Abstract

We model two non-identical delay-line optoelectronic oscillators mutually coupled through delayed cross-feedback. The system can generate multi-stable nanosecond periodic square-wave solutions which arise through a Hopf instability. We show that for suitable ratios between self and cross delay times, the two oscillators generate square waves with different amplitude but synchronized in phase, out of phase or with a dephasing of a quarter of the period. We also show that the synchronization is robust to small mismatches in the delay times.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  26. M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 13827 (2013).
    [Crossref]
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  28. J. Martínez-Llinàs, P. Colet, and T. Erneux, “Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems,” Phys. Rev. E 89, 42908 (2014).
    [Crossref]
  29. J. Martínez-Llinàs, P. Colet, and T. Erneux, “Synchronization of tunable asymmetric square-wave pulses in delay-coupled optoelectronic oscillators,” Phys. Rev. E 91, 032911 (2015).
    [Crossref]
  30. M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins, “Symmetry in locomotor central pattern generators and animal gaits,” Nature,  401, 693–695 (1999).
    [Crossref] [PubMed]
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    [Crossref]
  33. T. Yoshida, L.E. Jones, S.P. Ellner, G.F. Fussmann, and N.G. Hairston, “Rapid evolution drives ecological dynamics in a predator - prey system,” Nature 424, 303–306 (2003).
    [Crossref] [PubMed]
  34. L.E. Jones and S.P. Ellner, “Effects of rapid prey evolution on predator-prey cycles,” J. Math. Biol. 55, 541–573 (2007).
    [Crossref] [PubMed]
  35. T. Hiltunen, L. Jones, S. Ellner, and H.G. Hairston, “Temporal dynamics of a simple community with intraguild predation: an experimental test,” Ecology 94, 773–779 (2013).
    [Crossref]
  36. R. Gobbelé, T.D. Waberski, H. Simon, E. Peters, F. Klostermann, G. Curio, and H. Buchner, “Different origins of low- and high-frequency components (600 Hz) of human somatosensory evoked potentials,” Clin. Neurophysiol. 115, 927–937 (2004).
    [Crossref] [PubMed]
  37. S.A. Campbell, R. Edwards, and P. van den Driessche, “Delayed coupling between two neural network loops,” SIAM Appl. J. Math. 65, 316–335 (2004).
    [Crossref]
  38. T. Wang, L. Ofman, J.M. Davila, and Y. Su, “Growing transverse oscillations of a multistranded loop observed By Sdo/Aia,” Astrophys. J. Lett. 751, L27 (2012).
    [Crossref]
  39. N. Gastaud, S. Poinsot, L. Larger, J.-M. Merolla, M. Hanna, J.P. Goedgebuer, and F. Malassenet, “Electro-optical chaos for multi-10 Gbit/s optical transmissions,” Electron. Lett. 40, 898–899 (2004).
    [Crossref]
  40. M. Peil, M. Jacquot, Y.K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79, 26208 (2009).
    [Crossref]
  41. X.S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
    [Crossref]
  42. Y.K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
    [Crossref]

2015 (2)

S. García and I. Gasulla, “Multi-cavity optoelectronic oscillators using multicore fibers,” Opt. Express 232403–2415 (2015).
[Crossref] [PubMed]

J. Martínez-Llinàs, P. Colet, and T. Erneux, “Synchronization of tunable asymmetric square-wave pulses in delay-coupled optoelectronic oscillators,” Phys. Rev. E 91, 032911 (2015).
[Crossref]

2014 (4)

2013 (4)

A. Coillet, R. Henriet, P. Salzenstein, K.P. Huy, L. Larger, and Y.K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 1–12 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slowfast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 13827 (2013).
[Crossref]

T. Hiltunen, L. Jones, S. Ellner, and H.G. Hairston, “Temporal dynamics of a simple community with intraguild predation: an experimental test,” Ecology 94, 773–779 (2013).
[Crossref]

2012 (4)

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201 (2012).
[Crossref]

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012).
[Crossref] [PubMed]

X. Zhang, C. Gu, G. Chen, B. Sun, L. Xu, A. Wang, and H. Ming, “Square-wave pulse with ultra-wide tuning range in a passively mode-locked fiber laser,” Opt. Lett. 37, 1334–1336 (2012).
[Crossref] [PubMed]

T. Wang, L. Ofman, J.M. Davila, and Y. Su, “Growing transverse oscillations of a multistranded loop observed By Sdo/Aia,” Astrophys. J. Lett. 751, L27 (2012).
[Crossref]

2011 (3)

C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: Theory and experiment,” Phys. Rev. A 84, 23838 (2011).
[Crossref]

S. Ura, S. Shoda, K. Nishio, and Y. Awatsuji, “In-line rotation sensor based on VCSEL behavior under polarization-rotating optical feedback,” Opt. Express 19, 23683–23688 (2011).
[Crossref]

T. Morbiato, R. Vitaliani, and A. Saetta, “Numerical analysis of a synchronization phenomenon: Pedestrian-structure interaction,” Comput. Struct. 89, 1649–1663 (2011).
[Crossref]

2010 (1)

D.W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81, 025206 (2010).
[Crossref]

2009 (1)

M. Peil, M. Jacquot, Y.K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79, 26208 (2009).
[Crossref]

2007 (4)

Y.K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444–446 (2007).
[Crossref]

J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A 76, 43801 (2007).
[Crossref]

L.E. Jones and S.P. Ellner, “Effects of rapid prey evolution on predator-prey cycles,” J. Math. Biol. 55, 541–573 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

Y.C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref] [PubMed]

2004 (4)

R. Gobbelé, T.D. Waberski, H. Simon, E. Peters, F. Klostermann, G. Curio, and H. Buchner, “Different origins of low- and high-frequency components (600 Hz) of human somatosensory evoked potentials,” Clin. Neurophysiol. 115, 927–937 (2004).
[Crossref] [PubMed]

S.A. Campbell, R. Edwards, and P. van den Driessche, “Delayed coupling between two neural network loops,” SIAM Appl. J. Math. 65, 316–335 (2004).
[Crossref]

E.A. Viktorov, A.M. Yacomotti, and P. Mandel, “Semiconductor lasers coupled face-to-face,” J. Opt. B: Quantum Semiclass. Opt. 6, L9–L12 (2004).
[Crossref]

N. Gastaud, S. Poinsot, L. Larger, J.-M. Merolla, M. Hanna, J.P. Goedgebuer, and F. Malassenet, “Electro-optical chaos for multi-10 Gbit/s optical transmissions,” Electron. Lett. 40, 898–899 (2004).
[Crossref]

2003 (1)

T. Yoshida, L.E. Jones, S.P. Ellner, G.F. Fussmann, and N.G. Hairston, “Rapid evolution drives ecological dynamics in a predator - prey system,” Nature 424, 303–306 (2003).
[Crossref] [PubMed]

2002 (1)

J.P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W.T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

1999 (1)

M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins, “Symmetry in locomotor central pattern generators and animal gaits,” Nature,  401, 693–695 (1999).
[Crossref] [PubMed]

1998 (1)

B. Sartorius, C. Bornholdt, O. Brox, H. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett. 34, 1664–1665 (1998).
[Crossref]

1996 (3)

Z. Hong, Z. Feizhou, Y. Jie, and W. Yinghai, “Nonlinear differential delay equations using the Poincaré section technique,” Phys. Rev. E 54, 6925–6928 (1996).
[Crossref]

X.S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

X.S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

1992 (1)

T. Aida and P. Davis, “Oscillation modes of laser diode pumped hybrid bistable system with large delay and application to dynamical memory,” IEEE J. Quantum Electron. 28, 686–699 (1992).
[Crossref]

1987 (1)

K. Ikeda and K. Matsumoto, “High-dimensional chaotic behavior in systems with time-delayed feedback,” Physica D 29, 223–235 (1987).
[Crossref]

1980 (1)

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

1979 (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 257–261 (1979).
[Crossref]

Aida, T.

T. Aida and P. Davis, “Oscillation modes of laser diode pumped hybrid bistable system with large delay and application to dynamical memory,” IEEE J. Quantum Electron. 28, 686–699 (1992).
[Crossref]

Akimoto, O.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

Amonette, J.

Awatsuji, Y.

Balakireva, I.V.

Balle, S.

M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 13827 (2013).
[Crossref]

J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A 76, 43801 (2007).
[Crossref]

Barland, S.

M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 13827 (2013).
[Crossref]

Bendoula, R.

Bornholdt, C.

B. Sartorius, C. Bornholdt, O. Brox, H. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett. 34, 1664–1665 (1998).
[Crossref]

Brox, O.

B. Sartorius, C. Bornholdt, O. Brox, H. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett. 34, 1664–1665 (1998).
[Crossref]

Buchner, H.

R. Gobbelé, T.D. Waberski, H. Simon, E. Peters, F. Klostermann, G. Curio, and H. Buchner, “Different origins of low- and high-frequency components (600 Hz) of human somatosensory evoked potentials,” Clin. Neurophysiol. 115, 927–937 (2004).
[Crossref] [PubMed]

Buono, P.-L.

M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins, “Symmetry in locomotor central pattern generators and animal gaits,” Nature,  401, 693–695 (1999).
[Crossref] [PubMed]

Burner, G.

Campbell, S.A.

S.A. Campbell, R. Edwards, and P. van den Driessche, “Delayed coupling between two neural network loops,” SIAM Appl. J. Math. 65, 316–335 (2004).
[Crossref]

Chembo, Y.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slowfast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201 (2012).
[Crossref]

Chembo, Y.K.

K. Saleh, R. Henriet, S. Diallo, G. Lin, R. Martinenghi, I.V. Balakireva, P. Salzenstein, A. Coillet, and Y.K. Chembo, “Phase noise performance comparison between optoelectronic oscillators based on optical delay lines and whispering gallery mode resonators,” Opt. Express 22, 32158–32173 (2014).
[Crossref]

A. Coillet, R. Henriet, P. Salzenstein, K.P. Huy, L. Larger, and Y.K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 1–12 (2013).
[Crossref]

M. Peil, M. Jacquot, Y.K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79, 26208 (2009).
[Crossref]

Y.K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

Chen, C.-C.

J.P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W.T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
[Crossref]

Chen, G.

Coillet, A.

K. Saleh, R. Henriet, S. Diallo, G. Lin, R. Martinenghi, I.V. Balakireva, P. Salzenstein, A. Coillet, and Y.K. Chembo, “Phase noise performance comparison between optoelectronic oscillators based on optical delay lines and whispering gallery mode resonators,” Opt. Express 22, 32158–32173 (2014).
[Crossref]

A. Coillet, R. Henriet, P. Salzenstein, K.P. Huy, L. Larger, and Y.K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 1–12 (2013).
[Crossref]

Colet, P.

J. Martínez-Llinàs, P. Colet, and T. Erneux, “Synchronization of tunable asymmetric square-wave pulses in delay-coupled optoelectronic oscillators,” Phys. Rev. E 91, 032911 (2015).
[Crossref]

J. Martínez-Llinàs, P. Colet, and T. Erneux, “Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems,” Phys. Rev. E 89, 42908 (2014).
[Crossref]

Y.K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).
[Crossref]

Y.C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95, 203903 (2005).
[Crossref] [PubMed]

Collins, J.J.

M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins, “Symmetry in locomotor central pattern generators and animal gaits,” Nature,  401, 693–695 (1999).
[Crossref] [PubMed]

Curio, G.

R. Gobbelé, T.D. Waberski, H. Simon, E. Peters, F. Klostermann, G. Curio, and H. Buchner, “Different origins of low- and high-frequency components (600 Hz) of human somatosensory evoked potentials,” Clin. Neurophysiol. 115, 927–937 (2004).
[Crossref] [PubMed]

D’Huys, O.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slowfast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201 (2012).
[Crossref]

Daido, H.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[Crossref]

Danckaert, J.

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slowfast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
[Crossref]

L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square waves in a time-delayed system,” Phys. Rev. E 86, 055201 (2012).
[Crossref]

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012).
[Crossref] [PubMed]

Davila, J.M.

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M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 13827 (2013).
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Z. Hong, Z. Feizhou, Y. Jie, and W. Yinghai, “Nonlinear differential delay equations using the Poincaré section technique,” Phys. Rev. E 54, 6925–6928 (1996).
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Jones, L.

T. Hiltunen, L. Jones, S. Ellner, and H.G. Hairston, “Temporal dynamics of a simple community with intraguild predation: an experimental test,” Ecology 94, 773–779 (2013).
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L.E. Jones and S.P. Ellner, “Effects of rapid prey evolution on predator-prey cycles,” J. Math. Biol. 55, 541–573 (2007).
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L. Weicker, T. Erneux, O. D’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slowfast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371, 20120459 (2013).
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D.W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81, 025206 (2010).
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J.P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W.T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
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C. Zhang, R.D. Guy, B. Mulloney, Q. Zhang, and T.J. Lewis, “Neural mechanism of optimal limb coordination in crustacean swimming.,” Proc. Nat. Acad. Sci. 111, 13840–13845 (2014).
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Ludwig, R.

B. Sartorius, C. Bornholdt, O. Brox, H. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett. 34, 1664–1665 (1998).
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N. Gastaud, S. Poinsot, L. Larger, J.-M. Merolla, M. Hanna, J.P. Goedgebuer, and F. Malassenet, “Electro-optical chaos for multi-10 Gbit/s optical transmissions,” Electron. Lett. 40, 898–899 (2004).
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Martínez-Llinàs, J.

J. Martínez-Llinàs, P. Colet, and T. Erneux, “Synchronization of tunable asymmetric square-wave pulses in delay-coupled optoelectronic oscillators,” Phys. Rev. E 91, 032911 (2015).
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K. Ikeda and K. Matsumoto, “High-dimensional chaotic behavior in systems with time-delayed feedback,” Physica D 29, 223–235 (1987).
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D.W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81, 025206 (2010).
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N. Gastaud, S. Poinsot, L. Larger, J.-M. Merolla, M. Hanna, J.P. Goedgebuer, and F. Malassenet, “Electro-optical chaos for multi-10 Gbit/s optical transmissions,” Electron. Lett. 40, 898–899 (2004).
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Ming, H.

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B. Sartorius, C. Bornholdt, O. Brox, H. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett. 34, 1664–1665 (1998).
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D.W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81, 025206 (2010).
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J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A 76, 43801 (2007).
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C. Zhang, R.D. Guy, B. Mulloney, Q. Zhang, and T.J. Lewis, “Neural mechanism of optimal limb coordination in crustacean swimming.,” Proc. Nat. Acad. Sci. 111, 13840–13845 (2014).
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Ofman, L.

T. Wang, L. Ofman, J.M. Davila, and Y. Su, “Growing transverse oscillations of a multistranded loop observed By Sdo/Aia,” Astrophys. J. Lett. 751, L27 (2012).
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M. Peil, M. Jacquot, Y.K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79, 26208 (2009).
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N. Gastaud, S. Poinsot, L. Larger, J.-M. Merolla, M. Hanna, J.P. Goedgebuer, and F. Malassenet, “Electro-optical chaos for multi-10 Gbit/s optical transmissions,” Electron. Lett. 40, 898–899 (2004).
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J.P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W.T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38, 1178–1183 (2002).
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Saetta, A.

T. Morbiato, R. Vitaliani, and A. Saetta, “Numerical analysis of a synchronization phenomenon: Pedestrian-structure interaction,” Comput. Struct. 89, 1649–1663 (2011).
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E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444–446 (2007).
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K. Saleh, R. Henriet, S. Diallo, G. Lin, R. Martinenghi, I.V. Balakireva, P. Salzenstein, A. Coillet, and Y.K. Chembo, “Phase noise performance comparison between optoelectronic oscillators based on optical delay lines and whispering gallery mode resonators,” Opt. Express 22, 32158–32173 (2014).
[Crossref]

A. Coillet, R. Henriet, P. Salzenstein, K.P. Huy, L. Larger, and Y.K. Chembo, “Time-domain dynamics and stability analysis of optoelectronic oscillators based on whispering-gallery mode resonators,” IEEE J. Sel. Top. Quantum Electron. 19, 1–12 (2013).
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B. Sartorius, C. Bornholdt, O. Brox, H. Ehrke, D. Hoffmann, R. Ludwig, and M. Mohrle, “All-optical clock recovery module based on self-pulsating DFB laser,” Electron. Lett. 34, 1664–1665 (1998).
[Crossref]

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C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: Theory and experiment,” Phys. Rev. A 84, 23838 (2011).
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Simon, H.

R. Gobbelé, T.D. Waberski, H. Simon, E. Peters, F. Klostermann, G. Curio, and H. Buchner, “Different origins of low- and high-frequency components (600 Hz) of human somatosensory evoked potentials,” Clin. Neurophysiol. 115, 927–937 (2004).
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M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins, “Symmetry in locomotor central pattern generators and animal gaits,” Nature,  401, 693–695 (1999).
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T. Wang, L. Ofman, J.M. Davila, and Y. Su, “Growing transverse oscillations of a multistranded loop observed By Sdo/Aia,” Astrophys. J. Lett. 751, L27 (2012).
[Crossref]

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C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: Theory and experiment,” Phys. Rev. A 84, 23838 (2011).
[Crossref]

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D.W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81, 025206 (2010).
[Crossref]

A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. 31, 2006–2008 (2006).
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E.A. Viktorov, A.M. Yacomotti, and P. Mandel, “Semiconductor lasers coupled face-to-face,” J. Opt. B: Quantum Semiclass. Opt. 6, L9–L12 (2004).
[Crossref]

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T. Morbiato, R. Vitaliani, and A. Saetta, “Numerical analysis of a synchronization phenomenon: Pedestrian-structure interaction,” Comput. Struct. 89, 1649–1663 (2011).
[Crossref]

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R. Gobbelé, T.D. Waberski, H. Simon, E. Peters, F. Klostermann, G. Curio, and H. Buchner, “Different origins of low- and high-frequency components (600 Hz) of human somatosensory evoked potentials,” Clin. Neurophysiol. 115, 927–937 (2004).
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Wang, T.

T. Wang, L. Ofman, J.M. Davila, and Y. Su, “Growing transverse oscillations of a multistranded loop observed By Sdo/Aia,” Astrophys. J. Lett. 751, L27 (2012).
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Figures (15)

Fig. 1
Fig. 1 Diagram of the system modeled integrated by two mutually delay-coupled OEOs. Each OEO consists of a Mach-Zehnder interferometer (MZI), a fiber loop with delay time Tf, a photodiode (PD) and a RF amplifier (G) whose output modulates an arm of the MZI. OEOs are fed by a laser diode (LD) whose output is split in two parts by a 50/50 fiber splitter. The OEOs are mutually coupled with cross-feedback delay time Tc.
Fig. 2
Fig. 2 Pc as given by Eqs. (16)(17) for γij = 0.5 and Φ0 = 0. (a) k′ odd and k even, (b) k′ even and k even, (c) k′ odd and k odd, (d) k′ even and k odd. Parameter regions in which Pc is negative or imaginary are plotted in white and grey, respectively.
Fig. 3
Fig. 3 Value of Pc as in Fig. 2 for γ11 = 0.5, γ22 = 0.3, γ12 = 0.2 and γ21 = 0.4.
Fig. 4
Fig. 4 Hopf bifurcations with ω < 10π for Φ1 = 0.2π, Φ2 = 0.3π, Φ0 = 0, γij = 0.5, ε = 4.17 × 10−4 and δ = 1.2 × 10−2. In (b) lines are plotted only in the range where Pc < 2. Red and black lines correspond to in- and out-of-phase oscillations, respectively.
Fig. 5
Fig. 5 Out-of-phase oscillations with symmetric duty cycle for Φ1 = 0.2π, Φ2 = 0.25π, Φ0 = 0, γ11 = 0.5, γ22 = 0.3, γ12 = 0.2, γ21 = 0.4, Tf = 30ns, Tc = 40ns, ε = 6.25 × 10−4, δ = 8 × 10−3 and P = 2.117 (a), P = 2.12 (b), and P = 2.3 (c). Black and red lines correspond to x1 and x2 respectively.
Fig. 6
Fig. 6 Hopf bifurcations for Φ1 = −0.2π and Φ2 = −0.3π. Other parameters as in Fig. 4. The green line corresponds to microsecond oscillations.
Fig. 7
Fig. 7 Fundamental out-of-phase solution for Φ1 = −0.1π, Φ2 = −0.4π, Φ0 = 0, γii = 0.5, Tf = 40ns, Tc = 60ns, ε = 4.17 × 10−4, δ = 1.2 × 10−2, P = 1.702 (a), P = 1.72 (b), and P = 1.8 (c). Pc = 1.7013.
Fig. 8
Fig. 8 Fundamental out-of-phase solution for Φ1 = −0.25π, Φ2 = −0.15π, Tf = 40ns, Tc = 60ns, ε = 4.17×10−4, δ = 1.2×10−2, P = 1.616 (a), P = 1.6271 (just below threshold) (b), P = 1.7 (c), and P = 2 (d). Other parameters as in Fig. 5.
Fig. 9
Fig. 9 Microsecond solution obtained for Φ1 = −0.15π, Φ2 = −0.1π, and P = 1.444768 (a), P = 1.4538 (b), and P = 1.454 (c). Panel (d) shows a zoom of c) close to the maximum. Other parameters as in Fig. 7. The threshold is Pc = 1.4392 and at threshold ω0 = 0.1215. Time traces for x1 and x2 overlap in panels (a)–(c).
Fig. 10
Fig. 10 Hopf lines for Φ1 = −0.25π and Φ2 = 0.15π. Other parameters as in Fig. 4. Yellow and blue lines correspond to bifurcations leading to synchronized solutions dephased +T/4 and −T/4, respectively.
Fig. 11
Fig. 11 Coexistence of T/4 square waves for the parameters of Fig. 10, P = 1.3, and Tc = 60ns. Fundamental, 1 st , 2 nd and 30 th harmonics for Tf = 80ns (s0 = 4/3) are shown in (a)–(d) while (e)–(f) show the corresponding ones for Tf = 40ns (s0 = 2/3). Black and red lines correspond to x1 and x2 respectively.
Fig. 12
Fig. 12 Higher harmonics for the parameters of Fig. 11 but τ = 10ns and s0 = 4/3: (a) 40 th , T = 4/243 (0.99ns), (b) 60 th , T = 4/363 (0.66ns) and (c) 80 th , T = 4/483 (0.50ns).
Fig. 13
Fig. 13 Square-waves dephased T/4 for Tc = 60ns, Tf = 80ns, P = 1.119 (a), P = 1.13 (b), and P = 1.25 (c). Other parameters as in Fig. 11.
Fig. 14
Fig. 14 (a)–(d): Robustness of the T/4 dephased fundamental solution changing Tf : 80ns (s0 = 4/3) (a), 81ns (s0 = 1.350) (b), 82ns (s0 = 1.367) (c), and 83ns (s0 = 1.383) (d). P = 1.25, other parameters as in Fig. 11. We also show the frequency of x1 (e) and the phase difference between x2 and x1 (f) as a function of s0. Red dots correspond to numerical simulations of (3) and solid black lines to the theoretical prediction from (9) and (10).
Fig. 15
Fig. 15 As Fig. 14 for the first harmonic with Tf = 80ns (s0 = 4/3) (a), Tf = 80.5ns (s0 = 1.342) (b), Tf = 81ns (s0 = 1.350) (c) and Tf = 81.5ns (s0 = 1.358) (d).

Equations (23)

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τ i x ˙ i ( t ) = x i ( t ) θ i 1 y i ( t ) + P { γ i i 2 cos 2 [ x i ( t T i i ) + Φ i ] + γ j i 2 cos 2 [ x j ( t T j i ) + Φ j ] + 2 γ i i γ j i cos [ x j ( t T j i ) + Φ j ] cos [ x i ( t T i i ) + Φ i ] × cos [ x i ( t T i i ) x j ( t T j i ) + Φ i Φ j + ( 1 ) i Φ 0 ] } y ˙ i ( t ) = x i ( t ) ,
x i st = 0 y i st = θ P { γ i i 2 cos 2 Φ i + γ i i 2 cos 2 Φ j + 2 γ i i γ j i cos Φ j cos Φ i cos [ Φ 1 Φ 2 Φ 0 ] } , j i .
ε x i ( s ) = x i ( s ) δ Y i ( s ) + P γ i i 2 { cos 2 [ x i ( s s 0 ) + Φ i ] cos 2 Φ i } + P γ j i 2 { cos 2 [ x j ( s 1 ) + Φ j ] cos 2 Φ j } + 2 P γ i i γ j i { cos [ x i ( s s 0 ) + Φ i ] cos [ x j ( s 1 ) + Φ j ] × cos [ x i ( s s 0 ) + x j ( s 1 ) + Φ j Φ i + ( 1 ) i Φ 0 ] cos Φ i cos Φ j cos ( Φ 1 Φ 2 Φ 0 ) } Y i ( s ) = x i ( s ) ,
s 0 = T f / T c , ε = τ T c 1 , δ = T c θ 1 .
ε U i ( s ) = U i ( s ) δ V i P [ F i U i ( s s 0 ) + K i U j ( s 1 ) ] V i ( s ) = U i ( s ) ,
F i = γ i i 2 sin 2 Φ i + 2 γ i i γ j i cos Φ j sin [ 2 Φ i Φ j + ( 1 ) i Φ 0 ] , K i = γ j i 2 sin 2 Φ j + 2 γ i i γ j i cos Φ i sin [ 2 Φ j Φ i ( 1 ) i Φ 0 ] .
0 = [ 1 + ε ( λ + i ω ) + δ ( λ + i ω ) 1 + P F i e ( λ + i ω ) s 0 ] u i + P K i e ( λ + i ω ) u j ,
0 = [ 1 + ε ( λ + i ω ) + δ ( λ + i ω ) 1 + P F 1 e ( λ + i ω ) s 0 ] × [ 1 + ε ( λ + i ω ) + δ ( λ + i ω ) 1 + P F 2 e ( λ + i ω ) s 0 ] P 2 K 1 K 2 e 2 ( λ + i ω ) .
0 = 1 + P c ( F 1 + F 2 ) [ cos ( ω s 0 ) + ( ε ω δ ω ) 1 sin ( ω s 0 ) ] + P c 2 [ F 1 F 2 cos ( 2 ω s 0 ) K 1 K 2 cos ( 2 ω ) ] ( ε ω δ ω 1 ) 2 ,
0 = P c ( F 1 + F 2 ) [ ( ε ω δ ω 1 ) cos ( ω s 0 ) sin ( ω s 0 ) ] P c 2 [ F 1 F 2 sin ( 2 ω s 0 ) K 1 K 2 sin ( 2 ω ) ] + 2 ( ε ω δ ω 1 ) .
ω s 0 = k π ,
ω = k π / 2.
s 0 = 2 k k .
T = 2 ( 1 s 0 ) 2 k k = 2 s 0 k = 4 k .
1 + P c ( F 1 + F 2 ) ( 1 ) k + P c 2 ( F 1 F 2 K 1 K 2 ) ( 1 ) k ) = 0.
P c = ( 1 ) k F 1 + F 2 ,
P c = ( F 1 + F 2 ) ( 1 ) k ( F 1 F 2 ) 2 + 4 K 1 K 2 ( 1 ) k 2 [ F 1 F 2 K 1 K 2 ( 1 ) k ] .
u 2 = Q u 1 , Q = ( i ) k 1 + P c F 1 ( 1 ) k P c K 1 .
| Q | = [ 1 + ( ε ω δ ω 1 ) 2 + P c 2 F 1 2 + 2 P c F 1 [ cos ( ω s 0 ) ( ε ω δ ω 1 ) sin ( ω s 0 ) ] 1 / 2 P c | K 1 | ,
φ 2 φ 1 = arctan sin ( ω ) + ( ε ω δ ω 1 ) cos ( ω ) + P c F 1 sin [ ω ( 1 s 0 ) ] cos ( ω ) ( ε ω δ ω 1 ) sin ( ω ) + P c F 1 cos [ ω ( 1 s 0 ) ] .
ω k = k π s 0 + α k ,
0 = P c ( F 1 + F 2 ) ( ε α 0 δ α 0 α 0 s 0 ) 2 P c 2 α 0 ( F 1 F 2 s 0 K 1 K 2 ) + 2 ( ε α 0 δ α 0 ) ,
α 0 2 = [ P c ( F 1 + F 2 ) + 2 ] δ P c ( F 1 + F 2 ) ( ε s 0 ) 2 P c 2 ( F 1 F 2 s 0 K 1 K 2 ) + 2 ε .

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