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 are starting 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) | 966-982 |
| Number of pages | 17 |
| Journal | SIAM Journal on Applied Mathematics |
| Volume | 61 |
| Issue number | 3 |
| Publication status | Published - 2000 |
| Externally published | Yes |
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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
Hopf bifurcation subject to a large delay in a laser system. / Pieroux, Didier; Erneux, Thomas; Gavrielides, Athanasios; Kovanis, Vassilios.
In: SIAM Journal on Applied Mathematics, Vol. 61, No. 3, 2000, p. 966-982.Research output: Contribution to journal › Article
}
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 - 2000
Y1 - 2000
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 are starting 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 are starting 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
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M3 - Article
VL - 61
SP - 966
EP - 982
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
SN - 0036-1399
IS - 3
ER -