Abstract
We consider the dynamics of an electro-optic oscillator described by a system of differential equations with delay. An essential feature of this model is that one of the derivatives is multiplied by a small parameter, which allows us to draw conclusions about the action of processes with rates of different orders. The local dynamics of a singularly perturbed system near a zero steady state is analyzed. When values of the parameters are close to critical, the characteristic equation of the linearized problem has an asymptotically large number of roots with close-to-zero real parts. The bifurcations taking place in the system are studied by constructing special normalized equations for slow amplitudes, which describe the behavior of close-to-zero solutions of the initial problem. An important feature of these equations is that they do not depend on the small parameter. The structure of roots of the characteristic equation and the order of supercriticality determine the normal form, which can be represented by a partial differential equation. The “fast” time, for which periodicity conditions are fulfilled, acts as a “spatial” variable. The high sensitivity of dynamic properties of normalized equations to the change of a small parameter is noted, which is a sign of a possible unlimited process of direct and inverse bifurcations. Also, the obtained equations exhibit multistability of solutions.
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REFERENCES
Ikeda, K. and Matsumoto, K., High-dimensional chaotic behavior in systems with time-delayed feedback, Phys. D: Nonlinear Phenom., 1987, vol. 29, nos. 1–2, pp. 223–235.
Vallée, R. and Marriott, C., Analysis of an Nth-order nonlinear differential-delay equation, Phys. Rev. A, 1989, vol. 39, no. 1, pp. 197–205.
Kouomou, C., et al., Chaotic breathers in delayed electro-optical systems, Phys. Rev. Lett., 2005, vol. 95, no. 20, p. 203903.
Weicker, L., et al., Multirhythmicity in an optoelectronic oscillator with large delay, Phys. Rev. E, 2015, vol. 91, no. 1, p. 012910.
Weicker, L., et al., Strongly asymmetric square waves in time-delayed system, Phys. Rev. E, 2012, vol. 86, no. 5, p. 055201(R).
Peil, M., et al., Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators, Phys. Rev. E, 2009, vol. 79, no. 2, p. 026208.
Talla Mbé, J.H., et al., Mixed-mode oscillations in slow-fast delayed optoelectronic systems, Phys. Rev. E, 2015, vol. 91, no. 1, p. 012902.
Marquez, B.A., et al., Interaction between Lienard and Ikeda dynamics in a nonlinear electro-optical oscillator with delayed bandpass feedback, Phys. Rev. E, 2016, vol. 94, no. 6,p. 062208.
Kaschenko, I.S. and Kaschenko, S.A., Local dynamics of the two-component singular perturbed systems of parabolic type, Int. J. Bifurcation Chaos, 2015, vol. 25, no. 11, p. 1550142.
Giacomelli, G. and Politi, A., Relationship between delayed and spatially extended dynamical systems, Phys. Rev. Lett., 1996, vol. 76, no. 15, pp. 2686–2689.
Yanchuk, S. and Giacomelli, G., Dynamical systems with multiple long-delayed feedbacks: Multiscale analysis and spatiotemporal equivalence, Phys. Rev. E, 2015, vol. 92, no. 4, p. 042903.
Marconi, M., et al., Vectorial dissipative solitons in vertical-cavity surface-emitting lasers with delays, Nat. Photonics, 2015, vol. 9, no. 7, pp. 450–455. doi 10.1038/nphoton.2015.92
Pimenov, A., et al., Dispersive time-delay dynamical systems, Phys. Rev. Lett., 2017, vol. 118, no. 19, p. 193901.
Bestehorn, M., Grigorieva, E.V., Haken, H., and Kaschenko, S.A., Order parameters for class-B lasers with a long time delayed feedback, Phys. D: Nonlinear Phenom., 2000, vol. 145, nos. 1–2, pp. 110–129.
Grigorieva, E.V., Kaschenko, I.S., and Kaschenko, S.A., Dynamics of Lang–Kobayashi equations with large control coefficient, Nonlinear Phenom. Complex Syst., 2012, vol. 15, no. 4, pp. 403–409.
Akhromeeva, T.S., et al., Nestatsionarnye struktury i diffuzionnyi khaos (Nonstationary Structures and Diffusion Chaos), Moscow: Nauka, 1992.
Kashchenko, I.S., Local dynamics of equations with large delay, Comput. Math. Math. Phys., 2008, vol. 48, no. 12, pp. 2172–2181.
Kashchenko, I.S., Dynamics of an equation with a large coefficient of delayed control, Dokl. Math., 2011, vol. 83, pp. 258–261.
ACKNOWLEDGMENTS
This study was performed within the state assignment of the Ministry of Education and Science of the Russian Federation, project no. 1.10160.2017/5.1.
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Translated by V. A. Alekseev
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Grigorieva, E.V., Kashchenko, S.A. & Glazkov, D.V. Local Dynamics of a Model of an Opto-Electronic Oscillator with Delay. Aut. Control Comp. Sci. 52, 700–707 (2018). https://doi.org/10.3103/S014641161807012X
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DOI: https://doi.org/10.3103/S014641161807012X