisfy boundary conditions of the type of periodicity in one of the variables. It is convenient to consider such a variable as a spatial one, although the “fast” time plays this role. We determine amplitudes and frequencies of oscillatory components of the solutions. We formulate results about the correspondence between local solutions of the original system and nonlocal solutions of the partial differential equations playing the role of normal forms.
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References
G. Giacomelli and A. Politi, “Relationship between delayed and spatially extended dynamical systems,” Phys. Rev. Lett., 76, 2686–2689 (1996).
S. Yanchuk and G. Giacomelli, “Dynamical systems with multiple long-delayed feedbacks: multiscale analysis and spatiotemporal equivalence,” Phys. Rev. E, 92, 042903 (2015) 12.
A. Pimenov, S. Slepneva, G. Huyet, and A. G. Vladimirov, “Dispersive time-delay dynamical systems,” Phys. Rev. Lett., 118, 193901 (2017) 6.
M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, “Order parameters for class-B lasers with a long time delayed feedback,” Phys. D, 145, 110–129 (2000).
D. V. Glazkov, “Local dynamics of complex DDE with large delay feedback,” Nonlinear Phenom. Complex Syst., 13, 432–437 (2010).
D. V. Glazkov, “Local dynamics of a second order equation with large exponentially distributed delay and considerable friction,” Model. Anal. Inform. Sist., 22, 65–73 (2015).
E. V. Grigorieva, S. A. Kaschenko, and D. V. Glazkov, “Local dynamics of a model of an opto- electronic oscillator with delay,” Aut. Control Comp. Sci., 52, 700–707 (2018).
E. V. Grigorieva and S. A. Kashchenko, “Slow and fast oscillations in a model of an optoelectronic oscillator with delay,” Dokl. Math., 99, 95–98 (2019).
E. V. Grigorieva and S. A. Kashchenko, “Normalized boundary value problems in the model of optoelectronic oscillator delayed,” Izvestiya VUZ. Applied Nonlinear Dynamics, 28, 361–382 (2020).
E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York (1980).
E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov, and N. Kh. Rozov, Periodic Motions and Bifurcation Processes in Singularly Perturbed Systems, Fizmatlit, Moscow (1995).
V. I. Arnol’d, V. S. Afraimovich, Yu. S. Ilyashenko, and L. P. Shilnikov, “Bifurcation theory,” in: Dynamical Systems, Bifurcation Theory and Catastrophe Theory (Encyclopaedia of Mathematical Sciences, Vol. 5, V. I. Arnold, ed.), Springer, Berlin–Heidelberg, pp. 1–205 (1989).
A. L. Prigorovskiy, V. M. Sandalov, T. A. Tananaeva, Zadachi po teorii kolebaniy, ustoychivosti dvizheniya i kachestvennoy teorii differentsial’nykh uravneniy, Chast’‘6. “Bystrye” i “medlennye” dvizheniya. Razryvnye kolebaniya i avtokolebaniya (in Russian), Nizhegorodskiy gos. un-t, Nizhniy Novgorod (2017).
E. Hairer and G. Vanner, Solving Ordinary Differential Equations II. Stiff and differential-algebraic problems (Springer Series in Computational Mathematics, Vol. 14), Springer (1991, 1996).
K. Ikeda and K. Matsumoto, “High-dimensional chaotic behavior in systems with time-delayed feedback,” Phys. D, 29, 223–235 (1987).
M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E, 79, 026208 (2009) 15.
J. H. Talla Mbé, A. F. Talla, G. R. G. Chengui, A. Coillet, L. Larger, P. Woafo, and Y. K. Chembo, “Mixed-mode oscillations in slow-fast delay optoelectronic systems,” Phys. Rev. E, 91, 012902 (2015) 6.
I. S. Kaschenko and S. A. Kaschenko, “Local dynamics of the two-component singular perturbed systems of parabolic type,” Internat. J. Bifur. Chaos, 25, 1550142 (2015) 27.
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This paper was supported by the Russian Foundation for Basic Research, project No 18-29-10043.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 207, pp. 376-388 https://doi.org/10.4213/tmf10037.
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Glazkov, D.V. Local solutions of the fast–slow model of an optoelectronic oscillator with delay. Theor Math Phys 207, 727–737 (2021). https://doi.org/10.1134/S0040577921060040
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DOI: https://doi.org/10.1134/S0040577921060040