## Modeling synchronization in networks of delay-coupled fiber ring lasers |

Optics Express, Vol. 19, Issue 24, pp. 24460-24467 (2011)

http://dx.doi.org/10.1364/OE.19.024460

Acrobat PDF (816 KB)

### Abstract

We study the onset of synchronization in a network of *N* delay-coupled stochastic fiber ring lasers with respect to various parameters when the coupling power is weak. In particular, for groups of three or more ring lasers mutually coupled to a central hub laser, we demonstrate a robust tendency toward out-of-phase (achronal) synchronization between the *N* – 1 outer lasers and the single inner laser. In contrast to the achronal synchronization, we find the outer lasers synchronize with zero-lag (isochronal) with respect to each other, thus forming a set of *N* – 1 coherent fiber lasers.

© 2011 OSA

## 1. Introduction

1. M. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science **197**, 289–289 (1977). [CrossRef]

2. M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, “Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic co oxidation on pt(110),” Science **292**, 1357–1360 (2001). [CrossRef] [PubMed]

3. 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]

4. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

5. D. V. R. Reddy, A. Sen, and G. L. Johnston, “Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators,” Phys. Rev. Lett. **85**, 3381–3384 (2000). [CrossRef]

6. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. **76**, 1804–1807 (1996). [CrossRef] [PubMed]

7. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Phys. Rev. Lett. **78**, 4193–4196 (1997). [CrossRef]

11. T. Heil, I. Fischer, W. Elsasser, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” *Phys. Rev. Lett.*86, 795–798 (2001). [CrossRef] [PubMed]

12. J. K. White, M. Matus, and J. V. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E **65**, 036229 (2002). [CrossRef]

13. Q. L. Williams and R. Roy, “Fast polarization dynamics of an erbium-doped fiber ring laser,” Opt. Lett. **21**, 1478–1480 (1996). [CrossRef] [PubMed]

15. G. D. Vanwiggeren and R. Roy, “Chaotic communication using time-delayed optical systems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. **9**, 2129–2156 (1999). [CrossRef]

16. R. Wang and K. Shen, “Synchronization of chaotic erbium-doped fiber dual-ring lasers by using the method of another chaotic system to drive them,” Phys. Rev. E **65**, 016207 (2002). [CrossRef]

17. Y. Imai, H. Murakawa, and T. Imoto, “Chaos synchronization characteristics in erbium-doped fiber laser systems,” Opt. Commun. **217**, 415–420 (2003). [CrossRef]

18. D. J. DeShazer, B. P. Tighe, M. Kurths, and R. Roy, “Experimental observation of noise-induced synchronization of bursting dynamical systems,” IEEE J. Sel. Top. Quantum Electron. **10**, 906–910 (2004). [CrossRef]

19. L. B. Shaw, I. B. Schwartz, E. A. Rogers, and R. Roy, “Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers,” Chaos **16**, 01511 (2006). [CrossRef]

21. A. L. Franz, R. Roy, L. B. Shaw, and I. B. Schwartz, “Changing dynamical complexity with time delay in coupled fiber laser oscillators,” Phys. Rev. Lett. **99**, 053905 (2007). [CrossRef] [PubMed]

22. C. Massoller and A. Marti, “Random delays and the sychronization of chaotic maps,” Phys. Rev. Lett. **94**, 134102 (2005). [CrossRef]

4. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

*N*, is sufficiently large. Motivated by this result, we examined similar coupling arrangements for EDFRL lasers. In contrast to the chaotic semiconductor result, we show that there is very weak dependence on

*N*. However, we observe a uniform scaling of synchronization onset as a function of weak coupling. The scaling appears to be independent of the number of lasers,

*N*.

## 2. Mathematical model

14. Q. L. Williams, J. Garcia-Ojalvo, and R. Roy, “Fast intracavity polarization dynamics of an erbium-doped fiber ring laser: Inclusion of stochastic effects,” Phys. Rev. A **55**, 2376–2386 (1997). [CrossRef]

24. E. A. Rogers-Dakin, J. Garcia-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E **73**, 045201 (2006). [CrossRef]

19. L. B. Shaw, I. B. Schwartz, E. A. Rogers, and R. Roy, “Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers,” Chaos **16**, 01511 (2006). [CrossRef]

*N*ring lasers coupled via

*N*– 1 injection lines comprised of a passive single mode optical fiber, a splitter, and a variable attenuator. The model is characterized by the total population inversion,

*W*(

_{i}*t*), of each

*i*laser (averaged over the length of the fiber amplifier) and the electric field

^{th}*E*(

_{i}*t*) in each laser. Two delay times occur in the model:

*τ*is the cavity round-trip time in the ring, and

_{R}*τ*is the delay in the coupling between lasers. These delay terms correspond to the travel time of light in various components of the system, and thus resolve the spatial dependencies in the system normally modeled by the Maxwell-Bloch equations. The equations for the delay-coupled dynamics, including noise terms, are as follows: where

_{d}*E*(

_{j}*t*) denotes the complex envelope of the electric field in laser

*j*, measured at a given reference point inside of the cavity.

*j*and includes optical feedback within laser

*j*and optical coupling with other lasers. Time is dimensionless, measured in units of the decay time of the atomic transition,

*γ*

_{||}. The active medium is characterized by the dimensionless detuning

*α*between the transition and lasing frequencies, and by the dimensionless gain

_{j}*a*is the material gain,

*L*the length of the active fiber, and

_{a}*N*

_{0}the population inversion at transparency. Each ring cavity is characterized by its return coefficient

*R*, which represents the fraction of light remaining in the cavity after one round-trip, and the average phase change Δ

*ϕ*= 2

*πnL*/

_{p}*λ*due to propagation of light with wavelength

*λ*along the passive fiber of length

*L*and index of refraction

_{p}*n*. The energy input for each laser is given by the pump parameter

*q*. The normalization term

*b*is the number of lasers connected to laser

_{j}*j*.

*B*. Matrix

*B*has components

*B*

_{j,k}= 1, if node

*j*is directly connected to node

*k*, and zero otherwise. For the star topology,

*B*is symmetric since the coupling is assumed to be mutual, and given by the

*NxN*matrix: By altering

*B*in various ways, a variety of different connection schemes for coupled laser systems can be considered. The

*ξ*terms model stimulated emission and are mean zero Gaussian noise applied to the system with standard deviation

_{j}*D*. Coupling between the lasers is characterized by the coupling strength

*κ*, the ratio of the power of light in the injection line to the power of light in the source ring, and only symmetric coupling is considered at the present time.

_{j}## 3. Numerical methods

19. L. B. Shaw, I. B. Schwartz, E. A. Rogers, and R. Roy, “Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers,” Chaos **16**, 01511 (2006). [CrossRef]

*τ*is calculated from the length of the active and passive fibers, where

_{R}*c*is the the speed of light. For the purposes of discretization, the ring cavity is divided into

*N*spatial elements, giving us a time discretization Δ

_{s}*t*=

*τ*/

_{R}*N*.

_{s}*W*(

_{j}*t*+ Δ

*t*) is approximated using a second-order modified Euler method. New values of

*E*and the feedback are computed directly from the formulas 1 and 3. The raw data is saved in a truncated time series from which the intensity

_{j}*I*= |

_{j}*E*|

_{j}^{2}is computed and then passed through a low-pass filter. For these simulations, a second order Butterworth filter from Matlab was used, and set with a threshold of 125 MHz, in keeping with experiments that use a bandwidth photo-detector set at that frequency [19

**16**, 01511 (2006). [CrossRef]

24. E. A. Rogers-Dakin, J. Garcia-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E **73**, 045201 (2006). [CrossRef]

*E*, and initializing the inversion near a steady state. Simulations are run for a minimum of 3000–5000 round-trips in the ring so transients are removed. The parameter values used for all simulations listed in Table 1 are based on previous experimental and modeling efforts as reported in [19

_{j}**16**, 01511 (2006). [CrossRef]

## 4. Results

*C*and

_{ab}*R*terms for more than a few lasers, and so we introduce

_{ab}*C*and

_{out}*C*as the mean value of the cross correlation of all outer lasers with one another, and all outer lasers with the inner hub, respectively. Analogous quantities are also defined on

_{in}*R*. To measure synchronization lag between between the central laser and the outer lasers, we examine which values of time shift,

*τ*, maximize the cross correlation

_{s}*C*and

_{out}*C*.

_{in}*τ*. The results were generated with coupling coefficient

_{s}*κ*= 0.009 and a network of five lasers, four on the outside and one central hub. In this simulation the four outer lasers are detuned from the inner laser, with each laser having the identical detuning factor

*α*= .0352, while the detuning factor for the inner laser is

_{j}*α*= .0202. Note that when considering

_{j}*C*, as in figure 1, with respect to

_{in}*τ*we adopt the convention that the inner laser time series is the one being shifted backward, thus

_{s}*C*here can be viewed as the cross correlation when the inner laser is leading. Note that in Figure 1, the outer lasers have an average peak cross and phase correlation at

_{in}*τ*= 0, meaning that the outer lasers synchronize with each other. Concurrently the outer lasers synchronize with the inner hub at

_{s}*τ*=

_{s}*τ*. This is consistent with the case of two connected fiber lasers, where the synchronization is offset by one factor of the delay time [11

_{d}11. T. Heil, I. Fischer, W. Elsasser, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” *Phys. Rev. Lett.*86, 795–798 (2001). [CrossRef] [PubMed]

**16**, 01511 (2006). [CrossRef]

25. A. S. Landsman and I. B. Schwartz, “Complete chaotic synchronization in mutually coupled time-delay systems,” Phys. Rev. E **75**, 026201 (2007). [CrossRef]

*C*and

_{out}*R*occurs at

_{out}*τ*= 2

_{s}*τ*. This agrees with intuition, since the total time it takes for a signal from one outer laser to reach another is exactly two delay cycles. From repeated numerical experiments illustrating the behavior described above, we can conclude that the outer lasers tend to synchronize with one another precisely, while the outer lasers tend to be offset by a factor of one delay with the inner laser.

_{d}*τ*, and its peaks line up with the outer lasers, all of which are synchronized.

_{d}*N*= 3, we run simulations with identical initial conditions and detuning parameters and examine the final state as a function of

*κ*. Figure 3 shows how increasing the coupling strength causes the lasers to go from an unsynchronized to a synchronized state. Since the system (1–3) behaves like a two laser system when the outer lasers are synchronized, the initial conditions used in generating figure 3 are given by the final state of a two laser simulation (which is simulated long enough for transient effects to pass). In each experiment, all the outer lasers are given the same exact initial condition and prehistory as determined by one of the lasers in the two laser simulation, and the inner laser is given the prehistory associated with the other laser. The error bars shown are derived from doing batches of 20 runs with different random number seeds for the noise terms. Here, the values of

*C*are taken with zero lag, and were computed over a time series of ten round-trips (

_{out}*τ*). Note, as more lasers are added, the scaling behavior of the cross correlation does not change, but the run to run variance decreases, as evidenced by the smaller error bars on the cross-correlation of the outer lasers. We can conclude that adding lasers appears to diminish the variance due to noise, but does not enhance the overall synchronization scaling as a function of

_{R}*κ*and

*N*. Larger values of

*N*were run (up to

*N*= 200), with no noticeable improvement on the overall synchronization (for fixed

*κ*), though these are not shown in Figure 3.

26. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. **105**, 264101 (2010). [CrossRef]

26. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. **105**, 264101 (2010). [CrossRef]

## 6. Conclusion

*N*coupled fiber ring lasers in the weak coupling limit. The model we considered replaced the spatial component of the ring laser with a time delay. In addition, delay was included in coupling fibers used for communication. Using this model, we have shown synchronization in both amplitude and phase of the measured intensities can occur by measuring the cross correlation between inner hub and outer lasers. When examining cross correlation between outer lasers only, we find that synchronization occurs with zero-lag. This is consistent with the results reported in [27

27. D. Tsygankov and K. Wiesenfeld, “Weak link synchronization,” Phys. Rev. E **73**, 026222 (2006). [CrossRef]

**16**, 01511 (2006). [CrossRef]

21. A. L. Franz, R. Roy, L. B. Shaw, and I. B. Schwartz, “Changing dynamical complexity with time delay in coupled fiber laser oscillators,” Phys. Rev. Lett. **99**, 053905 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science |

2. | M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, “Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic co oxidation on pt(110),” Science |

3. | K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. |

4. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. |

5. | D. V. R. Reddy, A. Sen, and G. L. Johnston, “Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators,” Phys. Rev. Lett. |

6. | M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. |

7. | M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Phys. Rev. Lett. |

8. | A. Wagemakers, J. M. Buldu, and M. Sanjuan, “Isochronous synchronization in mutually couple chaotic circuits,” Chaos |

9. | O. D. Huys, R. Vicente, J. Danckaert, and I. Fischer, “Amplitude and phase effects on the synchronication of delay-couple oscillators,” Chaos |

10. | J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronication scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclass. Opt. |

11. | T. Heil, I. Fischer, W. Elsasser, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” |

12. | J. K. White, M. Matus, and J. V. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E |

13. | Q. L. Williams and R. Roy, “Fast polarization dynamics of an erbium-doped fiber ring laser,” Opt. Lett. |

14. | Q. L. Williams, J. Garcia-Ojalvo, and R. Roy, “Fast intracavity polarization dynamics of an erbium-doped fiber ring laser: Inclusion of stochastic effects,” Phys. Rev. A |

15. | G. D. Vanwiggeren and R. Roy, “Chaotic communication using time-delayed optical systems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. |

16. | R. Wang and K. Shen, “Synchronization of chaotic erbium-doped fiber dual-ring lasers by using the method of another chaotic system to drive them,” Phys. Rev. E |

17. | Y. Imai, H. Murakawa, and T. Imoto, “Chaos synchronization characteristics in erbium-doped fiber laser systems,” Opt. Commun. |

18. | D. J. DeShazer, B. P. Tighe, M. Kurths, and R. Roy, “Experimental observation of noise-induced synchronization of bursting dynamical systems,” IEEE J. Sel. Top. Quantum Electron. |

19. | L. B. Shaw, I. B. Schwartz, E. A. Rogers, and R. Roy, “Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers,” Chaos |

20. | I. B. Schwartz and L. B. Shaw, “Isochronal synchronization of delay-coupled systems,” Phys. Rev. E |

21. | A. L. Franz, R. Roy, L. B. Shaw, and I. B. Schwartz, “Changing dynamical complexity with time delay in coupled fiber laser oscillators,” Phys. Rev. Lett. |

22. | C. Massoller and A. Marti, “Random delays and the sychronization of chaotic maps,” Phys. Rev. Lett. |

23. | E. A. Rogers, “Synchronization of high dimensional dynamical systems,” Ph.D. thesis, University of Maryland College Park (2005). |

24. | E. A. Rogers-Dakin, J. Garcia-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E |

25. | A. S. Landsman and I. B. Schwartz, “Complete chaotic synchronization in mutually coupled time-delay systems,” Phys. Rev. E |

26. | J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. |

27. | D. Tsygankov and K. Wiesenfeld, “Weak link synchronization,” Phys. Rev. E |

28. | L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. |

**OCIS Codes**

(140.1540) Lasers and laser optics : Chaos

(140.3290) Lasers and laser optics : Laser arrays

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 11, 2011

Revised Manuscript: October 21, 2011

Manuscript Accepted: October 24, 2011

Published: November 15, 2011

**Citation**

Brandon S. Lindley and Ira B. Schwartz, "Modeling synchronization in networks of delay-coupled fiber ring lasers," Opt. Express **19**, 24460-24467 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24460

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### References

- M. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science197, 289–289 (1977). [CrossRef]
- M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, “Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic co oxidation on pt(110),” Science292, 1357–1360 (2001). [CrossRef] [PubMed]
- 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]
- R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.16, 347–355 (1980). [CrossRef]
- D. V. R. Reddy, A. Sen, and G. L. Johnston, “Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators,” Phys. Rev. Lett.85, 3381–3384 (2000). [CrossRef]
- M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett.76, 1804–1807 (1996). [CrossRef] [PubMed]
- M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Phys. Rev. Lett.78, 4193–4196 (1997). [CrossRef]
- A. Wagemakers, J. M. Buldu, and M. Sanjuan, “Isochronous synchronization in mutually couple chaotic circuits,” Chaos17, 023128 (2007). [CrossRef] [PubMed]
- O. D. Huys, R. Vicente, J. Danckaert, and I. Fischer, “Amplitude and phase effects on the synchronication of delay-couple oscillators,” Chaos20, 043127 (2010). [CrossRef]
- J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronication scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclass. Opt.6, 97–105 (2004). [CrossRef]
- T. Heil, I. Fischer, W. Elsasser, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett.86, 795–798 (2001). [CrossRef] [PubMed]
- J. K. White, M. Matus, and J. V. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E65, 036229 (2002). [CrossRef]
- Q. L. Williams and R. Roy, “Fast polarization dynamics of an erbium-doped fiber ring laser,” Opt. Lett.21, 1478–1480 (1996). [CrossRef] [PubMed]
- Q. L. Williams, J. Garcia-Ojalvo, and R. Roy, “Fast intracavity polarization dynamics of an erbium-doped fiber ring laser: Inclusion of stochastic effects,” Phys. Rev. A55, 2376–2386 (1997). [CrossRef]
- G. D. Vanwiggeren and R. Roy, “Chaotic communication using time-delayed optical systems,” Int. J. Bifurcation Chaos Appl. Sci. Eng.9, 2129–2156 (1999). [CrossRef]
- R. Wang and K. Shen, “Synchronization of chaotic erbium-doped fiber dual-ring lasers by using the method of another chaotic system to drive them,” Phys. Rev. E65, 016207 (2002). [CrossRef]
- Y. Imai, H. Murakawa, and T. Imoto, “Chaos synchronization characteristics in erbium-doped fiber laser systems,” Opt. Commun.217, 415–420 (2003). [CrossRef]
- D. J. DeShazer, B. P. Tighe, M. Kurths, and R. Roy, “Experimental observation of noise-induced synchronization of bursting dynamical systems,” IEEE J. Sel. Top. Quantum Electron.10, 906–910 (2004). [CrossRef]
- L. B. Shaw, I. B. Schwartz, E. A. Rogers, and R. Roy, “Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers,” Chaos16, 01511 (2006). [CrossRef]
- I. B. Schwartz and L. B. Shaw, “Isochronal synchronization of delay-coupled systems,” Phys. Rev. E75, 046207 (2007). [CrossRef]
- A. L. Franz, R. Roy, L. B. Shaw, and I. B. Schwartz, “Changing dynamical complexity with time delay in coupled fiber laser oscillators,” Phys. Rev. Lett.99, 053905 (2007). [CrossRef] [PubMed]
- C. Massoller and A. Marti, “Random delays and the sychronization of chaotic maps,” Phys. Rev. Lett.94, 134102 (2005). [CrossRef]
- E. A. Rogers, “Synchronization of high dimensional dynamical systems,” Ph.D. thesis, University of Maryland College Park (2005).
- E. A. Rogers-Dakin, J. Garcia-Ojalvo, D. J. Deshazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E73, 045201 (2006). [CrossRef]
- A. S. Landsman and I. B. Schwartz, “Complete chaotic synchronization in mutually coupled time-delay systems,” Phys. Rev. E75, 026201 (2007). [CrossRef]
- J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105, 264101 (2010). [CrossRef]
- D. Tsygankov and K. Wiesenfeld, “Weak link synchronization,” Phys. Rev. E73, 026222 (2006). [CrossRef]
- L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett.76, 1816–1819 (1996). [CrossRef] [PubMed]

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