### 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 language | English |
---|---|

Pages (from-to) | 523-540 |

Number of pages | 18 |

Journal | SIAM Review |

Volume | 45 |

Issue number | 3 |

Publication status | Published - Sep 2003 |

Externally published | Yes |

### Fingerprint

### 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

*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.

Research output: Contribution to journal › Article

*SIAM Review*, vol. 45, no. 3, pp. 523-540.

}

TY - JOUR

T1 - Hopf bifurcation subject to a large delay in a laser system

AU - Pieroux, Didier

AU - Erneux, Thomas

AU - Gavrielides, Athanasios

AU - Kovanis, Vassilios

PY - 2003/9

Y1 - 2003/9

N2 - 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.

AB - 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.

KW - Bifurcation to periodic and quasi-periodic oscillations

KW - Semiconductor laser instabilities

KW - System of delay-differential equations

KW - Two-time solution

UR - http://www.scopus.com/inward/record.url?scp=0242415475&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242415475&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0242415475

VL - 45

SP - 523

EP - 540

JO - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 3

ER -