Hopf bifurcation subject to a large delay in a laser system

Didier Pieroux, Thomas Erneux, Athanasios Gavrielides, Vassilios Kovanis

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities start from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.

Original languageEnglish
Pages (from-to)523-540
Number of pages18
JournalSIAM Review
Volume45
Issue number3
Publication statusPublished - Sep 2003
Externally publishedYes

Fingerprint

Hopf bifurcation
Hopf Bifurcation
Laser
Lasers
Bifurcation Point
Bifurcation
Optical Feedback
Optical feedback
Amplitude Equations
Bifurcation (mathematics)
Bistability
Semiconductor Lasers
Bifurcation Theory
Bifurcation Diagram
Semiconductor lasers
Periodic Solution
Branch
Feedback
Approximation

Keywords

  • Bifurcation to periodic and quasi-periodic oscillations
  • Semiconductor laser instabilities
  • System of delay-differential equations
  • Two-time solution

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Pieroux, D., Erneux, T., Gavrielides, A., & Kovanis, V. (2003). Hopf bifurcation subject to a large delay in a laser system. SIAM Review, 45(3), 523-540.

Hopf bifurcation subject to a large delay in a laser system. / Pieroux, Didier; Erneux, Thomas; Gavrielides, Athanasios; Kovanis, Vassilios.

In: SIAM Review, Vol. 45, No. 3, 09.2003, p. 523-540.

Research output: Contribution to journalArticle

Pieroux, D, Erneux, T, Gavrielides, A & Kovanis, V 2003, 'Hopf bifurcation subject to a large delay in a laser system', SIAM Review, vol. 45, no. 3, pp. 523-540.
Pieroux D, Erneux T, Gavrielides A, Kovanis V. Hopf bifurcation subject to a large delay in a laser system. SIAM Review. 2003 Sep;45(3):523-540.
Pieroux, Didier ; Erneux, Thomas ; Gavrielides, Athanasios ; Kovanis, Vassilios. / Hopf bifurcation subject to a large delay in a laser system. In: SIAM Review. 2003 ; Vol. 45, No. 3. pp. 523-540.
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