Classification of Codimension-One Bifurcations in a Symmetric Laser System

  • Juancho A. Collera
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)


We consider a class symmetric laser system, particularly rings of n identical semiconductor lasers coupled bidirectionally. The model is described using the Lang-Kobayashi rate equations where the finite propagation time of the light from one laser to another is reflected by a constant delay time parameter in the laser optical fields. Due to the network structure, the resulting system of delay differential equations has symmetry group \(\mathbb {D}_n\times \textbf {S}^1\). We follow the discussions in Buono and Collera (SIAM J Appl Dyn Syst 14:1868–1898, 2015) for the general case, then give the important example where n = 4 to explicitly illustrate the general method which is a classification of codimension-one bifurcations into regular and symmetry-breaking. This particular example complements the recent works on rings with unidirectional coupling (Domogo and Collera, AIP Conf Proc 1787:080002, 2016), and on laser networks with all-to-all coupling (Collera, Mathematical and computational approaches in advancing modern science and engineering. Springer, Cham, 2016). Numerical continuation using DDE-Biftool are carried out to corroborate our classification results.



This work was funded by the UP System Emerging Interdisciplinary Research Program (OVPAA-EIDR-C03-011). The author also acknowledged the support of the UP Baguio through RLCs during the A.Y. 2015–2016.


  1. 1.
    Alsing, P.M., Kovanis, V., Gavrielides, A., Erneux, T.: Lang and Kobayashi phase equation. Phys. Rev. A 53, 4429–4434 (1996)Google Scholar
  2. 2.
    Buono, P.-L., Collera, J.A.: Symmetry-breaking bifurcations in rings of delay-coupled semiconductor lasers. SIAM J. Appl. Dyn. Syst. 14, 1868–1898 (2015)Google Scholar
  3. 3.
    Collera, J.A.: Symmetry-breaking bifurcations in two mutually delay-coupled lasers. Int. J. Philipp. Sci. Technol. 8, 17–21 (2015)Google Scholar
  4. 4.
    Collera, J.A.: Symmetry-breaking bifurcations in laser systems with all-to-all coupling. In: Belair, J. et al., (eds.) Mathematical and Computational Approaches in Advancing Modern Science and Engineering, pp. 81–88. Springer, Cham (2016)Google Scholar
  5. 5.
    Domogo, A.A., Collera, J.A.: Classification of codimension-one bifurcations in a tetrad of delay-coupled lasers with feed forward coupling. AIP Conf. Proc. 1787, 080002 (2016)Google Scholar
  6. 6.
    Engelborghs, K., Luzyanina, T., Samaey, G.: DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical report TW-330. Department of Computer Science, K.U. Leuven, Leuven (2001)Google Scholar
  7. 7.
    Erzgräber, H., Krauskopf, B., Lenstra, D.: Compound laser modes of mutually delay-coupled lasers. SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Golubitsky, M., Schaeffer, D., Stewart, I.: Singularities and Groups in Bifurcation Theory Volume II. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lang, R., Kobayashi, K.: External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron. 16, 347–355 (1980)CrossRefGoogle Scholar
  10. 10.
    Lunel, S.M.V., Krauskopf, B.: The mathematics of delay equations with an application to the Lang-Kobayashi equations. AIP Conf. Proc. 548, 66–86 (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of the Philippines BaguioBaguio CityPhilippines

Personalised recommendations