Abstract

We examine experimentally the consequence of frequency detuning an actively mode-locked external-cavity semiconductor laser from resonance. We observe a transition of the laser system from a periodic oscillation to a nonperiodic state with broadened spectral tones. By estimating the fractal dimension of the corresponding phase-space attractors, we show the presence of low-dimensional chaos. The route to chaos is a well-defined regime of three-frequency quasi-periodicity preceded by a two-frequency quasi-periodicity.

© 1993 Optical Society of America

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  1. K. Y. Lau, IEEE J. Quantum Electron. 26, 250 (1990).
    [CrossRef]
  2. S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
    [CrossRef]
  3. A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
    [CrossRef]
  4. H. G. Winful, Y. C. Chen, and J. M. Liu, Appl. Phys. Lett. 48, 616 (1986).
    [CrossRef]
  5. J. Mork, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990).
    [CrossRef] [PubMed]
  6. L. A. Melnikov, E. M. Rabinovich, and V. V. Tuchin, J. Opt. Soc. Am. B 5, 1134 (1988).
    [CrossRef]
  7. D. L. MacFarlane and L. W. Casperson, J. Opt. Soc. Am. B 7, 285 (1990).
    [CrossRef]
  8. D. L. MacFarlane and L. W. Casperson, J. Opt. Soc. Am. B 4, 1777 (1987).
    [CrossRef]
  9. An attractor is a set of points in phase space that the state of a system approaches after transients die out. The phase space is a mathematical Euclidean space whose coordinates are the dynamical variables or the state variables of a dynamical system. See Ref. 10.
  10. F. C. Moon, Chaotic Vibrations (Wiley, New York, 1987), p. 56.
  11. Quasi-periodicity refers to a dynamical state of oscillation involving a finite number of discrete frequencies wherein the ratio of any two frequencies is an irrational number. See Ref. 12.
  12. T. S. Parker and L. O. Chua, Practical Numerical Algorithm for Chaotic Systems (Springer-Verlag, Berlin, 1989), p. 167.
    [CrossRef]
  13. S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978), p. 100.
    [CrossRef]
  14. O. P. MacDuff and S. E. Harris, IEEE J. Quantum Electron. QE-3, 101 (1967).
    [CrossRef]
  15. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
  16. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. (to be published).
  17. P. Grassberger and I. Procaccia, Physica D 9, 189 (1983).
    [CrossRef]
  18. N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
    [CrossRef]
  19. N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
    [CrossRef]
  20. J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
    [CrossRef]
  21. F. Takens, Detecting Strange Attractors in Turbulence, Vol. 898 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1980), p. 366.
  22. J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
    [CrossRef]
  23. J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
    [CrossRef]
  24. R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
    [CrossRef] [PubMed]
  25. A. Ben-Mizrachi, I. Procaccia, and P. Grassberger, Phys. Rev. A 29, 975 (1984).
    [CrossRef]

1992 (1)

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

1990 (4)

J. Mork, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990).
[CrossRef] [PubMed]

K. Y. Lau, IEEE J. Quantum Electron. 26, 250 (1990).
[CrossRef]

D. L. MacFarlane and L. W. Casperson, J. Opt. Soc. Am. B 7, 285 (1990).
[CrossRef]

J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
[CrossRef]

1989 (1)

A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
[CrossRef]

1988 (2)

L. A. Melnikov, E. M. Rabinovich, and V. V. Tuchin, J. Opt. Soc. Am. B 5, 1134 (1988).
[CrossRef]

R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (2)

H. G. Winful, Y. C. Chen, and J. M. Liu, Appl. Phys. Lett. 48, 616 (1986).
[CrossRef]

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

1985 (1)

J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

1984 (1)

A. Ben-Mizrachi, I. Procaccia, and P. Grassberger, Phys. Rev. A 29, 975 (1984).
[CrossRef]

1983 (1)

P. Grassberger and I. Procaccia, Physica D 9, 189 (1983).
[CrossRef]

1980 (1)

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

1978 (1)

S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978), p. 100.
[CrossRef]

1967 (1)

O. P. MacDuff and S. E. Harris, IEEE J. Quantum Electron. QE-3, 101 (1967).
[CrossRef]

Abarbanel, H. D. I.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. (to be published).

Abraham, N. B.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Albano, A. M.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Badii, R.

R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Ben-Mizrachi, A.

A. Ben-Mizrachi, I. Procaccia, and P. Grassberger, Phys. Rev. A 29, 975 (1984).
[CrossRef]

Bowers, J. E.

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

Broggi, G.

R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Brown, R.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. (to be published).

Campion, D. C.

J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
[CrossRef]

Casperson, L. W.

Chen, Y. C.

H. G. Winful, Y. C. Chen, and J. M. Liu, Appl. Phys. Lett. 48, 616 (1986).
[CrossRef]

Chesnoy, J.

J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
[CrossRef]

Chua, L. O.

T. S. Parker and L. O. Chua, Practical Numerical Algorithm for Chaotic Systems (Springer-Verlag, Berlin, 1989), p. 167.
[CrossRef]

Chusseau, L.

J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
[CrossRef]

Costa, J. M.

J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
[CrossRef]

Crutchfield, J. P.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Das, B.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

De Guzman, G.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Derighetti, B.

R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Eckmann, J. P.

J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

Farmer, J. D.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Gioggia, R. S.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Grassberger, P.

A. Ben-Mizrachi, I. Procaccia, and P. Grassberger, Phys. Rev. A 29, 975 (1984).
[CrossRef]

P. Grassberger and I. Procaccia, Physica D 9, 189 (1983).
[CrossRef]

Harris, S. E.

O. P. MacDuff and S. E. Harris, IEEE J. Quantum Electron. QE-3, 101 (1967).
[CrossRef]

Humphrey, P.

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

Kellner, A. L.

A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
[CrossRef]

J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
[CrossRef]

Klein, M. C.

J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
[CrossRef]

Lam, B. C.

A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
[CrossRef]

J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
[CrossRef]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Lau, K. Y.

K. Y. Lau, IEEE J. Quantum Electron. 26, 250 (1990).
[CrossRef]

Levy, S.

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

Liu, J. M.

H. G. Winful, Y. C. Chen, and J. M. Liu, Appl. Phys. Lett. 48, 616 (1986).
[CrossRef]

Lourtioz, J. M.

J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
[CrossRef]

MacDuff, O. P.

O. P. MacDuff and S. E. Harris, IEEE J. Quantum Electron. QE-3, 101 (1967).
[CrossRef]

MacFarlane, D. L.

Mar, A.

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

Mark, J.

J. Mork, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990).
[CrossRef] [PubMed]

Melnikov, L. A.

Moon, F. C.

F. C. Moon, Chaotic Vibrations (Wiley, New York, 1987), p. 56.

Mork, J.

J. Mork, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990).
[CrossRef] [PubMed]

Nagarajan, R.

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

Newhouse, S.

S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978), p. 100.
[CrossRef]

Packard, N. H.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Parker, T. S.

T. S. Parker and L. O. Chua, Practical Numerical Algorithm for Chaotic Systems (Springer-Verlag, Berlin, 1989), p. 167.
[CrossRef]

Procaccia, I.

A. Ben-Mizrachi, I. Procaccia, and P. Grassberger, Phys. Rev. A 29, 975 (1984).
[CrossRef]

P. Grassberger and I. Procaccia, Physica D 9, 189 (1983).
[CrossRef]

Puccioni, G. P.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Rabinovich, E. M.

Ravani, M.

R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

Ruelle, D.

J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978), p. 100.
[CrossRef]

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Shaw, R. S.

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

Sidorowich, J. J.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. (to be published).

Takens, F.

S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978), p. 100.
[CrossRef]

F. Takens, Detecting Strange Attractors in Turbulence, Vol. 898 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1980), p. 366.

Tredicce, J. R.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Tromborg, B.

J. Mork, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990).
[CrossRef] [PubMed]

Tsimring, L. S.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. (to be published).

Tuchin, V. V.

Winful, H. G.

H. G. Winful, Y. C. Chen, and J. M. Liu, Appl. Phys. Lett. 48, 616 (1986).
[CrossRef]

Yan, G. S.

A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
[CrossRef]

Yong, S.

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Yu, P. K. L.

A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
[CrossRef]

J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

H. G. Winful, Y. C. Chen, and J. M. Liu, Appl. Phys. Lett. 48, 616 (1986).
[CrossRef]

Commun. Math. Phys. (1)

S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978), p. 100.
[CrossRef]

Electron. Lett. (2)

S. Levy, R. Nagarajan, A. Mar, P. Humphrey, and J. E. Bowers, Electron. Lett. 28, 2103 (1992).
[CrossRef]

A. L. Kellner, B. C. Lam, G. S. Yan, and P. K. L. Yu, Electron. Lett. 25, 1291 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

O. P. MacDuff and S. E. Harris, IEEE J. Quantum Electron. QE-3, 101 (1967).
[CrossRef]

K. Y. Lau, IEEE J. Quantum Electron. 26, 250 (1990).
[CrossRef]

J. Appl. Phys. (1)

J. Chesnoy, M. C. Klein, L. Chusseau, and J. M. Lourtioz, J. Appl. Phys. 67, 7615 (1990).
[CrossRef]

J. Opt. Soc. Am. B (3)

Phys. Lett. A (1)

N. B. Abraham, A. M. Albano, B. Das, G. De Guzman, S. Yong, R. S. Gioggia, G. P. Puccioni, and J. R. Tredicce, Phys. Lett. A 114, 217 (1986).
[CrossRef]

Phys. Rev. A (1)

A. Ben-Mizrachi, I. Procaccia, and P. Grassberger, Phys. Rev. A 29, 975 (1984).
[CrossRef]

Phys. Rev. Lett. (3)

R. Badii, G. Broggi, B. Derighetti, and M. Ravani, Phys. Rev. Lett. 60, 979 (1988).
[CrossRef] [PubMed]

N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980).
[CrossRef]

J. Mork, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990).
[CrossRef] [PubMed]

Physica D (1)

P. Grassberger and I. Procaccia, Physica D 9, 189 (1983).
[CrossRef]

Rev. Mod. Phys. (1)

J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
[CrossRef]

Other (8)

F. Takens, Detecting Strange Attractors in Turbulence, Vol. 898 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1980), p. 366.

J. M. Costa, B. C. Lam, A. L. Kellner, D. C. Campion, and P. K. L. Yu, in Optical Technology for Microwave Applications V, S. Yao, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1476, 74 (1991).
[CrossRef]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. (to be published).

An attractor is a set of points in phase space that the state of a system approaches after transients die out. The phase space is a mathematical Euclidean space whose coordinates are the dynamical variables or the state variables of a dynamical system. See Ref. 10.

F. C. Moon, Chaotic Vibrations (Wiley, New York, 1987), p. 56.

Quasi-periodicity refers to a dynamical state of oscillation involving a finite number of discrete frequencies wherein the ratio of any two frequencies is an irrational number. See Ref. 12.

T. S. Parker and L. O. Chua, Practical Numerical Algorithm for Chaotic Systems (Springer-Verlag, Berlin, 1989), p. 167.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Power (microwave) spectra of the laser in (a) the properly mode-locked state at 205 MHz; (b) the two-frequency regime, where fm = 192 MHz and f1 = 208 MHz; (c) the three-frequency regime, where fm = 190 MHz, f1 = 208 MHz, and f2 = 194 MHz; (d) the low-dimensional chaotic state, where fm = 171 MHz. The vertical scale is 10 dB/division; the horizontal scale is from 0 to 500 MHz. The noise floor of all the traces is at −112 dBm. Input rf power is 1 mW The other tones in (a)–(d) are beat combinations and higher-order harmonics of the tones at fm, f1, and f2.

Fig. 2
Fig. 2

Dynamical states of the actively mode-locked semiconductor laser when mode-locking frequency is tuned below resonance, relative to the resonant frequency of 205 MHz.

Fig. 3
Fig. 3

Time history of the laser while it is in the chaotic state that is shown in Fig. 1(d).

Fig. 4
Fig. 4

Phase-space plot of the filtered chaotic signal. The coordinates of the plot are x(t) and x(tT), where x(t) is the time history that was obtained from the oscilloscope. The time delay T that we used is three data points (corresponding to 0.75 ns).

Fig. 5
Fig. 5

Correlation-dimension estimates of the laser in (a) the properly mode-locked state and (b) the broadband state, (c) Linear fit to determine the slopes in (a), (d) Linear fit to determine the slopes in (b). dim, Dimension.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

d E n d t = ν 2 χ n E n ( ν 2 Q + 2 α c T R ) E n + α c T R × [ E n + 1 cos ( ϕ n + 1 ϕ n ) + E n 1 cos ( ϕ n ϕ n 1 ) ] ,
d ϕ n d t = ( f n ν n ) ν 2 χ n α c T R E n × [ E n + 1 sin ( ϕ n + 1 ϕ n ) + E n 1 sin ( ϕ n ϕ n 1 ) ] ,
C ( r ) = lim N 1 N 2 i N j N H ( r | x i x j | ) ,
η = [ x n , x n + T , x n + 2 T , , x n + ( D e 1 ) T ] ,

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