Abstract

We demonstrate a chaos generator involving two time delays and two nonlinear functions. Dynamic behaviors are numerically and experimentally observed. The complexity of the dynamics is discussed in terms of Lyapunov exponents and dimensions. The setup can provide a new architecture for enhancing message security in chaos encryption systems.

© 2004 Optical Society of America

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  8. S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
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  9. J. Wei and S. Ruan, Physica D 130, 255 (1999).
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    [CrossRef]
  13. J. D. Farmer, Physica D 4, 366 (1982).
    [CrossRef]
  14. J. Kaplan and J. Yorke, in Functional Differential Equations and Approximation of Fixed Points, H.-O. Peitgen and H.-O. Walther, eds. (Springer-Verlag, Berlin, 1979).

2003

S. Tang and J.-M. Liu, Phys. Rev. Lett. 90, 194101 (2003).
[CrossRef]

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

M. W. Lee, L. Larger, and J.-P. Goedgebuer, IEEE J. Quantum Electron. 39, 931 (2003).
[CrossRef]

2002

J.-M. Liu, H.-F. Chen, and S. Tang, IEEE J. Quantum Electron. 38, 1184 (2002).
[CrossRef]

2001

2000

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2000).
[CrossRef]

S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
[CrossRef]

1999

1998

R. Hegger, M. J. Bünner, and H. Kantz, Phys. Rev. Lett. 81, 558 (1998).
[CrossRef]

J.-P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

G. D. VanWiggeren and R. Roy, Phys. Rev. Lett. 81, 3547 (1998).
[CrossRef]

1982

J. D. Farmer, Physica D 4, 366 (1982).
[CrossRef]

Bouysse, P.

S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
[CrossRef]

Bünner, M. J.

R. Hegger, M. J. Bünner, and H. Kantz, Phys. Rev. Lett. 81, 558 (1998).
[CrossRef]

Chen, H.-F.

J.-M. Liu, H.-F. Chen, and S. Tang, IEEE J. Quantum Electron. 38, 1184 (2002).
[CrossRef]

Cuenot, J.-B.

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Elflein, W.

Erneux, T.

L. Larger, M. W. Lee, J.-P. Goedgebuer, W. Elflein, and T. Erneux, J. Opt. Soc. Am. B 18, 1063 (2001).
[CrossRef]

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2000).
[CrossRef]

Farmer, J. D.

J. D. Farmer, Physica D 4, 366 (1982).
[CrossRef]

Gavrielides, A.

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2000).
[CrossRef]

Goedgebuer, J.-P.

M. W. Lee, L. Larger, and J.-P. Goedgebuer, IEEE J. Quantum Electron. 39, 931 (2003).
[CrossRef]

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

L. Larger, M. W. Lee, J.-P. Goedgebuer, W. Elflein, and T. Erneux, J. Opt. Soc. Am. B 18, 1063 (2001).
[CrossRef]

J.-P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

Guittard, J.

S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
[CrossRef]

Hegger, R.

R. Hegger, M. J. Bünner, and H. Kantz, Phys. Rev. Lett. 81, 558 (1998).
[CrossRef]

Kantz, H.

R. Hegger, M. J. Bünner, and H. Kantz, Phys. Rev. Lett. 81, 558 (1998).
[CrossRef]

Kaplan, J.

J. Kaplan and J. Yorke, in Functional Differential Equations and Approximation of Fixed Points, H.-O. Peitgen and H.-O. Walther, eds. (Springer-Verlag, Berlin, 1979).

Kovanis, V.

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2000).
[CrossRef]

Larger, L.

M. W. Lee, L. Larger, and J.-P. Goedgebuer, IEEE J. Quantum Electron. 39, 931 (2003).
[CrossRef]

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

L. Larger, M. W. Lee, J.-P. Goedgebuer, W. Elflein, and T. Erneux, J. Opt. Soc. Am. B 18, 1063 (2001).
[CrossRef]

J.-P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

Lee, M. W.

M. W. Lee, L. Larger, and J.-P. Goedgebuer, IEEE J. Quantum Electron. 39, 931 (2003).
[CrossRef]

L. Larger, M. W. Lee, J.-P. Goedgebuer, W. Elflein, and T. Erneux, J. Opt. Soc. Am. B 18, 1063 (2001).
[CrossRef]

Levy, P.

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Liu, J.-M.

S. Tang and J.-M. Liu, Phys. Rev. Lett. 90, 194101 (2003).
[CrossRef]

J.-M. Liu, H.-F. Chen, and S. Tang, IEEE J. Quantum Electron. 38, 1184 (2002).
[CrossRef]

Penaud, S.

S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
[CrossRef]

Porte, H.

J.-P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

Quéré, R.

S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
[CrossRef]

Rhodes, W. T.

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Rogister, F.

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2000).
[CrossRef]

Roy, R.

G. D. VanWiggeren and R. Roy, Phys. Rev. Lett. 81, 3547 (1998).
[CrossRef]

Ruan, S.

J. Wei and S. Ruan, Physica D 130, 255 (1999).
[CrossRef]

Shore, K. A.

Sivaprakasam, S.

Tang, S.

S. Tang and J.-M. Liu, Phys. Rev. Lett. 90, 194101 (2003).
[CrossRef]

J.-M. Liu, H.-F. Chen, and S. Tang, IEEE J. Quantum Electron. 38, 1184 (2002).
[CrossRef]

Udaltsov, V. S.

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

VanWiggeren, G. D.

G. D. VanWiggeren and R. Roy, Phys. Rev. Lett. 81, 3547 (1998).
[CrossRef]

Wei, J.

J. Wei and S. Ruan, Physica D 130, 255 (1999).
[CrossRef]

Yorke, J.

J. Kaplan and J. Yorke, in Functional Differential Equations and Approximation of Fixed Points, H.-O. Peitgen and H.-O. Walther, eds. (Springer-Verlag, Berlin, 1979).

Electron. Lett.

S. Penaud, J. Guittard, P. Bouysse, and R. Quéré, Electron. Lett. 36, 366 (2000).
[CrossRef]

IEEE J. Quantum Electron.

J.-M. Liu, H.-F. Chen, and S. Tang, IEEE J. Quantum Electron. 38, 1184 (2002).
[CrossRef]

M. W. Lee, L. Larger, and J.-P. Goedgebuer, IEEE J. Quantum Electron. 39, 931 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2000).
[CrossRef]

Opt. Lett.

Phys. Lett. A

V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Phys. Rev. Lett.

R. Hegger, M. J. Bünner, and H. Kantz, Phys. Rev. Lett. 81, 558 (1998).
[CrossRef]

S. Tang and J.-M. Liu, Phys. Rev. Lett. 90, 194101 (2003).
[CrossRef]

J.-P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

G. D. VanWiggeren and R. Roy, Phys. Rev. Lett. 81, 3547 (1998).
[CrossRef]

Physica D

J. Wei and S. Ruan, Physica D 130, 255 (1999).
[CrossRef]

J. D. Farmer, Physica D 4, 366 (1982).
[CrossRef]

Other

J. Kaplan and J. Yorke, in Functional Differential Equations and Approximation of Fixed Points, H.-O. Peitgen and H.-O. Walther, eds. (Springer-Verlag, Berlin, 1979).

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Figures (4)

Fig. 1
Fig. 1

Schematic experimental setup: SLD, superluminescent diode; BS, beam splitter; PLs, polarizers; RP1, RP2, retardation plates; LPF, low-pass filter; MUL, multiplier.

Fig. 2
Fig. 2

Numerical bifurcation diagrams. (a) Case of Φ1=π/2 and Φ2=0. (b) Case of Φ1=π and Φ2=0.

Fig. 3
Fig. 3

Dynamic complexity for the double and single time delay. (a) Spectra of Lyapunov exponents. (b) Accumulated Lyapunov exponents.

Fig. 4
Fig. 4

Experimental bifurcation diagrams for the double time delay. (a) Case of Φ1=π/2 and Φ2=0. (b) Case of Φ1=π and Φ2=0.

Tables (1)

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Table 1 Comparison Between Experimental and Numerical Bifurcation Pointsa

Equations (1)

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δDt+τδD·t=FδDt-T1,β1,Φ1+FδDt-T2,β2,Φ2,

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