March  2015, 20(2): 519-586. doi: 10.3934/dcdsb.2015.20.519

Mode structure of a semiconductor laser with feedback from two external filters

1. 

Mathematics Research Institute, CEMPS, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom

2. 

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142

3. 

Department of Applied Mathematics, University College Cork, Western Gateway Building, Cork, Ireland

Received  April 2014 Revised  May 2014 Published  January 2015

We investigate the solution structure and stability of a semiconductor laser receiving time-delayed and frequency-filtered optical feedback from two external filters. This system is referred to as the 2FOF laser, and it has been used as pump laser in optical telecommunication and as light source in sensor applications. The underlying idea is that the two filter loops provide a means of stabilizing and controling the laser output. The mathematical model takes the form of delay differential equations for the (real-valued) population inversion of the laser active medium and for the (complex-valued) electric fields of the laser cavity and of the two filters. There are two time delays, which are the travel times of the light from the laser to each of the filters and back.
    Our analysis of the 2FOF laser focuses on the basic solutions, known as continuous waves or external filtered modes (EFMs), which correspond to laser output with steady amplitude and frequency. Specifically, we consider the EFM-surface in the $(\omega_s,\,N_s,\,dC_p)$-space of steady frequency $\omega_s$, the corresponding steady population inversion $N_s$, and the feedback phase difference $dC_p$. This surface emerges as the natural object for the study of the 2FOF laser because it conveniently catalogues information about available frequency ranges of the EFMs. We identify five transitions, through four different singularities and a cubic tangency, which change the type of the EFM-surface locally and determine the EFM-surface bifurcation diagram in the $(\Delta_1,\,\Delta_2)$-plane. In this way, we classify the possible types of the EFM-surface, which consist of a combination of bands (covering the entire $dC_p$-range) and islands (covering only a finite range of $dC_p$).
    We also investigate the stability of the EFMs, where we focus on saddle-node and Hopf bifurcation curves that bound regions of stable EFMs on the EFM-surface. It is shown how these stability regions evolve when parameters are changed along a chosen path in the $(\Delta_1,\,\Delta_2)$-plane. From a viewpoint of practical interests, we find various bands and islands of stability on the EFM-surface that may be accessible experimentally.
    Beyond their relevance for the 2FOF laser system, the results presented here also showcase how advanced tools from bifurcation theory and singularity theory can be employed to uncover and represent the complex solution structure of a delay differential equation model that depends on a considerable number of input parameters, including two time delays.
Citation: Piotr Słowiński, Bernd Krauskopf, Sebastian Wieczorek. Mode structure of a semiconductor laser with feedback from two external filters. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 519-586. doi: 10.3934/dcdsb.2015.20.519
References:
[1]

V. I. Arnold, The Theory of Singularities and Its Applications, Accademia Nazionale dei Lincei, Pisa, 1991.

[2]

A. Asok, About the cover: The mathematics imagery of Lun-Yi Tsai, Bulletin of the AMS, 47 (2010), 695-696. doi: 10.1090/S0273-0979-2010-01316-1.

[3]

J. Chen, X. Wu, J. Ge, A. Hermerschmidt and H. J. Eichler, Broad-are laser diode with 0.02 nm bandwidth and diffraction limited output due to double external cavity feedback, Appl. Phys. Lett., 85 (2004), 525-527. doi: 10.1063/1.1774248.

[4]

M. Chi, N-S. Bogh, B. Thestrup and P. M. Petersen, Improvement of the beam quality of a broad-area diode laser using double feedback from two external mirrors, Appl. Phys. Lett., 85 (2004), 1107-1109. doi: 10.1063/1.1783017.

[5]

J. S. Cohen, R. R. Drenten and B. H. Verbeeck, The effect of optical feedback on the relaxation oscillation in semiconductor lasers, IEEE J. Quantum Electron., 24 (1988), 1989-1995. doi: 10.1109/3.8533.

[6]

B. Dahmani, L. Hollberg and R. Drullinger, Frequency stabilization of semiconductor lasers by resonant optical feedback, Opt. Lett., 12 (1987), 876-878. doi: 10.1364/OL.12.000876.

[7]

D. H. DeTienne, G. R. Gray, G. P. Agrawal and D. Lenstra, Semiconductor laser dynamics for feedback from a finite-penetration-depth phase-conjugate mirror, IEEE J. Quantum Electron., 33 (1997), 838-844. doi: 10.1109/3.572159.

[8]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[9]

B. Docter, J. Pozo, S. Beri, I. V. Ermakov, J. Danckaert, M. K. Smit and F. Karouta, Discretely tunable laser based on filtered feedback for telecommunication applications, IEEE J. Sel. Topics Qunat. Electronics, 16 (2010), 1405-1412. doi: 10.1109/JSTQE.2009.2038072.

[10]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math. 1480, Springer-Verlag, Berlin, 1991.

[11]

K. Engelborghs, T. Luyzanina and G. Samaey, DDE-Biftool v. 2.00 User Manual: A Matlab Package for Bifurcation Analysis of Delay Differential Equations, Tech. report tw-330, Department of Computer Science, K. U. Leuven, 2001.

[12]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, 28 (2002), 1-21. doi: 10.1145/513001.513002.

[13]

I. V. Ermakov, V. Z. Tronciu, P. Colet and C. R. Mirasso, Controlling the unstable emission of a semiconductor laser subject to conventional optical feedback with a filtered feedback branch, Optics Express, 17 (2009), 8749-8755. doi: 10.1364/OE.17.008749.

[14]

T. Erneux, A. Gavrielides, K. Green and B. Krauskopf, External cavity modes of semiconductor lasers with phase-conjugate feedback, Phys. Rev. E, 68 (2003), 066205. doi: 10.1103/PhysRevE.68.066205.

[15]

T. Erneux, M. Yousefi and D. Lenstra, The Injection Laser Limit of Lasers Subject to Filtered Optical Feedback, Proc. European Quantum Electronics Conf., 2003.

[16]

H. Erzgräber and B. Krauskopf, Dynamics of a filtered-feedback laser: Influence of the filter width, Optics Letters, 32 (2007), 2441-2443. doi: 10.1364/OL.32.002441.

[17]

H. Erzgräber, B. Krauskopf and D. Lenstra, Bifurcation analysis of a semiconductor laser with filtered optical feedback, SIAM J. Appl. Dyn. Sys., 6 (2007), 1-28. doi: 10.1137/060656656.

[18]

H. Erzgräber, B. Krauskopf, D. Lenstra, A. P. A. Fischer and G. Vemuri, Frequency versus relaxation oscillations in semiconductor laser with coherent filtered optical feedback, Phys. Rev. E, 73 (2006), 055201(R). doi: 10.1103/PhysRevE.73.055201.

[19]

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. A. Fischer and G. Vemuri, Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory, Phys. Rev. E, 76 (2007), 026212. doi: 10.1103/PhysRevE.76.026212.

[20]

B. Farias, T. P. de Silans, M. Chevrollier and M. Oriá, Frequency bistability of a semiconductor laser under a frequency-dependent feedback, Phys. Rev. Lett., 94 (2005), 173902. doi: 10.1103/PhysRevLett.94.173902.

[21]

S. G. Fischer, M. Ahmed, T. Okamoto, W. Ishimori and M. Yamada, An improved analysis of semiconductor laser dynamics under strong optical feedback, IEEE J. Quantum Electron., 9 (2003), 1265-1274.

[22]

A. P. A. Fischer, O. Andersen, M. Yousefi, S. Stolte and D. Lenstra, Experimental and theoretical study of semiconductor laser dynamics due to filtered optical feedback, IEEE J. Quantum Electron., 36 (2000), 375-384.

[23]

A. P. A. Fischer, M. Yousefi, D. Lenstra, M. Carter and G. Vemuri, Experimental and theoretical study of semiconductor laser dynamics due to filtered optical feedback, IEEE J. Sel. Top. Quantum Electron., 10 (2004), 944-954. doi: 10.1109/JSTQE.2004.835997.

[24]

A. Fischer, M. Yousefi, D. Lenstra, M. Carter and G. Vemuri, Filtered optical feedback induced frequency dynamics in semiconductor lasers, Phys. Rev. Lett., 92 (2004), 023901. doi: 10.1103/PhysRevLett.92.023901.

[25]

I. Fischer, G. H. M. van Tartwijk, A. M. Levine, W. Elsässer, E. Göbel and D. Lenstra, Fast pulsing and chaotic itinerancy with a drift in the coherence collapse of semiconductor lasers, Phys. Rev. Lett., 76 (1996), 220-223. doi: 10.1103/PhysRevLett.76.220.

[26]

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, volume 1, Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5034-0.

[27]

K. Green and B. Krauskopf, Bifurcation analysis of a semiconductor laser subject to non-instantaneous phase-conjugate feedback, Opt. Commun., 231 (2004), 383-393. doi: 10.1016/j.optcom.2003.12.026.

[28]

K. Green and B. Krauskopf, Mode structure of a semiconductor laser subject to filtered optical feedback, Opt. Commun., 258 (2006), 243-255. doi: 10.1016/j.optcom.2005.08.005.

[29]

B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux and T. Erneux, Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback, Phys. Rev. E, 66 (2002), 046216. doi: 10.1103/PhysRevE.66.046216.

[30]

T. Heil, I. Fischer, W. Elsäßer, B. Krauskopf, K. Green and A. Gavrielides, Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms, Phys. Rev. E, 67 (2003), 066214, 11pp. doi: 10.1103/PhysRevE.67.066214.

[31]

G. Hek and V. Rottschäfer, Semiconductor laser with filtered optical feedback: From optical injection to conventional feedback, IMA J. Appl. Math., 72 (2007), 420-450. doi: 10.1093/imamat/hxm019.

[32]

C. Henry, Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron., 18 (1982), 259-264. doi: 10.1109/JQE.1982.1071522.

[33]

D. M. Kane and K. A. Shore (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, Wiley, 2005. doi: 10.1002/0470856211.

[34]

B. Krauskopf, Bifurcation analysis of lasers with delay, in D. Kane and K. Shore (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, Wiley, (2005), 147-183.

[35]

B. Krauskopf, H. Erzgräber and D. Lenstra, Dynamics of semiconductor lasers with filtered optical feedback, in D. Lenstra, M. Pessa and I. H. White (Eds.), Semiconductor Lasers and Laser Dynamics II, Proceedings of SPIE, 6184, 2006, 61840V.

[36]

B. Krauskopf and C. Rousseau, Codimension-three unfoldings of reflectionally symmetric planar vector fields, Nonlinearity, 10 (1997), 1115-1150. doi: 10.1088/0951-7715/10/5/007.

[37]

B. Krauskopf, G. R. Gray and D. Lenstra, Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations, Phys. Rev. E, 58 (1998), 7190-7197. doi: 10.1103/PhysRevE.58.7190.

[38]

B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, 548, AIP, Melville, New York, 2000.

[39]

B. Krauskopf, H. Osinga and J. Galán-Vioque (Eds.), Numerical Continuation Methods for Dynamical Systems: Pathfollowing and Boundary Value Problems, Springer, Dordrecht, 2007. doi: 10.1007/978-1-4020-6356-5.

[40]

B. Krauskopf, G. H. M. van Tartwijk and G. R. Gray, Symmetry properties of lasers subject to optical feedback, Opt. Commun., 177 (2006), 347-353. doi: 10.1016/S0030-4018(00)00574-5.

[41]

B. Krauskopf, C. M. Lee and H. Osinga, Codimension-one tangency bifurcations of global Poincaré maps of four-dimensional vector fields, Nonlinearity, 22 (2009), 1091-1121. doi: 10.1088/0951-7715/22/5/008.

[42]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[43]

R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection laser properties, IEEE J. Quantum Electron., 16 (1980), 347-355. doi: 10.1109/JQE.1980.1070479.

[44]

L. Larger, P.-A. Lacourt, S. Poinsot and M. Hanna, From flow to map in an experimental high-dimensional electro-optic nonlinear delay oscillator, Phys. Rev. Lett., 95 (2005), 043903. doi: 10.1103/PhysRevLett.95.043903.

[45]

D. Lenstra, M. Van Vaalen and B. Jaskorzyńska, On the theory of a single-mode laser with weak optical feedback, Physica B+C, 125 (1984), 255-264. doi: 10.1016/0378-4363(84)90009-3.

[46]

D. Lenstra and M. Yousefi, Theory of delayed optical feedback in lasers, in B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, AIP, Melville, New York, 548 (2000), 87-111.

[47]

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra and T. Erneux, Diode lasers with optical feedback: Stability of the maximum gain mode, Phys. Rev. A, 52 (1995), R3436-R3439. doi: 10.1103/PhysRevA.52.R3436.

[48]

K. Lüdge (Ed.), Nonlinear Laser Dynamics: From Quantum Dots to Cryptography, Wiley-VCH, 2012.

[49]

B. E. Martínez-Zérega, R. Jaimes-Reategui, A. N. Pisarchik and J. M. Liu, Experimental study of self-oscillation frequency in a semiconductor laser with optical injection, J. Phys.: Conf. Ser., 23 (2005), 62-67.

[50]

J. Mørk, B. Tromborg and J. Mark, Chaos in semiconductor lasers with optical feedback: Theory and experiment, IEEE J. Quantum Electron., 28 (1992), 93-108. doi: 10.1109/3.119502.

[51]

A. Naumenko, P. Besnard, N. Loiko, G. Ughetto and J. C. Bertreux, Characteristics of a semiconductor laser coupled with a fiber Bragg grating with arbitrary amount of feedback, IEEE J. Quantum Electron., 39 (2003), 1216-1228. doi: 10.1109/JQE.2003.817669.

[52]

A. Naumenko, N. A. Loiko and T. Ackemann, Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation, Phys. Rev. A, 76 (2007), 023802. doi: 10.1103/PhysRevA.76.023802.

[53]

M. Nizette and T. Erneux, Optical frequency dynamics and relaxation oscillations of a semiconductor laser subject to filtered optical feedback, in D. Lenstra, M. Pessa and I. H. White (Eds.), Semiconductor Lasers and Laser Dynamics II, Proceedings of SPIE, 6184 (2006), p6184.

[54]

Oclaro, data sheets for grating-stabilized 980nm pump laser modules, LC94 300mW, PLC94 Rev 3.1 August 2009; LC96 600mW, PLC96 Rev 4.1 August 2009.

[55]

V. Pal, J. S. Suelzer, A. Prasad, G. Vemuri and R. Ghosh, Semiconductor laser dynamics with two filtered optical feedbacks, IEEE J. Quantum Electron., 49 (2013), 340-349. doi: 10.1109/JQE.2013.2244559.

[56]

T. Poston and I. Stewart, Catastrophe Theory and its Applications, Pitman Publishing Ltd., 1978.

[57]

F. Rogister, P. Mégret, O. Deparis, M. Blondel and T. Erneux, Suppression of low-frequency fluctuations and stabilization of a semiconductor laser subjected to optical feedback from a double cavity: theoretical results, Opt. Lett., 24 (1999), 1218-1220. doi: 10.1364/OL.24.001218.

[58]

F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis and M. Blondel, Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external-cavity laser diode, Opt. Lett., 25 (2000), 808-810. doi: 10.1364/OL.25.000808.

[59]

D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in B. Krauskopf, H. Osinga and J. Galán-Vioque (Eds.), Numerical Continuation Methods for Dynamical Systems, Pathfollowing and boundary value problems, Springer, Dordrecht, (2007), 359-399. doi: 10.1007/978-1-4020-6356-5_12.

[60]

V. Rottschäfer and B. Krauskopf, The ECM-backbone of the Lang-Kobayashi equations: A geometric picture, Int. J. Bifurcation and Chaos, 17 (2007), 1575-1588. doi: 10.1142/S0218127407017914.

[61]

T. Sano, Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback, Phys. Rev. A, 50 (1994), 2719-2726. doi: 10.1103/PhysRevA.50.2719.

[62]

J. Sonksen, M. Ahmad, N. Storch, H. Krause, S. Blom, A. Pötzl and H. Hillmer, Controlling and tuning the emission of a semiconductor optical amplifier for sensor application by means of fiber Bragg gratings, Proceedings of the 8th WSEAS Int. Conf. on Microelectronic, Nanoelectronics, Optoelectronics, (2009), 59-62.

[63]

J. Sonksen, M. Ahmad, N. Storch, H. Krause, S. Blom, A. Pötzl and H. Hillmer, Aufbau eines faserbasierten Laserresonators mit zwei Fabri-Pérot Kavitäten und einer gemeinsamen aktiven Zone für Sensonrikanwendungen, DGaO Proceedings 2008; ISSN: 1614-8436.

[64]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470. doi: 10.1103/RevModPhys.85.421.

[65]

G. H. M. Van Tartwijk and D. Lenstra, Semiconductor lasers with optical injection and feedback, Quantum Semiclass. Opt., 7 (1995), 87-143. doi: 10.1088/1355-5111/7/2/003.

[66]

B. Tromborg, J. Osmundsen and H. Olesen, Stability analysis for a semiconductor laser in an external cavity, IEEE J Quantum Electron., 20 (1984), 1023-1032. doi: 10.1109/JQE.1984.1072508.

[67]

V. Z. Tronciu, C. R. Mirasso and P. Colet, Chaos-based communications using semiconductor lasers subject to feedback from an integrated double cavity, J. Phys. B: At. Mol. Opt. Phys, 41 (2008), 155401. doi: 10.1088/0953-4075/41/15/155401.

[68]

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum and M. Radziunas, Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation, Phys. Rev. E, 73 (2006), 046205. doi: 10.1103/PhysRevE.73.046205.

[69]

S. Valling, B. Krauskopf, T. Fordell and A. Lindberg, Experimental bifurcation diagram of a solid state laser with optical injection, Opt. Commun., 271 (2007), 532-542. doi: 10.1016/j.optcom.2006.10.086.

[70]

H. Venghaus, ed., Wavelength Filters in Fibre Optics, Springer, Berlin, 2006.

[71]

S. M. Verduyn Lunel and B. Krauskopf, The Mathematics of Delay Equations with an Application to the Lang-Kobayashi Equations, in B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, AIP, Melville, New York, 548 (2000), 66-87. doi: 10.1063/1.1337759.

[72]

S. Wieczorek, B. Krauskopf and D. Lenstra, A unifying view of bifurcations in a semiconductor laser subject to optical injection, Opt. Commun., 172 (1999), 279-295. doi: 10.1016/S0030-4018(99)00603-3.

[73]

S. Wieczorek, B. Krauskopf and D. Lenstra, Sudden chaotic transitions in an optically injected semiconductor laser, Opt. Lett., 26 (2001), 816-818. doi: 10.1364/OL.26.000816.

[74]

S. Wieczorek, B. Krauskopf and D. Lenstra, Unnested islands of period doublings in an injected semiconductor laser, Phys. Rev. E, 64 (2001), 056204, 9pp. doi: 10.1103/PhysRevE.64.056204.

[75]

S. Wieczorek, T. B. Simpson, B. Krauskopf and D. Lenstra, Bifurcation transitions in an optically injected diode laser: theory and experiment, Opt. Commun., 215 (2003), 125-134. doi: 10.1016/S0030-4018(02)02191-0.

[76]

S. Wieczorek, B. Krauskopf, T. B. Simpson and D. Lenstra, The dynamical complexity of optically injected semiconductor lasers, Physics Reports, 416 (2005), 1-128. doi: 10.1016/j.physrep.2005.06.003.

[77]

J. Ye, H. Li and J. G. McInerney, Period-doubling route to chaos in a semiconductor laser with weak optical feedback, Phys. Rev. A, 47 (1993), 2249-2252. doi: 10.1103/PhysRevA.47.2249.

[78]

M. Yousefi and D. Lenstra, Dynamical behaviour of a semiconductor laser with filtered external optical feedback, IEEE J. Quantum Electron., 35 (1999), 970-976. doi: 10.1109/3.766841.

[79]

M. Yousefi, D. Lenstra, A. Fischer and G. Vemuri, Simulations of a semiconductor laser with filtered optical feedback: deterministic dynamics and transitions to chaos, in P. Blood, M. Osinski and Y. Arakawa (Eds.), "Physics and Simulation of Optoelectronic Devices X,'' Proceedings of SPIE, 4646 (2002), 447-452. doi: 10.1117/12.470547.

[80]

M. Yousefi, D. Lenstra and G. Vemuri, Nonlinear dynamics of a semiconductor laser with filtered optical feedback and the influence of noise, Phys. Rev. E, 67 (2003), 046213. doi: 10.1103/PhysRevE.67.046213.

[81]

M. Yousefi, D. Lenstra, G. Vemuri and A. P. A. Fischer, Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback, IEEE Proc. Optoelectron., 148 (2001), 233-237. doi: 10.1049/ip-opt:20010721.

show all references

References:
[1]

V. I. Arnold, The Theory of Singularities and Its Applications, Accademia Nazionale dei Lincei, Pisa, 1991.

[2]

A. Asok, About the cover: The mathematics imagery of Lun-Yi Tsai, Bulletin of the AMS, 47 (2010), 695-696. doi: 10.1090/S0273-0979-2010-01316-1.

[3]

J. Chen, X. Wu, J. Ge, A. Hermerschmidt and H. J. Eichler, Broad-are laser diode with 0.02 nm bandwidth and diffraction limited output due to double external cavity feedback, Appl. Phys. Lett., 85 (2004), 525-527. doi: 10.1063/1.1774248.

[4]

M. Chi, N-S. Bogh, B. Thestrup and P. M. Petersen, Improvement of the beam quality of a broad-area diode laser using double feedback from two external mirrors, Appl. Phys. Lett., 85 (2004), 1107-1109. doi: 10.1063/1.1783017.

[5]

J. S. Cohen, R. R. Drenten and B. H. Verbeeck, The effect of optical feedback on the relaxation oscillation in semiconductor lasers, IEEE J. Quantum Electron., 24 (1988), 1989-1995. doi: 10.1109/3.8533.

[6]

B. Dahmani, L. Hollberg and R. Drullinger, Frequency stabilization of semiconductor lasers by resonant optical feedback, Opt. Lett., 12 (1987), 876-878. doi: 10.1364/OL.12.000876.

[7]

D. H. DeTienne, G. R. Gray, G. P. Agrawal and D. Lenstra, Semiconductor laser dynamics for feedback from a finite-penetration-depth phase-conjugate mirror, IEEE J. Quantum Electron., 33 (1997), 838-844. doi: 10.1109/3.572159.

[8]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[9]

B. Docter, J. Pozo, S. Beri, I. V. Ermakov, J. Danckaert, M. K. Smit and F. Karouta, Discretely tunable laser based on filtered feedback for telecommunication applications, IEEE J. Sel. Topics Qunat. Electronics, 16 (2010), 1405-1412. doi: 10.1109/JSTQE.2009.2038072.

[10]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math. 1480, Springer-Verlag, Berlin, 1991.

[11]

K. Engelborghs, T. Luyzanina and G. Samaey, DDE-Biftool v. 2.00 User Manual: A Matlab Package for Bifurcation Analysis of Delay Differential Equations, Tech. report tw-330, Department of Computer Science, K. U. Leuven, 2001.

[12]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, 28 (2002), 1-21. doi: 10.1145/513001.513002.

[13]

I. V. Ermakov, V. Z. Tronciu, P. Colet and C. R. Mirasso, Controlling the unstable emission of a semiconductor laser subject to conventional optical feedback with a filtered feedback branch, Optics Express, 17 (2009), 8749-8755. doi: 10.1364/OE.17.008749.

[14]

T. Erneux, A. Gavrielides, K. Green and B. Krauskopf, External cavity modes of semiconductor lasers with phase-conjugate feedback, Phys. Rev. E, 68 (2003), 066205. doi: 10.1103/PhysRevE.68.066205.

[15]

T. Erneux, M. Yousefi and D. Lenstra, The Injection Laser Limit of Lasers Subject to Filtered Optical Feedback, Proc. European Quantum Electronics Conf., 2003.

[16]

H. Erzgräber and B. Krauskopf, Dynamics of a filtered-feedback laser: Influence of the filter width, Optics Letters, 32 (2007), 2441-2443. doi: 10.1364/OL.32.002441.

[17]

H. Erzgräber, B. Krauskopf and D. Lenstra, Bifurcation analysis of a semiconductor laser with filtered optical feedback, SIAM J. Appl. Dyn. Sys., 6 (2007), 1-28. doi: 10.1137/060656656.

[18]

H. Erzgräber, B. Krauskopf, D. Lenstra, A. P. A. Fischer and G. Vemuri, Frequency versus relaxation oscillations in semiconductor laser with coherent filtered optical feedback, Phys. Rev. E, 73 (2006), 055201(R). doi: 10.1103/PhysRevE.73.055201.

[19]

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. A. Fischer and G. Vemuri, Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory, Phys. Rev. E, 76 (2007), 026212. doi: 10.1103/PhysRevE.76.026212.

[20]

B. Farias, T. P. de Silans, M. Chevrollier and M. Oriá, Frequency bistability of a semiconductor laser under a frequency-dependent feedback, Phys. Rev. Lett., 94 (2005), 173902. doi: 10.1103/PhysRevLett.94.173902.

[21]

S. G. Fischer, M. Ahmed, T. Okamoto, W. Ishimori and M. Yamada, An improved analysis of semiconductor laser dynamics under strong optical feedback, IEEE J. Quantum Electron., 9 (2003), 1265-1274.

[22]

A. P. A. Fischer, O. Andersen, M. Yousefi, S. Stolte and D. Lenstra, Experimental and theoretical study of semiconductor laser dynamics due to filtered optical feedback, IEEE J. Quantum Electron., 36 (2000), 375-384.

[23]

A. P. A. Fischer, M. Yousefi, D. Lenstra, M. Carter and G. Vemuri, Experimental and theoretical study of semiconductor laser dynamics due to filtered optical feedback, IEEE J. Sel. Top. Quantum Electron., 10 (2004), 944-954. doi: 10.1109/JSTQE.2004.835997.

[24]

A. Fischer, M. Yousefi, D. Lenstra, M. Carter and G. Vemuri, Filtered optical feedback induced frequency dynamics in semiconductor lasers, Phys. Rev. Lett., 92 (2004), 023901. doi: 10.1103/PhysRevLett.92.023901.

[25]

I. Fischer, G. H. M. van Tartwijk, A. M. Levine, W. Elsässer, E. Göbel and D. Lenstra, Fast pulsing and chaotic itinerancy with a drift in the coherence collapse of semiconductor lasers, Phys. Rev. Lett., 76 (1996), 220-223. doi: 10.1103/PhysRevLett.76.220.

[26]

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, volume 1, Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5034-0.

[27]

K. Green and B. Krauskopf, Bifurcation analysis of a semiconductor laser subject to non-instantaneous phase-conjugate feedback, Opt. Commun., 231 (2004), 383-393. doi: 10.1016/j.optcom.2003.12.026.

[28]

K. Green and B. Krauskopf, Mode structure of a semiconductor laser subject to filtered optical feedback, Opt. Commun., 258 (2006), 243-255. doi: 10.1016/j.optcom.2005.08.005.

[29]

B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux and T. Erneux, Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback, Phys. Rev. E, 66 (2002), 046216. doi: 10.1103/PhysRevE.66.046216.

[30]

T. Heil, I. Fischer, W. Elsäßer, B. Krauskopf, K. Green and A. Gavrielides, Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms, Phys. Rev. E, 67 (2003), 066214, 11pp. doi: 10.1103/PhysRevE.67.066214.

[31]

G. Hek and V. Rottschäfer, Semiconductor laser with filtered optical feedback: From optical injection to conventional feedback, IMA J. Appl. Math., 72 (2007), 420-450. doi: 10.1093/imamat/hxm019.

[32]

C. Henry, Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron., 18 (1982), 259-264. doi: 10.1109/JQE.1982.1071522.

[33]

D. M. Kane and K. A. Shore (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, Wiley, 2005. doi: 10.1002/0470856211.

[34]

B. Krauskopf, Bifurcation analysis of lasers with delay, in D. Kane and K. Shore (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, Wiley, (2005), 147-183.

[35]

B. Krauskopf, H. Erzgräber and D. Lenstra, Dynamics of semiconductor lasers with filtered optical feedback, in D. Lenstra, M. Pessa and I. H. White (Eds.), Semiconductor Lasers and Laser Dynamics II, Proceedings of SPIE, 6184, 2006, 61840V.

[36]

B. Krauskopf and C. Rousseau, Codimension-three unfoldings of reflectionally symmetric planar vector fields, Nonlinearity, 10 (1997), 1115-1150. doi: 10.1088/0951-7715/10/5/007.

[37]

B. Krauskopf, G. R. Gray and D. Lenstra, Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations, Phys. Rev. E, 58 (1998), 7190-7197. doi: 10.1103/PhysRevE.58.7190.

[38]

B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, 548, AIP, Melville, New York, 2000.

[39]

B. Krauskopf, H. Osinga and J. Galán-Vioque (Eds.), Numerical Continuation Methods for Dynamical Systems: Pathfollowing and Boundary Value Problems, Springer, Dordrecht, 2007. doi: 10.1007/978-1-4020-6356-5.

[40]

B. Krauskopf, G. H. M. van Tartwijk and G. R. Gray, Symmetry properties of lasers subject to optical feedback, Opt. Commun., 177 (2006), 347-353. doi: 10.1016/S0030-4018(00)00574-5.

[41]

B. Krauskopf, C. M. Lee and H. Osinga, Codimension-one tangency bifurcations of global Poincaré maps of four-dimensional vector fields, Nonlinearity, 22 (2009), 1091-1121. doi: 10.1088/0951-7715/22/5/008.

[42]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[43]

R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection laser properties, IEEE J. Quantum Electron., 16 (1980), 347-355. doi: 10.1109/JQE.1980.1070479.

[44]

L. Larger, P.-A. Lacourt, S. Poinsot and M. Hanna, From flow to map in an experimental high-dimensional electro-optic nonlinear delay oscillator, Phys. Rev. Lett., 95 (2005), 043903. doi: 10.1103/PhysRevLett.95.043903.

[45]

D. Lenstra, M. Van Vaalen and B. Jaskorzyńska, On the theory of a single-mode laser with weak optical feedback, Physica B+C, 125 (1984), 255-264. doi: 10.1016/0378-4363(84)90009-3.

[46]

D. Lenstra and M. Yousefi, Theory of delayed optical feedback in lasers, in B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, AIP, Melville, New York, 548 (2000), 87-111.

[47]

A. M. Levine, G. H. M. van Tartwijk, D. Lenstra and T. Erneux, Diode lasers with optical feedback: Stability of the maximum gain mode, Phys. Rev. A, 52 (1995), R3436-R3439. doi: 10.1103/PhysRevA.52.R3436.

[48]

K. Lüdge (Ed.), Nonlinear Laser Dynamics: From Quantum Dots to Cryptography, Wiley-VCH, 2012.

[49]

B. E. Martínez-Zérega, R. Jaimes-Reategui, A. N. Pisarchik and J. M. Liu, Experimental study of self-oscillation frequency in a semiconductor laser with optical injection, J. Phys.: Conf. Ser., 23 (2005), 62-67.

[50]

J. Mørk, B. Tromborg and J. Mark, Chaos in semiconductor lasers with optical feedback: Theory and experiment, IEEE J. Quantum Electron., 28 (1992), 93-108. doi: 10.1109/3.119502.

[51]

A. Naumenko, P. Besnard, N. Loiko, G. Ughetto and J. C. Bertreux, Characteristics of a semiconductor laser coupled with a fiber Bragg grating with arbitrary amount of feedback, IEEE J. Quantum Electron., 39 (2003), 1216-1228. doi: 10.1109/JQE.2003.817669.

[52]

A. Naumenko, N. A. Loiko and T. Ackemann, Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation, Phys. Rev. A, 76 (2007), 023802. doi: 10.1103/PhysRevA.76.023802.

[53]

M. Nizette and T. Erneux, Optical frequency dynamics and relaxation oscillations of a semiconductor laser subject to filtered optical feedback, in D. Lenstra, M. Pessa and I. H. White (Eds.), Semiconductor Lasers and Laser Dynamics II, Proceedings of SPIE, 6184 (2006), p6184.

[54]

Oclaro, data sheets for grating-stabilized 980nm pump laser modules, LC94 300mW, PLC94 Rev 3.1 August 2009; LC96 600mW, PLC96 Rev 4.1 August 2009.

[55]

V. Pal, J. S. Suelzer, A. Prasad, G. Vemuri and R. Ghosh, Semiconductor laser dynamics with two filtered optical feedbacks, IEEE J. Quantum Electron., 49 (2013), 340-349. doi: 10.1109/JQE.2013.2244559.

[56]

T. Poston and I. Stewart, Catastrophe Theory and its Applications, Pitman Publishing Ltd., 1978.

[57]

F. Rogister, P. Mégret, O. Deparis, M. Blondel and T. Erneux, Suppression of low-frequency fluctuations and stabilization of a semiconductor laser subjected to optical feedback from a double cavity: theoretical results, Opt. Lett., 24 (1999), 1218-1220. doi: 10.1364/OL.24.001218.

[58]

F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis and M. Blondel, Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external-cavity laser diode, Opt. Lett., 25 (2000), 808-810. doi: 10.1364/OL.25.000808.

[59]

D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in B. Krauskopf, H. Osinga and J. Galán-Vioque (Eds.), Numerical Continuation Methods for Dynamical Systems, Pathfollowing and boundary value problems, Springer, Dordrecht, (2007), 359-399. doi: 10.1007/978-1-4020-6356-5_12.

[60]

V. Rottschäfer and B. Krauskopf, The ECM-backbone of the Lang-Kobayashi equations: A geometric picture, Int. J. Bifurcation and Chaos, 17 (2007), 1575-1588. doi: 10.1142/S0218127407017914.

[61]

T. Sano, Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback, Phys. Rev. A, 50 (1994), 2719-2726. doi: 10.1103/PhysRevA.50.2719.

[62]

J. Sonksen, M. Ahmad, N. Storch, H. Krause, S. Blom, A. Pötzl and H. Hillmer, Controlling and tuning the emission of a semiconductor optical amplifier for sensor application by means of fiber Bragg gratings, Proceedings of the 8th WSEAS Int. Conf. on Microelectronic, Nanoelectronics, Optoelectronics, (2009), 59-62.

[63]

J. Sonksen, M. Ahmad, N. Storch, H. Krause, S. Blom, A. Pötzl and H. Hillmer, Aufbau eines faserbasierten Laserresonators mit zwei Fabri-Pérot Kavitäten und einer gemeinsamen aktiven Zone für Sensonrikanwendungen, DGaO Proceedings 2008; ISSN: 1614-8436.

[64]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470. doi: 10.1103/RevModPhys.85.421.

[65]

G. H. M. Van Tartwijk and D. Lenstra, Semiconductor lasers with optical injection and feedback, Quantum Semiclass. Opt., 7 (1995), 87-143. doi: 10.1088/1355-5111/7/2/003.

[66]

B. Tromborg, J. Osmundsen and H. Olesen, Stability analysis for a semiconductor laser in an external cavity, IEEE J Quantum Electron., 20 (1984), 1023-1032. doi: 10.1109/JQE.1984.1072508.

[67]

V. Z. Tronciu, C. R. Mirasso and P. Colet, Chaos-based communications using semiconductor lasers subject to feedback from an integrated double cavity, J. Phys. B: At. Mol. Opt. Phys, 41 (2008), 155401. doi: 10.1088/0953-4075/41/15/155401.

[68]

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum and M. Radziunas, Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation, Phys. Rev. E, 73 (2006), 046205. doi: 10.1103/PhysRevE.73.046205.

[69]

S. Valling, B. Krauskopf, T. Fordell and A. Lindberg, Experimental bifurcation diagram of a solid state laser with optical injection, Opt. Commun., 271 (2007), 532-542. doi: 10.1016/j.optcom.2006.10.086.

[70]

H. Venghaus, ed., Wavelength Filters in Fibre Optics, Springer, Berlin, 2006.

[71]

S. M. Verduyn Lunel and B. Krauskopf, The Mathematics of Delay Equations with an Application to the Lang-Kobayashi Equations, in B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, AIP, Melville, New York, 548 (2000), 66-87. doi: 10.1063/1.1337759.

[72]

S. Wieczorek, B. Krauskopf and D. Lenstra, A unifying view of bifurcations in a semiconductor laser subject to optical injection, Opt. Commun., 172 (1999), 279-295. doi: 10.1016/S0030-4018(99)00603-3.

[73]

S. Wieczorek, B. Krauskopf and D. Lenstra, Sudden chaotic transitions in an optically injected semiconductor laser, Opt. Lett., 26 (2001), 816-818. doi: 10.1364/OL.26.000816.

[74]

S. Wieczorek, B. Krauskopf and D. Lenstra, Unnested islands of period doublings in an injected semiconductor laser, Phys. Rev. E, 64 (2001), 056204, 9pp. doi: 10.1103/PhysRevE.64.056204.

[75]

S. Wieczorek, T. B. Simpson, B. Krauskopf and D. Lenstra, Bifurcation transitions in an optically injected diode laser: theory and experiment, Opt. Commun., 215 (2003), 125-134. doi: 10.1016/S0030-4018(02)02191-0.

[76]

S. Wieczorek, B. Krauskopf, T. B. Simpson and D. Lenstra, The dynamical complexity of optically injected semiconductor lasers, Physics Reports, 416 (2005), 1-128. doi: 10.1016/j.physrep.2005.06.003.

[77]

J. Ye, H. Li and J. G. McInerney, Period-doubling route to chaos in a semiconductor laser with weak optical feedback, Phys. Rev. A, 47 (1993), 2249-2252. doi: 10.1103/PhysRevA.47.2249.

[78]

M. Yousefi and D. Lenstra, Dynamical behaviour of a semiconductor laser with filtered external optical feedback, IEEE J. Quantum Electron., 35 (1999), 970-976. doi: 10.1109/3.766841.

[79]

M. Yousefi, D. Lenstra, A. Fischer and G. Vemuri, Simulations of a semiconductor laser with filtered optical feedback: deterministic dynamics and transitions to chaos, in P. Blood, M. Osinski and Y. Arakawa (Eds.), "Physics and Simulation of Optoelectronic Devices X,'' Proceedings of SPIE, 4646 (2002), 447-452. doi: 10.1117/12.470547.

[80]

M. Yousefi, D. Lenstra and G. Vemuri, Nonlinear dynamics of a semiconductor laser with filtered optical feedback and the influence of noise, Phys. Rev. E, 67 (2003), 046213. doi: 10.1103/PhysRevE.67.046213.

[81]

M. Yousefi, D. Lenstra, G. Vemuri and A. P. A. Fischer, Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback, IEEE Proc. Optoelectron., 148 (2001), 233-237. doi: 10.1049/ip-opt:20010721.

[1]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[2]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268

[3]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

[4]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[5]

Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355

[6]

Songbai Guo, Jing-An Cui, Wanbiao Ma. An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3831-3844. doi: 10.3934/dcdsb.2021208

[7]

Marion Weedermann. Analysis of a model for the effects of an external toxin on anaerobic digestion. Mathematical Biosciences & Engineering, 2012, 9 (2) : 445-459. doi: 10.3934/mbe.2012.9.445

[8]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215

[9]

Ruiqiang Guo, Lu Song. Optical chaotic secure algorithm based on space laser communication. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1355-1369. doi: 10.3934/dcdss.2019093

[10]

Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006

[11]

Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259

[12]

Lukas F. Lang, Otmar Scherzer. Optical flow on evolving sphere-like surfaces. Inverse Problems and Imaging, 2017, 11 (2) : 305-338. doi: 10.3934/ipi.2017015

[13]

Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1145-1185. doi: 10.3934/dcdss.2020068

[14]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052

[15]

Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021035

[16]

François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 261-310. doi: 10.3934/dcds.2004.11.261

[17]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[18]

Yu-Chi Chen. Security analysis of public key encryption with filtered equality test. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021053

[19]

Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324

[20]

Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (4)

[Back to Top]