Hopf bifurcation subject to a large delay in a laser system

Didier Pieroux, Thomas Erneux, Athanasios Gavrielides, Vassilios Kovanis

Research output: Contribution to journalArticlepeer-review

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
DOIs
Publication statusPublished - Sep 2003
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics

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