{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:07Z","timestamp":1787232667699,"version":"3.56.0"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The Lang\u2013Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so-called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In this case, both the number of ECMs and the number of critical eigenvalues is large. Applying a newly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.<\/jats:p>","DOI":"10.1137\/090751335","type":"journal-article","created":{"date-parts":[[2010,6,2]],"date-time":"2010-06-02T18:09:01Z","timestamp":1275502141000},"page":"519-535","source":"Crossref","is-referenced-by-count":58,"title":["A Multiple Time Scale Approach to the Stability of External Cavity Modes in the Lang\u2013Kobayashi System Using the Limit of Large Delay"],"prefix":"10.1137","volume":"9","author":[{"given":"Serhiy","family":"Yanchuk","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthias","family":"Wolfrum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,6,2]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.53.4429"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"R. L. Davidchack, Y.C. Lai, A. Gavrielides, and V. Kovanis,\n                      Regular dynamics of low-frequency fluctuations in external cavity semiconductor lasers\n                      , Phys. Rev. E, 63 (2001), paper 056206.","DOI":"10.1103\/PhysRevE.63.056206"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000397"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"K. Green,\n                      Stability near threshold in a semiconductor laser subject to optical feedback: A bifurcation analysis of the Lang-Kobayashi equations\n                      , Phys. Rev. E, 79 (2009), paper 036210.","DOI":"10.1103\/PhysRevE.79.036210"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux, and T. Erneux,\n                      Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback\n                      , Phys. Rev. E, 66 (2002), paper 046216.","DOI":"10.1103\/PhysRevE.66.046216"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.58.R2672"},{"key":"R7","first-page":"443","volume":"38","author":"Kashchenko S. A.","year":"1998","journal-title":"Comput. Math. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0965-5425","issn-type":"print"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02679755"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"B. Krauskopf and D. Lenstra, eds.\n                      Fundamental Issues of Nonlinear Laser Dynamics\n                      , AIP Conf. Proc. 548, AIP, New York, 2000.","DOI":"10.1063\/1.1337756"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1109\/JQE.1980.1070479"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.52.R3436"},{"key":"R12","unstructured":"M. Lichtner, M. Wolfrum, and S. Yanchuk,\n                      The spectrum of delay differential equations with large delay\n                      , SIAM J. Math. Anal., submitted."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1109\/3.119502"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.59.5400"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"M. Nizette,\n                      Stability of square oscillations in a delayed-feedback system\n                      , Phys. Rev. E, 70 (2004), paper 056204.","DOI":"10.1103\/PhysRevE.70.056204"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"M. Peil, T. Heil, I. Fischer, and W. Els\u00e4\u00dfer,\n                      Synchronization of chaotic semiconductor laser systems: A vectorial coupling-dependent scenario\n                      , Phys. Rev. Lett., 88 (2002), paper 174101.","DOI":"10.1103\/PhysRevLett.88.174101"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"D. Pieroux, T. Erneux, T. Luzyanina, and K. Engelborghs,\n                      Interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback\n                      , Phys. Rev. E, 63 (2001), paper 036211.","DOI":"10.1103\/PhysRevE.63.036211"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"J. Piprek, ed.\n                      Optoelectronic Devices. Advanced Simulation and Analysis\n                      , Springer-Verlag, New York, 2005.","DOI":"10.1007\/b138826"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2006.09.032"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAB.10.000130"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAB.10.000145"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127407017914"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1109\/JQE.1984.1072508"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"V. Z. Tronciu, H.J. W\u00fcnsche, M. Wolfrum, and M. Radziunas,\n                      Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation\n                      , Phys. Rev. E, 73 (2006), paper 046205.","DOI":"10.1103\/PhysRevE.73.046205"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/S0030-4018(02)01824-2"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"M. Wolfrum and S. Yanchuk,\n                      Eckhaus instability in systems with large delay\n                      , Phys. Rev. Lett., 96 (2006), paper 220201.","DOI":"10.1103\/PhysRevLett.96.220201"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1002\/mma.584"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"S. Yanchuk, K. R. Schneider, and L. Recke,\n                      Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit\n                      , Phys. Rev. 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