Abstract

Permutation entropy (PE) has a growing significance as a relative measure of complexity in nonlinear systems. It has been applied successfully to measuring complexity in nonlinear laser systems. Here, PE and weighted permutation entropy (WPE) are discovered to show an unexpected inversion to higher values, when characterizing the complexity at the characteristic frequencies of nonlinear drivers in laser systems, for output power sequences which are pulsed. The cause of this inversion is explained and its presence can be used to identify when irregular dynamics transform into a fairly regular pulsed signal (with amplitude and timing jitter). When WPE is calculated from experimental output power time series from various nonlinear laser systems as a function of delay time, both the minimum value of WPE, and the width of the peak in the WPE plot are shown to be indicative of the level of amplitude variation and timing jitter present in the pulsed output. Links are made with analysis using simulated time series data with systematic variation in timing jitter and/or amplitude variations.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2014

2013

A. Aragoneses, N. Rubido, J. Tiana-Alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013).
[CrossRef]

B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(2), 022911 (2013).
[CrossRef] [PubMed]

2011

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

L. Zunino, O. A. Rosso, and M. C. Soriano, “Characterizing the hyperchaotic dynamics of a semiconductor laser subject to optical feedback via permutation entropy,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1250–1257 (2011).
[CrossRef]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

D. M. Kane and J. P. Toomey, “Variable pulse repetition frequency output from an optically injected solid state laser,” Opt. Express 19(5), 4692–4702 (2011).
[CrossRef] [PubMed]

2010

J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18(16), 16955–16972 (2010).
[CrossRef] [PubMed]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

2009

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
[CrossRef] [PubMed]

N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. 45(11), 1388–1395 (2009).
[CrossRef]

2008

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100(1), 013907 (2008).
[CrossRef] [PubMed]

2007

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[CrossRef]

2006

M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A 369(2), 439–462 (2006).
[CrossRef]

2005

S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
[CrossRef]

2004

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004).
[CrossRef] [PubMed]

2002

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[CrossRef] [PubMed]

1994

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[CrossRef]

1993

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

1983

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

Ackemann, T.

N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. 45(11), 1388–1395 (2009).
[CrossRef]

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100(1), 013907 (2008).
[CrossRef] [PubMed]

Aragoneses, A.

A. Aragoneses, N. Rubido, J. Tiana-Alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013).
[CrossRef]

Bandt, C.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[CrossRef] [PubMed]

Cao, Y. H.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004).
[CrossRef] [PubMed]

Chen, B.

B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(2), 022911 (2013).
[CrossRef] [PubMed]

Collins, J. J.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

Deluca, C. J.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

Fadlallah, B.

B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(2), 022911 (2013).
[CrossRef] [PubMed]

Firth, W. J.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100(1), 013907 (2008).
[CrossRef] [PubMed]

Fischer, I.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

Fordell, T.

S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
[CrossRef]

Gao, J. B.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004).
[CrossRef] [PubMed]

Garcia-Ojalvo, J.

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

Grassberger, P.

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

Hively, L. M.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004).
[CrossRef] [PubMed]

Jäger, R.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100(1), 013907 (2008).
[CrossRef] [PubMed]

Kane, D. M.

Kantz, H.

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[CrossRef]

Keil, A.

B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(2), 022911 (2013).
[CrossRef] [PubMed]

Lee, M. W.

Lehnertz, K.

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[CrossRef]

Lindberg, A. M.

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
[CrossRef] [PubMed]

S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
[CrossRef]

Martin, M. T.

M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A 369(2), 439–462 (2006).
[CrossRef]

Masoller, C.

A. Aragoneses, N. Rubido, J. Tiana-Alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013).
[CrossRef]

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

Mirasso, C. R.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

Plastino, A.

M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A 369(2), 439–462 (2006).
[CrossRef]

Pompe, B.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[CrossRef] [PubMed]

Príncipe, J.

B. Fadlallah, B. Chen, A. Keil, and J. Príncipe, “Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(2), 022911 (2013).
[CrossRef] [PubMed]

Procaccia, I.

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

Protopopescu, V. A.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004).
[CrossRef] [PubMed]

Radwell, N.

N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. 45(11), 1388–1395 (2009).
[CrossRef]

Rosenstein, M. T.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

Rosso, O. A.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

L. Zunino, O. A. Rosso, and M. C. Soriano, “Characterizing the hyperchaotic dynamics of a semiconductor laser subject to optical feedback via permutation entropy,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1250–1257 (2011).
[CrossRef]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

M. T. Martin, A. Plastino, and O. A. Rosso, “Generalized statistical complexity measures: Geometrical and analytical properties,” Physica A 369(2), 439–462 (2006).
[CrossRef]

Rubido, N.

A. Aragoneses, N. Rubido, J. Tiana-Alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013).
[CrossRef]

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

Shore, K. A.

Soriano, M. C.

L. Zunino, O. A. Rosso, and M. C. Soriano, “Characterizing the hyperchaotic dynamics of a semiconductor laser subject to optical feedback via permutation entropy,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1250–1257 (2011).
[CrossRef]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

Staniek, M.

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[CrossRef]

Tanguy, Y.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. 100(1), 013907 (2008).
[CrossRef] [PubMed]

Tiana-Alsina, J.

A. Aragoneses, N. Rubido, J. Tiana-Alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013).
[CrossRef]

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

Toomey, J. P.

Torrent, M. C.

A. Aragoneses, N. Rubido, J. Tiana-Alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013).
[CrossRef]

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

Tung, W. W.

Y. H. Cao, W. W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046217 (2004).
[CrossRef] [PubMed]

Valling, S.

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
[CrossRef] [PubMed]

S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
[CrossRef]

Zunino, L.

L. Zunino, O. A. Rosso, and M. C. Soriano, “Characterizing the hyperchaotic dynamics of a semiconductor laser subject to optical feedback via permutation entropy,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1250–1257 (2011).
[CrossRef]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

IEEE J. Quantum Electron.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[CrossRef]

N. Radwell and T. Ackemann, “Characteristics of Laser Cavity Solitons in a Vertical-Cavity Surface-Emitting Laser With Feedback From a Volume Bragg Grating,” IEEE J. Quantum Electron. 45(11), 1388–1395 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

L. Zunino, O. A. Rosso, and M. C. Soriano, “Characterizing the hyperchaotic dynamics of a semiconductor laser subject to optical feedback via permutation entropy,” IEEE J. Sel. Top. Quantum Electron. 17(5), 1250–1257 (2011).
[CrossRef]

Int. J. Bifurcat. Chaos

M. Staniek and K. Lehnertz, “Parameter selection for permutation entropy measurements,” Int. J. Bifurcat. Chaos 17(10), 3729–3733 (2007).
[CrossRef]

Opt. Express

Phys. Lett. A

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[CrossRef]

Phys. Rev. A

S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
[CrossRef]

J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia-Ojalvo, “Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 82(1), 013819 (2010).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026202 (2011).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Permutation entropy (black) and weighted PE (red) for D = 5 as a function of ordinal pattern delay τ for a time series record of optical power (20GSa/s sampling rate) from a semiconductor laser with optical feedback (injection current = 68.7 mA, intra-cavity AOM transmission = 0.644, cavity round trip time τext = 4.5 ns and relaxation oscillation period = 150 ps) [10].

Fig. 2
Fig. 2

(a) Time series record of optical power from a single soliton in a VCSEL with frequency selective optical feedback, and (b) weighted PE (D = 5) as a function of ordinal pattern delay τ. The data are taken with an AC-coupled 12 GHz detector, amplified with a 15 GHz RF amplifier and digitized with a 18 GHz oscilloscope at 60 Gs/s (16.7 ps sampling time, 30000 points).

Fig. 3
Fig. 3

(a) Output power time series from an optically injected solid state laser (normalized frequency detuning Δω = −1.849 and normalized injection strength K = 0.226) [14], and (b) weighted PE (D = 5) as a function of ordinal pattern delay. Signal was sampled at 10ns and trace contained 5000 data points (50 µs).

Fig. 4
Fig. 4

(a) Simulated clean pulsed time series y = sin20(x). (b) Weighted PE (D = 5) as a function of ordinal pattern delay for a time series of clean simulated pulses (black trace) and the clean signal with each pulse multiplied by a small random number between 1 ± 10−5 (red trace).

Fig. 5
Fig. 5

(a) Weighted PE (D = 5) as a function of ordinal pattern delay for simulated pulses with added amplitude noise. (b) Magnified section around delay τ = 158 (pulse period).

Fig. 6
Fig. 6

(a) Minimum value of WPE either side of the peak at τ = 158 and linear fit. (b) Weighted permutation entropy (D = 4) for the random numbers used to generate pulse amplitude noise with τ = 1, and WPE of the pulses with τ = 158. Ordinal pattern length D = 4 was used to satisfy the requirement that ND! .

Fig. 7
Fig. 7

(a) Weighted PE (D = 5) as a function of ordinal pattern delay for simulated pulses with added temporal noise. (b) Magnified section around delay τ = 158.

Fig. 8
Fig. 8

Width of the WPE peak around τ = 158 for simulated pulses with added temporal noise and least-square fit to a straight line.

Fig. 9
Fig. 9

(a) Map of PE as a function of ordinal pattern delay and injection current from a semiconductor laser with optical feedback (reproduced from [10]). Two examples of measured time series and PE as a function of τ from regions of the map where no inversion is observed at 60 mA (b)-(c) and where inversion occurs due to pulsed operation (d)-(e).

Tables (1)

Tables Icon

Table 1 PE and weighted PE for the pulse amplitudes and pulse intervals of the experimental time series.a

Equations (5)

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S[ P ]= i=1 M p i ln p i .
X S ( x s( D1 )τ , x s( D2 )τ ,..., x sτ , x s ) .
x s r 0 τ x s r 1 τ ... x s r D2 τ x s r D1 τ .
S [ P ]= S[ P ] S max = i=1 D! p( π i ) lnp( π i ) lnD!
w s = 1 D k=1 D [ x s( k1 )τ X ¯ S ] 2

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