#### For citation:

Grigorieva E. V., Kashchenko S. A. Normalized boundary value problems in the model of optoelectronic oscillator delayed. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2020, vol. 28, iss. 4, pp. 361-382. DOI: 10.18500/0869-6632-2020-28-4-361-382

# Normalized boundary value problems in the model of optoelectronic oscillator delayed

Purpose of this work is reduction of differential-difference-model of optic-electronic oscillator to more simple normalized boundary value problems. We study the dynamics of an optoelectronic oscillator with delayed feedback in the vicinity of the zero equilibrium state. The differential-difference-model contains a small parameter with the derivative. It is shown that in a certain neighborhood of the bifurcation point, the number of roots of the characteristic equation that have a real part close to zero increases unlimitedly with decreasing small parameter. Partial boundary value problems are obtained that play the role of normal forms for the original system and which have stationary solutions in the form of symmetric or asymmetric rectangular structures. The multistability of rectangular structures with a different number and shape of steps is shown. The spatio-temporal representation of solutions of the initial equation with delay is substantiated. The frequencies and amplitudes of oscillating solutions of the delay equation are determined. Research methods. We apply standard methods of normal forms on central manifolds, as well as special methods for infinite-dimensional normalization. An algorithm is proposed for reducing the initial delayed equation to the boundary-value problem for slowly varying amplitudes. Results. Finite-dimensional and special infinite-dimensional equations – boundary value problem are constructed that play the role of normal forms. Their nonlocal dynamics determines the behavior of solutions to the original equation with delay from a small neighborhood of the equilibrium. Asymptotic formulas for solutions on the interval [t_{0}, ∞) are given.

- Yanchuk S., Giacomelli G. Spatio-temporal phenomena in complex systems with time delays // J. Phys. A: Math. Theor. 2017. Vol. 50. P. 103001.
- Cross M., Hohenberg P. Pattern formation outside of equilibrium // Rev. Mod. Phys. 1993. Vol. 65. P. 851–1112.
- Arecchi F.T., Giacomelli G., Lapucci A., Meucci R. Two-dimensional representation of a delayed dynamical system // Phys. Rev. A. 1992. Vol. 45. P. R4225–4228.
- Vladimirov A.G., Turaev D. Model for passive mode locking in semiconductor lasers // Phys. Rev. A 2005. Vol. 72. P. 033808.
- Marconi M., Javaloyes J., Barland S., Balle S. and Giudici M. Vectorial dissipative solitons in vertical-cavity surface-emitting lasers with delays // Nature Photonics. 2015. doi:10.1038/NPHOTON.2015.92.
- Pimenov A., Slepneva S., Huyet G., Vladimirov A. Dispersive time-delay dynamical systems // Phys. Rev. Lett. 2017. Vol. 118. 193901.
- Heiligenthal S., Dahms T., Yanchuk S., Jungling T., Flunkert V., Kanter I., Scholl E. and Kinzel W. Strong and weak chaos in nonlinear networks with time-delayed couplings // Phys. Rev. Lett. 2011. Vol. 107. P. 234.
- Kashchenko S.A. Application of the normalization method to the study of the dynamics of a differential-difference equation with a small factor multiplying the derivative. Differentsialnye Uravneniya, 1989, vol. 25, no. 8, pp. 1448–1451.
- Kashchenko S.A. The Ginzburg–Landau equation as a normal form for a second-order differencedifferential equation with a large delay. Computational Mathematics and Mathematical Physics, 1998, vol. 38, no. 3, pp. 443–451.
- Kashchenko S.A. Bifurcation singularities of a singularly perturbed equation with delay. Siberian Mathematical Journal, 1999, vol. 40, no. 3, pp. 483–487.
- Kashchenko S.A. Normalization in the systems with small diffusion // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 1996. Vol. 6, no. 6. P. 1093–1109.
- Grigorieva E.V., Kashchenko S.A. Slow and fast oscillations in a model of an optoelectronic oscillator with delay. Doklady Mathematics, 2019, vol. 99, no. 1, pp. 95–98.
- Giacomelli G., Politi A. Relationship between delayed and spatially extended dynamical Systems // Phys. Rev. Lett. 1996. Vol. 76. P. 2686–2689.
- Grigorieva E.V., Haken H., Kashchenko S.A. Theory of quasi-periodicity in model of lasers with delayed optoelectronic feedback // Optics Communications. 1999. Vol. 165. P. 279–292.
- Bestehorn M., Grigorieva E.V., Haken H., Kashchenko S.A. Order parameters for class-B lasers with a long time delayed feedback // Physica D. 2000. Vol. 145. P. 111–130.
- Grigorieva E.V., Haken H., Kashchenko S.A., Pelster A. Traveling wave dynamics in a nonlinear interferometer with spatial field transformer in feedback // Physica D. 1999. Vol. 125, no. 1–2. P. 123–141
- Grigorieva E.V., Kashchenko I.S., Kashchenko S.A. Dynamics of Lang-Kobayashi equations with large control coefficient // Nonlinear Phenomena in Complex Systems. 2012. Vol. 12. P. 403–409.
- Kashchenko I.S., Kashchenko S.A. Local Dynamics of the two-component singular perturbed systems of parabolic type // International Journal of Bifurcation and Chaos. 2015. Vol. 25, no. 11. P. 1550142.
- Ikeda K., Matsumoto K. High-dimensional chaotic behavior in systems with time-delayed feedback // Physica D. 1987. Vol. 29. P. 223–235.
- Kouomou C.Y., Colet P., Larger L., Gastaud N. Chaotic breathers in delayed electro-optical systems // Phys. Rev. Lett. 2005. Vol. 95. 203903.
- Weicker L., Erneux T., D'Huys O., Danckaert J., Jacquot M., Chembo Y. and Larger L. Strongly asymmetric square waves in time-delayed system // Phys. Rev. E. 2012. Vol. 86. 055201(R).
- Weicker L., Erneux T., Rosin D.P., Gauthier D.J. Multirhythmicity in an optoelectronic oscillator with large delay // Phys. Rev. E. 2015. Vol. 91. 012910.
- Peil M., Jacquot M, Chembo Y.C., Larger L., Erneux T. Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators // Phys. Rev. E. 2009. Vol. 79. 026208.
- Talla Mbe J.H., Talla A.F., Goune Chengui G.R., Coillet A., Larger L., Woafo P. Mixed-mode oscillations in slow-fast delay optoelectronic systems // Phys. Rev. E. 2015. Vol. 91. 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. 062208.
- Larger L., Penkovsky B. and Maistrenko Y. Virtual chimera states for delayed-feedback systems // Phys. Rev. Lett. 2013. Vol. 111. P. 054103.
- Grigorieva E.V., Kashchenko I.S., Glazkov D.V. Local dynamics of a model of an opto-electronic oscillator with delay // Automatic Control and Computer Sciences. 2018. Vol. 52. P. 700–707.
- Gibbs H.M. Optical Bistability: Controlling Light with Light. Academic Press, Inc., 1985.

- 1045 reads