ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Grigorieva E. V., Kashchenko S. A. Quasinormal forms at bifurcations of codimension two in the problem of the dynamics of a large chain of coupled lasers. Izvestiya VUZ. Applied Nonlinear Dynamics, 2026, vol. 34, iss. 4, pp. 497-516. DOI: 10.18500/0869-6632-003221, EDN: ZOPAHH

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon and_2026-4_grigorieva-kashchenko_497-516.pdf
(downloads: 26)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9, 535.8
DOI: 
10.18500/0869-6632-003221
EDN: 
ZOPAHH

Quasinormal forms at bifurcations of codimension two in the problem of the dynamics of a large chain of coupled lasers

Autors: 
Grigorieva Elena Viktorovna, Belarus State Economic University (BSEU)
Kashchenko Sergej Aleksandrovich, P. G. Demidov Yaroslavl State University
Abstract: 

Purpose. An integro-differential model of the generation dynamics of a large laser chain with two-way optoelectro nic coupling via pumping is investigated. Critical values of the coupling coefficient and delay time in the coupling lines are obtained, at which dual- and multi-frequency quasi-periodic oscillations of the radiation intensity can occur.

Methods. Local dynamics studies based on constructing normal forms on central manifolds are applied to critical cases of (asymptotically) infinite dimension. An algorithm for reducing the original boundary value problem to equations for slowly varying amplitudes is proposed. A system of complex Ginzburg–Landau equations is constructed as a quasi-normal form (QNF). The nonlocal dynamics of these equations determines the behavior of the solutions to the original boundary value problem.

Results. Homogeneous solutions of the quasi-normal form and corresponding solutions of the nonlinear chain model were obtained. These can be interpreted as quasi-periodic oscillations and a regime of alternating in-phase and anti-phase generation modes in bidirectionally coupled lasers. The frequencies and amplitudes of the laser intensity oscillations in the chain were calculated.

Key words: 
coupled lasers
delayed coupling
codimension two bifurcations
wave structures
Acknowledgments: 
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. 075-02-2026-1331).
Reference: 
  1. Soriano MC, Ojalvo JG, Mirasso CR, Fischer I. Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers. Rev. Mod. Phys. 2013;85(1):421–470. DOI: 10.1103/RevModPhys.85.421.
  2. Nixon M, Fridman M, Ronen E, Friesem AA, Davidson N, Kanter I. Controlling synchronization in large laser networks. Phys. Rev. Lett. 2012;108(21):214101. DOI: 10.1103/PhysRevLett.108.214101.
  3. Nixon M, Ronen E, Friesem AA, Davidson N. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett. 2013;110(1):184102. DOI: 10.1103/PhysRevLett.110.184102.
  4. Ji P, Ye J, Mu Yu, Lin W, Tian Y, Hens C, Perc M, Tang Y, Sun Jie, Kurths J. Signal propagation in complex networks. Phys. Rep. 2023;1017:1–96. DOI: 10.1016/j.physrep.2023.03.005.
  5. Shena J, Kominis Y, Bountis T, Kovanis V. Spatial control of localized oscillations in arrays of coupled laser dimmers. Phys. Rev. E. 2020;102:012201. DOI: 10.1103/PhysRevE.102.012201.
  6. Pando A, Gadasi S, Bernstein E, Stroev N, Friesem A, Davidson N. Synchronization in coupled laser arrays with correlated and uncorrelated disorder. Phys. Rev. Lett. 2024;133(11):113803. DOI: 10.1103/PhysRevLett.133.113803.
  7. Wieczorek S, Krauskopf B, Simpson TB, Lenstra D. The dynamical complexity of optically injected semiconductor lasers. Phys. Rep. 2005;416(1–2):1–128. DOI: 10.1016/j.physrep.2005.06.003.
  8. Grigorieva EV, Haken H, Kashchenko SA. Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback. Optics Communications. 1999;165(4–6):279–292. DOI: 10.1016/s0030-4018(99)00236-9.
  9. Kovalev AV, Islam MdS, Locquet A, Citrin DS, Viktorov EA, Erneux T. Resonances between fundamental frequencies for lasers with large delayed feedbacks. Phys. Rev. E. 2019;99(6):062219. DOI: 10.1103/PhysRevE.99.062219.
  10. Hohl A, Gavrielides A, Erneux T, Kovanis V. Quasiperiodic synchronization for two delay-coupled semiconductor lasers. Phys. Rev. A. 1999;59(5):3941–3949. DOI: 10.1103/PhysRevA.59.3941.
  11. Loose A, Wunsche H-J, Henneberger F. Synchronization of quasiperiodic oscillations to a periodic force studied with semiconductor lasers. Phys. Rev. E. 2010;82:035201. DOI: 10.1103/PhysRevE.82.035201.
  12. Fan Y, Shore KA, Chao X. Dynamics of electrically pumped semiconductor nano-laser arrays. Photonics. 2023;10(11):1249. DOI: 10.3390/photonics10111249.
  13. Kim M-Y, Roy R, Aron JL, Carr TW, Schwartz IB. Scaling behavior of laser population dynamics with time-delayed coupling: Theory and experiment. Phys. Rev. Lett. 2005;94(8):088101. DOI: 10.1103/PhysRevLett.94.088101.
  14. Vicente R, Tang S, Mulet J, Mirasso R, Liu JM. Dynamics of semiconductor lasers with bidirectional optoelectronic coupling: Stability, route to chaos, and entrainment. Phys. Rev. E. 2004;70:046216. DOI: 10.1103/PhysRevE.70.046216.
  15. Vicente R, Tang S, Mulet J, Mirasso R, Liu JM. Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling. Phys. Rev. E. 2006;73(4):047201. DOI: 10.1103/PhysRevE.73.047201.
  16. Kozyreff G, Vladimirov AG, Mandel P. Dynamics of a semiconductor laser array with delayed global coupling. Phys. Rev. E. 2001;64:016613. DOI: 10.1103/PhysRevE.64.016613.
  17. Mehrabbeik M, Jafari S, Meucci R, Perc M. Synchronization and multistability in a network of diffusively coupled laser models. Commun. Nonlinear Sci. Numer. Simulat. 2023;125:107380. DOI: 10.1016/j.cnsns.2023.107380.
  18. Shena J, Hizanidis J, Kouvaris NE, Tsironis GP. Localized patterns in star networks of class-B lasers with optoelectronic feedback. Phys. Rev. A. 2018;98:053817.
  19. DeWitte V, Govaerts W, Kuznetsov YuA, Meijer HGE. Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms. Physica D. 2014;269:126–141. DOI: 10.1016/j.physd.2013.12.002.
  20. Guckenheimer J, Holmes Ph. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. NY: Springer; 1983. 462 p. DOI: 10.1007/978-1-4612-1140-2.
  21. Emelianova YuP, Kuznetsov AP, Sataev IR, Turukina LV. Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D. 2013;244;36–49. DOI: 10.1016/j.physd.2012.10.012.
  22. Anishchenko V, Nikolaev S, Kurths J. Winding number locking on a two-dimensional torus: Synchronization of quasiperiodic motions. Phys. Rev. E. 2006;73(5):056202. DOI: 10.1103/PhysRevE.73.056202.
  23. Anishchenko VS, Nikolaev SM. Transition to chaos from quasiperiodic motions on a four-dimensional torus perturbed by external noise. Int. J. Bifurc. Chaos. 2008;18(9):2733–2741. DOI: 10.1142/S0218127408021956.
  24. Rosenblum M, Pikovsky A. Two types of quasiperiodic partial synchrony in oscilator ensembles. Phys. Rev. E. 2015;92:012919. DOI: 10.1103/PhysRevE.92.012919.
  25. Kuznetsov AP, Kuznetsov SP, Shchegoleva NA, Stankevich NV. Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena. Physica D. 2019;398:1–12. DOI: 10.1016/j.physd.2019.05.014.
  26. Kuznetsov AP, Sedova Yu, Stankevich N. Coupled systems with quasi-periodic and chaotic dynamics. Chaos, Solitons and Fractals. 2023;169:113278. DOI: 10.1016/j.chaos.2023.113278.
  27. Kashchenko SA. Dynamics of advectively coupled Van der Pol equations chain. Chaos. 2021;linebreak 31(3):033147. DOI: 10.1063/5.0040689.
  28. Grigorieva EV, Kashchenko SA. Phase-synchronized oscillations in a ring of pump-coupled lasers. Optics Communications. 2023;545:129688. DOI: 10.1016/j.optcom.2023.129688.
  29. Grigorieva EV, Kashchenko SA. Antiphase and in-phase dynamics in laser chain models with delayed bidirectional couplings. Physica D. 2023;467:134223. DOI: 10.1016/j.physd.2024.134223.
  30. Grigorieva EV, Kashchenko SA, Tolbey AO. Quasiperiodic dynamics in a large chain of pump-coupled lasers. Nonlinear Phenomena in Complex Systems. 2025;28:79–89.
  31. Khanin YaI. Fundamentals of Laser Dynamics. Moscow: Nauka; 1999. 368 p.
  32. Garcia-Morales V, Krischer K. The complex Ginzburg-Landau equation: An introduction. Contemporary Physics. 2012;53(2):79–95. DOI: 10.1080/00107514.2011.642554.
Received: 
10.03.2026
Accepted: 
11.05.2026
Available online: 
12.05.2026
Published: 
31.07.2026
  • 461 reads